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class="vector-toc-list"> <li id="toc-Early_years" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Early_years"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Early years</span> </div> </a> <ul id="toc-Early_years-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Education" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Education"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Education</span> </div> </a> <ul id="toc-Education-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Academia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Academia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Academia</span> </div> </a> <ul id="toc-Academia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Protestant_family_and_death_in_Italy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Protestant_family_and_death_in_Italy"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Protestant family and death in Italy</span> </div> </a> <ul id="toc-Protestant_family_and_death_in_Italy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Riemannian_geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Riemannian_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Riemannian geometry</span> </div> </a> <ul id="toc-Riemannian_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Complex_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Complex analysis</span> </div> </a> <ul id="toc-Complex_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Real_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Real analysis</span> </div> </a> <ul id="toc-Real_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Number_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Number theory</span> </div> </a> <ul id="toc-Number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Writings" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Writings"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Writings</span> </div> </a> <ul id="toc-Writings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Bernhard Riemann</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 100 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-100" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">100 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Afrikaans" lang="af" hreflang="af" data-title="Bernhard Riemann" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A8%D8%B1%D9%86%D8%A7%D8%B1%D8%AF_%D8%B1%D9%8A%D9%85%D8%A7%D9%86" title="برنارد ريمان – Arabic" lang="ar" hreflang="ar" data-title="برنارد ريمان" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Aragonese" lang="an" hreflang="an" data-title="Bernhard Riemann" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BE%E0%A7%B0%E0%A7%8D%E0%A6%A8%E0%A6%B9%E0%A6%BE%E0%A7%B0%E0%A7%8D%E0%A6%A1_%E0%A7%B0%E0%A6%BF%E0%A6%AE%E0%A6%BE%E0%A6%A8" title="বাৰ্নহাৰ্ড ৰিমান – Assamese" lang="as" hreflang="as" data-title="বাৰ্নহাৰ্ড ৰিমান" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Asturian" lang="ast" hreflang="ast" data-title="Bernhard Riemann" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Bernhard_Riman" title="Bernhard Riman – Azerbaijani" lang="az" hreflang="az" data-title="Bernhard Riman" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A8%D8%B1%D9%86%D9%87%D8%A7%D8%B1%D8%AA_%D8%B1%DB%8C%D9%85%D8%A7%D9%86" title="برنهارت ریمان – South Azerbaijani" lang="azb" hreflang="azb" data-title="برنهارت ریمان" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%87%E0%A6%B0%E0%A7%8D%E0%A6%A8%E0%A6%B9%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%9F_%E0%A6%B0%E0%A6%BF%E0%A6%AE%E0%A6%BE%E0%A6%A8" title="বের্নহার্ট রিমান – Bangla" lang="bn" hreflang="bn" data-title="বের্নহার্ট রিমান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Minnan" lang="nan" hreflang="nan" data-title="Bernhard Riemann" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D1%80%D0%B3_%D0%A4%D1%80%D1%8B%D0%B4%D1%80%D1%8B%D1%85_%D0%91%D0%B5%D1%80%D0%BD%D1%85%D0%B0%D1%80%D0%B4_%D0%A0%D1%8B%D0%BC%D0%B0%D0%BD" title="Георг Фрыдрых Бернхард Рыман – Belarusian" lang="be" hreflang="be" data-title="Георг Фрыдрых Бернхард Рыман" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%91%D1%8D%D1%80%D0%BD%D0%B3%D0%B0%D1%80%D0%B4_%D0%A0%D1%8B%D0%BC%D0%B0%D0%BD" title="Бэрнгард Рыман – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Бэрнгард Рыман" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D0%B5%D1%80%D0%BD%D1%85%D0%B0%D1%80%D0%B4_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD" title="Бернхард Риман – Bulgarian" lang="bg" hreflang="bg" data-title="Бернхард Риман" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Bosnian" lang="bs" hreflang="bs" data-title="Bernhard Riemann" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Georg_Friedrich_Bernhard_Riemann" title="Georg Friedrich Bernhard Riemann – Catalan" lang="ca" hreflang="ca" data-title="Georg Friedrich Bernhard Riemann" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%91%D0%B5%D1%80%D0%BD%D1%85%D0%B0%D1%80%D0%B4_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD" title="Бернхард Риман – Chuvash" lang="cv" hreflang="cv" data-title="Бернхард Риман" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Czech" lang="cs" hreflang="cs" data-title="Bernhard Riemann" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Welsh" lang="cy" hreflang="cy" data-title="Bernhard Riemann" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Danish" lang="da" hreflang="da" data-title="Bernhard Riemann" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – German" lang="de" hreflang="de" data-title="Bernhard Riemann" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Estonian" lang="et" hreflang="et" data-title="Bernhard Riemann" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CF%80%CE%AD%CF%81%CE%BD%CE%B1%CF%81%CE%BD%CF%84_%CE%A1%CE%AF%CE%BC%CE%B1%CE%BD" title="Μπέρναρντ Ρίμαν – Greek" lang="el" hreflang="el" data-title="Μπέρναρντ Ρίμαν" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Spanish" lang="es" hreflang="es" data-title="Bernhard Riemann" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Esperanto" lang="eo" hreflang="eo" data-title="Bernhard Riemann" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Basque" lang="eu" hreflang="eu" data-title="Bernhard Riemann" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%D8%B1%D9%86%D9%87%D8%A7%D8%B1%D8%AA_%D8%B1%DB%8C%D9%85%D8%A7%D9%86" title="برنهارت ریمان – Persian" lang="fa" hreflang="fa" data-title="برنهارت ریمان" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – French" lang="fr" hreflang="fr" data-title="Bernhard Riemann" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Western Frisian" lang="fy" hreflang="fy" data-title="Bernhard Riemann" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Georg_Friedrich_Bernhardt_Riemann" title="Georg Friedrich Bernhardt Riemann – Irish" lang="ga" hreflang="ga" data-title="Georg Friedrich Bernhardt Riemann" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Bernhard Riemann" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Galician" lang="gl" hreflang="gl" data-title="Bernhard Riemann" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D1%80%D0%B3_%D0%A4%D1%80%D0%B8%D0%B4%D1%80%D0%B8%D1%85_%D0%91%D0%B5%D1%80%D0%BD%D1%85%D0%B0%D1%80%D0%B4_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD" title="Георг Фридрих Бернхард Риман – Kalmyk" lang="xal" hreflang="xal" data-title="Георг Фридрих Бернхард Риман" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A0%EB%A5%B8%ED%95%98%EB%A5%B4%ED%8A%B8_%EB%A6%AC%EB%A7%8C" title="베른하르트 리만 – Korean" lang="ko" hreflang="ko" data-title="베른하르트 리만" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A5%D5%BC%D5%B6%D5%A1%D6%80%D5%A4_%D5%8C%D5%AB%D5%B4%D5%A1%D5%B6" title="Բեռնարդ Ռիման – Armenian" lang="hy" hreflang="hy" data-title="Բեռնարդ Ռիման" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A4%B0%E0%A5%8D%E0%A4%A8%E0%A4%B9%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A1_%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%BE%E0%A4%A8" title="बर्नहार्ड रीमान – Hindi" lang="hi" hreflang="hi" data-title="बर्नहार्ड रीमान" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Croatian" lang="hr" hreflang="hr" data-title="Bernhard Riemann" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Ido" lang="io" hreflang="io" data-title="Bernhard Riemann" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Georg_Friedrich_Bernhard_Riemann" title="Georg Friedrich Bernhard Riemann – Indonesian" lang="id" hreflang="id" data-title="Georg Friedrich Bernhard Riemann" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Interlingua" lang="ia" hreflang="ia" data-title="Bernhard Riemann" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Georg_Friedrich_Bernhard_Riemann" title="Georg Friedrich Bernhard Riemann – Icelandic" lang="is" hreflang="is" data-title="Georg Friedrich Bernhard Riemann" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Italian" lang="it" hreflang="it" data-title="Bernhard Riemann" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%91%D7%A8%D7%A0%D7%94%D7%A8%D7%93_%D7%A8%D7%99%D7%9E%D7%9F" title="ברנהרד רימן – Hebrew" lang="he" hreflang="he" data-title="ברנהרד רימן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Georg_Friedrich_Bernhard_Riemann" title="Georg Friedrich Bernhard Riemann – Javanese" lang="jv" hreflang="jv" data-title="Georg Friedrich Bernhard Riemann" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Kabiye" lang="kbp" hreflang="kbp" data-title="Bernhard Riemann" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AC%E0%B2%B0%E0%B3%8D%E0%B2%A8%E0%B3%8D%E2%80%8C%E0%B2%B9%E0%B2%BE%E0%B2%B0%E0%B3%8D%E0%B2%A1%E0%B3%8D_%E0%B2%B0%E0%B3%80%E0%B2%AE%E0%B2%A8%E0%B3%8D" title="ಬರ್ನ್‌ಹಾರ್ಡ್ ರೀಮನ್ – Kannada" lang="kn" hreflang="kn" data-title="ಬರ್ನ್‌ಹಾರ್ಡ್ ರೀಮನ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%91%E1%83%94%E1%83%A0%E1%83%9C%E1%83%B0%E1%83%90%E1%83%A0%E1%83%93_%E1%83%A0%E1%83%98%E1%83%9B%E1%83%90%E1%83%9C%E1%83%98" title="ბერნჰარდ რიმანი – Georgian" lang="ka" hreflang="ka" data-title="ბერნჰარდ რიმანი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D1%80%D0%B3_%D0%A4%D1%80%D0%B8%D0%B4%D1%80%D0%B8%D1%85_%D0%91%D0%B5%D1%80%D0%BD%D1%85%D0%B0%D1%80%D0%B4_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD" title="Георг Фридрих Бернхард Риман – Kazakh" lang="kk" hreflang="kk" data-title="Георг Фридрих Бернхард Риман" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Cornish" lang="kw" hreflang="kw" data-title="Bernhard Riemann" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Haitian Creole" lang="ht" hreflang="ht" data-title="Bernhard Riemann" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%91%D0%B5%D1%80%D0%BD%D1%85%D0%B0%D1%80%D0%B4_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD" title="Бернхард Риман – Kyrgyz" lang="ky" hreflang="ky" data-title="Бернхард Риман" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Bernardus_Riemann" title="Bernardus Riemann – Latin" lang="la" hreflang="la" data-title="Bernardus Riemann" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Bernhards_R%C4%ABmanis" title="Bernhards Rīmanis – Latvian" lang="lv" hreflang="lv" data-title="Bernhards Rīmanis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Lithuanian" lang="lt" hreflang="lt" data-title="Bernhard Riemann" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Ligurian" lang="lij" hreflang="lij" data-title="Bernhard Riemann" data-language-autonym="Ligure" data-language-local-name="Ligurian" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Georg_Friedrich_Bernhard_Riemann" title="Georg Friedrich Bernhard Riemann – Hungarian" lang="hu" hreflang="hu" data-title="Georg Friedrich Bernhard Riemann" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%B5%D1%80%D0%BD%D1%85%D0%B0%D1%80%D0%B4_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD" title="Бернхард Риман – Macedonian" lang="mk" hreflang="mk" data-title="Бернхард Риман" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Malagasy" lang="mg" hreflang="mg" data-title="Bernhard Riemann" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AC%E0%A5%87%E0%A4%B0%E0%A5%8D%E0%A4%A8%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A1_%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%BE%E0%A4%A8" title="बेर्नार्ड रीमान – Marathi" lang="mr" hreflang="mr" data-title="बेर्नार्ड रीमान" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%91%E1%83%94%E1%83%A0%E1%83%9C%E1%83%B0%E1%83%90%E1%83%A0%E1%83%93_%E1%83%A0%E1%83%98%E1%83%9B%E1%83%90%E1%83%9C%E1%83%98" title="ბერნჰარდ რიმანი – Mingrelian" lang="xmf" hreflang="xmf" data-title="ბერნჰარდ რიმანი" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%A8%D8%B1%D9%86%D8%A7%D8%B1%D8%AF_%D8%B1%D9%8A%D9%85%D8%A7%D9%86" title="برنارد ريمان – Egyptian Arabic" lang="arz" hreflang="arz" data-title="برنارد ريمان" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-mni mw-list-item"><a href="https://mni.wikipedia.org/wiki/%EA%AF%95%EA%AF%94%EA%AF%9F%EA%AF%8D%EA%AF%A5%EA%AF%94%EA%AF%97_%EA%AF%94%EA%AF%A5%EA%AF%8F%EA%AF%83%EA%AF%A6%EA%AF%9F" title="ꯕꯔꯟꯍꯥꯔꯗ ꯔꯥꯏꯃꯦꯟ – Manipuri" lang="mni" hreflang="mni" data-title="ꯕꯔꯟꯍꯥꯔꯗ ꯔꯥꯏꯃꯦꯟ" data-language-autonym="ꯃꯤꯇꯩ ꯂꯣꯟ" data-language-local-name="Manipuri" class="interlanguage-link-target"><span>ꯃꯤꯇꯩ ꯂꯣꯟ</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Mirandese" lang="mwl" hreflang="mwl" data-title="Bernhard Riemann" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%98%E1%80%94%E1%80%BA%E1%80%B8%E1%80%9F%E1%80%90%E1%80%BA_%E1%80%9B%E1%80%AE%E1%80%B8%E1%80%99%E1%80%94%E1%80%BA" title="ဘန်းဟတ် ရီးမန် – Burmese" lang="my" hreflang="my" data-title="ဘန်းဟတ် ရီးမန်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Dutch" lang="nl" hreflang="nl" data-title="Bernhard Riemann" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%AC%E0%A4%B0%E0%A5%8D%E0%A4%A8%E0%A5%8D%E0%A4%B9%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A1_%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%A8" title="बर्न्हार्ड रीम्यान – Newari" lang="new" hreflang="new" data-title="बर्न्हार्ड रीम्यान" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%83%AB%E3%83%B3%E3%83%8F%E3%83%AB%E3%83%88%E3%83%BB%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3" title="ベルンハルト・リーマン – Japanese" lang="ja" hreflang="ja" data-title="ベルンハルト・リーマン" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Bernhard Riemann" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Bernhard Riemann" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Occitan" lang="oc" hreflang="oc" data-title="Bernhard Riemann" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Riman_Georg_Fridrix_Bernhard" title="Riman Georg Fridrix Bernhard – Uzbek" lang="uz" hreflang="uz" data-title="Riman Georg Fridrix Bernhard" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%94%E1%9F%8A%E1%9F%82%E1%9E%93%E1%9E%A0%E1%9E%B6%E1%9E%8A_%E1%9E%9A%E1%9E%B8%E1%9E%98%E1%9F%89%E1%9E%B6%E1%9E%93" title="ប៊ែនហាដ រីម៉ាន – Khmer" lang="km" hreflang="km" data-title="ប៊ែនហាដ រីម៉ាន" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Piedmontese" lang="pms" hreflang="pms" data-title="Bernhard Riemann" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Polish" lang="pl" hreflang="pl" data-title="Bernhard Riemann" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Portuguese" lang="pt" hreflang="pt" data-title="Bernhard Riemann" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Romanian" lang="ro" hreflang="ro" data-title="Bernhard Riemann" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD,_%D0%91%D0%B5%D1%80%D0%BD%D1%85%D0%B0%D1%80%D0%B4" title="Риман, Бернхард – Russian" lang="ru" hreflang="ru" data-title="Риман, Бернхард" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%AC%E0%A4%B0%E0%A5%8D%E0%A4%A8%E0%A5%8D%E0%A4%B9%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A1_%E0%A4%B0%E0%A5%80%E0%A4%AE%E0%A4%A8" title="बर्न्हार्ड रीमन – Sanskrit" lang="sa" hreflang="sa" data-title="बर्न्हार्ड रीमन" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Georg_Friedrich_Bernhard_Riemann" title="Georg Friedrich Bernhard Riemann – Scots" lang="sco" hreflang="sco" data-title="Georg Friedrich Bernhard Riemann" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Albanian" lang="sq" hreflang="sq" data-title="Bernhard Riemann" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Georg_Friedrich_Bernhard_Riemann" title="Georg Friedrich Bernhard Riemann – Sicilian" lang="scn" hreflang="scn" data-title="Georg Friedrich Bernhard Riemann" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Simple English" lang="en-simple" hreflang="en-simple" data-title="Bernhard Riemann" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Georg_Friedrich_Bernhard_Riemann" title="Georg Friedrich Bernhard Riemann – Slovak" lang="sk" hreflang="sk" data-title="Georg Friedrich Bernhard Riemann" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Slovenian" lang="sl" hreflang="sl" data-title="Bernhard Riemann" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DB%8E%D8%B1%D9%86%D8%A7%D8%B1%D8%AF_%DA%95%DB%8C%D9%85%D8%A7%D9%86" title="بێرنارد ڕیمان – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بێرنارد ڕیمان" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Bernhard_Riman" title="Bernhard Riman – Serbian" lang="sr" hreflang="sr" data-title="Bernhard Riman" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Bernhard Riemann" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Finnish" lang="fi" hreflang="fi" data-title="Bernhard Riemann" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Swedish" lang="sv" hreflang="sv" data-title="Bernhard Riemann" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Tagalog" lang="tl" hreflang="tl" data-title="Bernhard Riemann" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%A9%E0%AE%BE%E0%AE%9F%E0%AF%8D_%E0%AE%B0%E0%AF%80%E0%AE%AE%E0%AE%A9%E0%AF%8D" title="பேர்னாட் ரீமன் – Tamil" lang="ta" hreflang="ta" data-title="பேர்னாட் ரீமன்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%9A%E0%B8%A3%E0%B9%8C%E0%B8%99%E0%B8%AE%E0%B8%B2%E0%B8%A3%E0%B9%8C%E0%B8%97_%E0%B8%A3%E0%B8%B5%E0%B8%A1%E0%B8%B1%E0%B8%99" title="แบร์นฮาร์ท รีมัน – Thai" lang="th" hreflang="th" data-title="แบร์นฮาร์ท รีมัน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Turkish" lang="tr" hreflang="tr" data-title="Bernhard Riemann" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D0%B5%D1%80%D0%BD%D0%B3%D0%B0%D1%80%D0%B4_%D0%A0%D1%96%D0%BC%D0%B0%D0%BD" title="Бернгард Ріман – Ukrainian" lang="uk" hreflang="uk" data-title="Бернгард Ріман" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A8%D8%B1%D9%86%DB%81%D8%A7%D8%B1%DA%88_%D8%B1%DB%8C%D9%85%D8%A7%D9%86" title="برنہارڈ ریمان – Urdu" lang="ur" hreflang="ur" data-title="برنہارڈ ریمان" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D8%B1%D9%89%D9%85%D8%A7%D9%86" title="رىمان – Uyghur" lang="ug" hreflang="ug" data-title="رىمان" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="Uyghur" class="interlanguage-link-target"><span>ئۇيغۇرچە / Uyghurche</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Vietnamese" lang="vi" hreflang="vi" data-title="Bernhard Riemann" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Waray" lang="war" hreflang="war" data-title="Bernhard Riemann" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%BC%AF%E6%81%A9%E5%93%88%E5%BE%B7%C2%B7%E9%BB%8E%E6%9B%BC" title="伯恩哈德·黎曼 – Wu" lang="wuu" hreflang="wuu" data-title="伯恩哈德·黎曼" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BC%AF%E6%81%A9%E5%93%88%E5%BE%B7%C2%B7%E9%BB%8E%E6%9B%BC" title="伯恩哈德·黎曼 – Cantonese" lang="yue" hreflang="yue" data-title="伯恩哈德·黎曼" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann – Zazaki" lang="diq" hreflang="diq" data-title="Bernhard Riemann" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BC%AF%E6%81%A9%E5%93%88%E5%BE%B7%C2%B7%E9%BB%8E%E6%9B%BC" title="伯恩哈德·黎曼 – Chinese" lang="zh" hreflang="zh" data-title="伯恩哈德·黎曼" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">German mathematician (1826–1866)</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output 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print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a 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div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Riemann" redirects here. For other people with the surname, see <a href="/wiki/Riemann_(surname)" title="Riemann (surname)">Riemann (surname)</a>. For other topics named after Bernhard Riemann, see <a href="/wiki/List_of_topics_named_after_Bernhard_Riemann" class="mw-redirect" title="List of topics named after Bernhard Riemann">List of topics named after Bernhard Riemann</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Bernhard_Raimann" title="Bernhard Raimann">Bernhard Raimann</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">Bernhard Riemann</div></th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Georg_Friedrich_Bernhard_Riemann.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Georg_Friedrich_Bernhard_Riemann.jpeg/220px-Georg_Friedrich_Bernhard_Riemann.jpeg" decoding="async" width="220" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Georg_Friedrich_Bernhard_Riemann.jpeg/330px-Georg_Friedrich_Bernhard_Riemann.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Georg_Friedrich_Bernhard_Riemann.jpeg/440px-Georg_Friedrich_Bernhard_Riemann.jpeg 2x" data-file-width="903" data-file-height="986" /></a></span><div class="infobox-caption">Riemann <abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1863</span></div></td></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><div style="display:inline" class="nickname">Georg Friedrich Bernhard Riemann</div><br />17 September 1826<span style="display:none"> (<span class="dtstart bday">1826-09-17</span>)</span><br /><div style="display:inline" class="birthplace"><a href="/wiki/Breselenz" class="mw-redirect" title="Breselenz">Breselenz</a>, <a href="/wiki/Kingdom_of_Hanover" title="Kingdom of Hanover">Kingdom of Hanover</a> (modern-day Germany)</div></td></tr><tr><th scope="row" class="infobox-label">Died</th><td class="infobox-data">20 July 1866<span style="display:none">(1866-07-20)</span> (aged 39)<br /><div style="display:inline" class="deathplace"><a href="/wiki/Verbania" title="Verbania">Selasca</a>, <a href="/wiki/Kingdom_of_Italy" title="Kingdom of Italy">Kingdom of Italy</a></div></td></tr><tr><th scope="row" class="infobox-label">Alma mater</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"><ul><li><a href="/wiki/University_of_G%C3%B6ttingen" title="University of Göttingen">University of Göttingen</a></li><li><a href="/wiki/University_of_Berlin" class="mw-redirect" title="University of Berlin">University of Berlin</a></li></ul></div></td></tr><tr><th scope="row" class="infobox-label">Known for</th><td class="infobox-data"><i><a href="/wiki/List_of_topics_named_after_Bernhard_Riemann" class="mw-redirect" title="List of topics named after Bernhard Riemann">See list</a></i></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><b>Scientific career</b></td></tr><tr><th scope="row" class="infobox-label">Fields</th><td class="infobox-data category"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><div class="hlist"><ul><li><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></li><li><a href="/wiki/Physics" title="Physics">Physics</a></li></ul></div></td></tr><tr><th scope="row" class="infobox-label">Institutions</th><td class="infobox-data"><a href="/wiki/University_of_G%C3%B6ttingen" title="University of Göttingen">University of Göttingen</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Thesis" title="Thesis">Thesis</a></th><td class="infobox-data"><i><a rel="nofollow" class="external text" href="http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/Grund.pdf">Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Größe</a></i> <span style="font-size:97%;">(1851)</span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Doctoral_advisor" title="Doctoral advisor">Doctoral advisor</a></th><td class="infobox-data"><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></td></tr><tr><th scope="row" class="infobox-label">Other academic advisors</th><td class="infobox-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><a href="/wiki/Gotthold_Eisenstein" title="Gotthold Eisenstein">Gotthold Eisenstein</a></li><li><a href="/wiki/Moritz_Abraham_Stern" title="Moritz Abraham Stern">Moritz A. Stern</a></li><li><a href="/wiki/Carl_Wolfgang_Benjamin_Goldschmidt" title="Carl Wolfgang Benjamin Goldschmidt">Carl W. B. Goldschmidt</a></li></ul></div></td></tr><tr><th scope="row" class="infobox-label">Notable students</th><td class="infobox-data"><a href="/wiki/Gustav_Roch" title="Gustav Roch">Gustav Roch</a><br /><a href="/wiki/Eduard_Selling" title="Eduard Selling">Eduard Selling</a></td></tr><tr style="display:none"><td colspan="2"> </td></tr><tr><th colspan="2" class="infobox-header">Signature</th></tr><tr><td colspan="2" class="infobox-full-data"><span class="infobox-signature skin-invert" typeof="mw:File"><a href="/wiki/File:Bernhard_Riemann_signature.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bernhard_Riemann_signature.png/150px-Bernhard_Riemann_signature.png" decoding="async" width="150" height="34" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bernhard_Riemann_signature.png/225px-Bernhard_Riemann_signature.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bernhard_Riemann_signature.png/300px-Bernhard_Riemann_signature.png 2x" data-file-width="487" data-file-height="112" /></a></span></td></tr></tbody></table> <p><b>Georg Friedrich Bernhard Riemann</b> (<style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style><span class="IPA-label IPA-label-small">German:</span> <span class="IPA nowrap" lang="de-Latn-fonipa"><a href="/wiki/Help:IPA/Standard_German" title="Help:IPA/Standard German">[ˈɡeːɔʁk<span class="wrap"> </span>ˈfʁiːdʁɪç<span class="wrap"> </span>ˈbɛʁnhaʁt<span class="wrap"> </span>ˈʁiːman]</a></span> <span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-1" class="ext-phonos-PhonosButton noexcerpt ext-phonos-PhonosButton-emptylabel oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/fb\/Georgfriedrichbernhardriemann.ogg\/Georgfriedrichbernhardriemann.ogg.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Georgfriedrichbernhardriemann.ogg"},"classes":["ext-phonos-PhonosButton","noexcerpt","ext-phonos-PhonosButton-emptylabel"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/fb/Georgfriedrichbernhardriemann.ogg/Georgfriedrichbernhardriemann.ogg.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label"></span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:Georgfriedrichbernhardriemann.ogg" title="File:Georgfriedrichbernhardriemann.ogg">ⓘ</a></sup></span></span>;<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> 17 September 1826 – 20 July 1866) was a German <a href="/wiki/Mathematician" title="Mathematician">mathematician</a> who made profound contributions to <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>, <a href="/wiki/Number_theory" title="Number theory">number theory</a>, and <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>. In the field of <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>, he is mostly known for the first rigorous formulation of the integral, the <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a>, and his work on <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>. His contributions to <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> include most notably the introduction of <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>, breaking new ground in a natural, geometric treatment of complex analysis. His <a href="/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude" title="On the Number of Primes Less Than a Given Magnitude">1859 paper</a> on the <a href="/wiki/Prime-counting_function" title="Prime-counting function">prime-counting function</a>, containing the original statement of the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, is regarded as a foundational paper of <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>. Through his pioneering <a href="/wiki/Riemannian_Geometry" class="mw-redirect" title="Riemannian Geometry">contributions to differential geometry</a>, Riemann laid the foundations of the mathematics of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> He is considered by many to be one of the greatest mathematicians of all time.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Biography">Biography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=1" title="Edit section: Biography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Early_years">Early years</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=2" title="Edit section: Early years"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Riemann was born on 17 September 1826 in <a href="/wiki/Breselenz" class="mw-redirect" title="Breselenz">Breselenz</a>, a village near <a href="/wiki/Dannenberg_(Elbe)" title="Dannenberg (Elbe)">Dannenberg</a> in the <a href="/wiki/Kingdom_of_Hanover" title="Kingdom of Hanover">Kingdom of Hanover</a>. His father, Friedrich Bernhard Riemann, was a poor <a href="/wiki/Lutheranism" title="Lutheranism">Lutheran</a> pastor in Breselenz who fought in the <a href="/wiki/Napoleonic_Wars" title="Napoleonic Wars">Napoleonic Wars</a>. His mother, Charlotte Ebell, died in 1846. Riemann was the second of six children. Riemann exhibited exceptional mathematical talent, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public. </p> <div class="mw-heading mw-heading3"><h3 id="Education">Education</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=3" title="Edit section: Education"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>During 1840, Riemann went to <a href="/wiki/Hanover" title="Hanover">Hanover</a> to live with his grandmother and attend <a href="/wiki/Lyceum" title="Lyceum">lyceum</a> (middle school years), because such a type of school was not accessible from his home village. After the death of his grandmother in 1842, he transferred to the <a href="https://de.wikipedia.org/wiki/Johanneum_L%C3%BCneburg" class="extiw" title="de:Johanneum Lüneburg">Johanneum Lüneburg</a>, a high school in <a href="/wiki/L%C3%BCneburg" title="Lüneburg">Lüneburg</a>. There, Riemann studied the <a href="/wiki/Bible" title="Bible">Bible</a> intensively, but he was often distracted by mathematics. His teachers were amazed by his ability to perform complicated mathematical operations, in which he often outstripped his instructor's knowledge. In 1846, at the age of 19, he started studying <a href="/wiki/Philology" title="Philology">philology</a> and <a href="/wiki/Christian_theology" title="Christian theology">Christian theology</a> in order to become a pastor and help with his family's finances. </p><p>During the spring of 1846, his father, after gathering enough money, sent Riemann to the <a href="/wiki/University_of_G%C3%B6ttingen" title="University of Göttingen">University of Göttingen</a>, where he planned to study towards a degree in <a href="/wiki/Theology" title="Theology">theology</a>. However, once there, he began studying <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> under <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (specifically his lectures on the <a href="/wiki/Method_of_least_squares" class="mw-redirect" title="Method of least squares">method of least squares</a>). Gauss recommended that Riemann give up his theological work and enter the mathematical field; after getting his father's approval, Riemann transferred to the <a href="/wiki/University_of_Berlin" class="mw-redirect" title="University of Berlin">University of Berlin</a> in 1847.<sup id="cite_ref-Hawking2005_6-0" class="reference"><a href="#cite_note-Hawking2005-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> During his time of study, <a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Carl Gustav Jacob Jacobi</a>, <a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a>, <a href="/wiki/Jakob_Steiner" title="Jakob Steiner">Jakob Steiner</a>, and <a href="/wiki/Gotthold_Eisenstein" title="Gotthold Eisenstein">Gotthold Eisenstein</a> were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849. </p> <div class="mw-heading mw-heading3"><h3 id="Academia">Academia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=4" title="Edit section: Academia"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Riemann held his first lectures in 1854, which founded the field of <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a> and thereby set the stage for <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s <a href="/wiki/General_theory_of_relativity" class="mw-redirect" title="General theory of relativity">general theory of relativity</a>.<sup id="cite_ref-:0_7-0" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In 1857, there was an attempt to promote Riemann to extraordinary professor status at the <a href="/wiki/University_of_G%C3%B6ttingen" title="University of Göttingen">University of Göttingen</a>. Although this attempt failed, it did result in Riemann finally being granted a regular salary. In 1859, following the death of Dirichlet (who held <a href="/wiki/Gauss" class="mw-redirect" title="Gauss">Gauss</a>'s chair at the University of Göttingen), he was promoted to head the mathematics department at the University of Göttingen. He was also the first to suggest using <a href="/wiki/Higher_dimensions" class="mw-redirect" title="Higher dimensions">dimensions higher than merely three or four</a> in order to describe physical reality.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_7-1" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>In 1862 he married Elise Koch; their daughter Ida Schilling was born on 22 December 1862.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Protestant_family_and_death_in_Italy">Protestant family and death in Italy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=5" title="Edit section: Protestant family and death in Italy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2009_11_riemann_008.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/2009_11_riemann_008.jpg/220px-2009_11_riemann_008.jpg" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/2009_11_riemann_008.jpg/330px-2009_11_riemann_008.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/2009_11_riemann_008.jpg/440px-2009_11_riemann_008.jpg 2x" data-file-width="2748" data-file-height="3664" /></a><figcaption>Riemann's tombstone in <a href="/wiki/Biganzolo" class="mw-redirect" title="Biganzolo">Biganzolo</a> in <a href="/wiki/Piedmont" title="Piedmont">Piedmont</a>, Italy</figcaption></figure><p>Riemann fled Göttingen when the armies of <a href="/wiki/Hanover" title="Hanover">Hanover</a> and <a href="/wiki/Prussia" title="Prussia">Prussia</a> clashed there in 1866.<sup id="cite_ref-duSautoy2003_10-0" class="reference"><a href="#cite_note-duSautoy2003-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> He died of <a href="/wiki/Tuberculosis" title="Tuberculosis">tuberculosis</a> during his third journey to Italy in Selasca (now a hamlet of <a href="/wiki/Verbania" title="Verbania">Verbania</a> on <a href="/wiki/Lake_Maggiore" title="Lake Maggiore">Lake Maggiore</a>), where he was buried in the cemetery in Biganzolo (Verbania).<br /> </p><p>Riemann was a dedicated Christian, the son of a Protestant minister, and saw his life as a mathematician as another way to serve God. During his life, he held closely to his Christian faith and considered it to be the most important aspect of his life. At the time of his death, he was reciting the <a href="/wiki/Lord%27s_Prayer" title="Lord's Prayer">Lord's Prayer</a> with his wife and died before they finished saying the prayer.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work, and some deep insights may have been lost.<sup id="cite_ref-duSautoy2003_10-1" class="reference"><a href="#cite_note-duSautoy2003-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Riemann's tombstone in <a href="/wiki/Biganzolo" class="mw-redirect" title="Biganzolo">Biganzolo</a> (Italy) refers to <a rel="nofollow" class="external text" href="https://www.biblica.com/bible/?osis=niv:Romans%208:28">Romans 8:28</a>:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Here rests in God<br /> <p>Georg Friedrich Bernhard Riemann<br /> Professor in Göttingen<br /> born in Breselenz, 17 September 1826<br /> died in Selasca, 20 July 1866<br /> </p> For those who love God, all things must work together for the best</div> <div 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size notpageimage" typeof="mw:File/Frameless"><a href="/wiki/File:Stereographic_projection_in_3D.svg" class="mw-file-description"><img alt="Stereographic projection from the top of a sphere onto a plane beneath it" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/220px-Stereographic_projection_in_3D.svg.png" decoding="async" width="220" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/330px-Stereographic_projection_in_3D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/440px-Stereographic_projection_in_3D.svg.png 2x" data-file-width="870" data-file-height="639" /></a></span><div class="sidebar-caption"><a href="/wiki/Projective_geometry" title="Projective geometry">Projecting</a> a <a href="/wiki/Sphere" title="Sphere">sphere</a> to a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">Branches</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean</a> <ul><li><a href="/wiki/Elliptic_geometry" title="Elliptic geometry">Elliptic</a> <ul><li><a href="/wiki/Spherical_geometry" title="Spherical geometry">Spherical</a></li></ul></li> <li><a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li></ul></li> <li><a href="/wiki/Non-Archimedean_geometry" title="Non-Archimedean geometry">Non-Archimedean geometry</a></li> <li><a href="/wiki/Projective_geometry" title="Projective geometry">Projective</a></li> <li><a href="/wiki/Affine_geometry" title="Affine geometry">Affine</a></li> <li><a href="/wiki/Synthetic_geometry" title="Synthetic geometry">Synthetic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a> <ul><li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine</a></li></ul></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a> <ul><li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li> <li><a href="/wiki/Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li> <li><a href="/wiki/Discrete_differential_geometry" title="Discrete differential geometry">Discrete differential</a></li></ul></li> <li><a href="/wiki/Complex_geometry" title="Complex geometry">Complex</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete/Combinatorial</a> <ul><li><a href="/wiki/Digital_geometry" title="Digital geometry">Digital</a></li></ul></li> <li><a href="/wiki/Convex_geometry" title="Convex geometry">Convex</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Incidence_geometry" title="Incidence geometry">Incidence </a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a> <ul><li><a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Concepts</li><li>Features</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><a href="/wiki/Dimension_(geometry)" class="mw-redirect" title="Dimension (geometry)">Dimension</a> <ul><li><a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass constructions</a></li></ul> <ul><li><a href="/wiki/Angle" title="Angle">Angle</a></li> <li><a href="/wiki/Curve" title="Curve">Curve</a></li> <li><a href="/wiki/Diagonal" title="Diagonal">Diagonal</a></li> <li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="/wiki/Perpendicular" title="Perpendicular">Perpendicular</a>)</li> <li><a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">Parallel</a></li> <li><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertex</a></li></ul> <ul><li><a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">Congruence</a></li> <li><a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></li> <li><a href="/wiki/Symmetry" title="Symmetry">Symmetry</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Zero-dimensional_space" title="Zero-dimensional space">Zero-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Point_(geometry)" title="Point (geometry)">Point</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/One-dimensional_space" title="One-dimensional space">One-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Line_(geometry)" title="Line (geometry)">Line</a> <ul><li><a href="/wiki/Line_segment" title="Line segment">segment</a></li> <li><a href="/wiki/Line_(geometry)#Ray" title="Line (geometry)">ray</a></li></ul></li> <li><a href="/wiki/Length" title="Length">Length</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Two-dimensional_space" title="Two-dimensional space">Two-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane</a></li> <li><a href="/wiki/Area" title="Area">Area</a></li> <li><a href="/wiki/Polygon" title="Polygon">Polygon</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Triangle" title="Triangle">Triangle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Altitude_(triangle)" title="Altitude (triangle)">Altitude</a></li> <li><a href="/wiki/Hypotenuse" title="Hypotenuse">Hypotenuse</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Square" title="Square">Square</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li> <li><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Circle" title="Circle">Circle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Diameter" title="Diameter">Diameter</a></li> <li><a href="/wiki/Circumference" title="Circumference">Circumference</a></li> <li><a href="/wiki/Area_of_a_circle" title="Area of a circle">Area</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Three-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Volume" title="Volume">Volume</a></li></ul> <ul><li><a href="/wiki/Cube" title="Cube">Cube</a> <ul><li><a href="/wiki/Cuboid" title="Cuboid">cuboid</a></li></ul></li> <li><a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">Cylinder</a></li> <li><a href="/wiki/Dodecahedron" title="Dodecahedron">Dodecahedron</a></li> <li><a href="/wiki/Icosahedron" title="Icosahedron">Icosahedron</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a></li> <li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">Pyramid</a></li> <li><a href="/wiki/Platonic_Solid" class="mw-redirect" title="Platonic Solid">Platonic Solid</a></li> <li><a href="/wiki/Sphere" title="Sphere">Sphere</a></li> <li><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">Four</a>- / other-dimensional</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Tesseract" title="Tesseract">Tesseract</a></li> <li><a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">Hypersphere</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.2em;"> <a href="/wiki/List_of_geometers" title="List of geometers">Geometers</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by name</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a class="mw-selflink selflink">Riemann</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/List_of_geometers" title="List of geometers">List of geometers</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by period</div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> <a href="/wiki/Before_Common_Era" class="mw-redirect" title="Before Common Era">BCE</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1–1400s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1400s–1700s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1700s–1900s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a class="mw-selflink selflink">Riemann</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Present day</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:General_geometry" title="Template:General geometry"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_geometry" title="Template talk:General geometry"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_geometry" title="Special:EditPage/Template:General geometry"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of the theories of <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, and <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifold</a> theory. The theory of <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a> was elaborated by <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> and particularly <a href="/wiki/Adolf_Hurwitz" title="Adolf Hurwitz">Adolf Hurwitz</a>. This area of mathematics is part of the foundation of <a href="/wiki/Topology" title="Topology">topology</a> and is still being applied in novel ways to <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a>. </p><p>In 1853, <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> asked Riemann, his student, to prepare a <i><a href="/wiki/Habilitationsschrift" class="mw-redirect" title="Habilitationsschrift">Habilitationsschrift</a></i> on the foundations of geometry. Over many months, Riemann developed his theory of <a href="/wiki/Higher_dimensions" class="mw-redirect" title="Higher dimensions">higher dimensions</a> and delivered his lecture at Göttingen on 10 June 1854, entitled <i>Ueber die Hypothesen, welche der Geometrie zu Grunde liegen</i>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> It was not published until twelve years later in 1868 by Dedekind, two years after his death. Its early reception appears to have been slow, but it is now recognized as one of the most important works in geometry. </p><p>The subject founded by this work is <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>. Riemann found the correct way to extend into <i>n</i> dimensions the <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> of surfaces, which Gauss himself proved in his <i><a href="/wiki/Theorema_egregium" class="mw-redirect" title="Theorema egregium">theorema egregium</a></i>. The fundamental objects are called the <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a> and the <a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a>. For the surface (two-dimensional) case, the curvature at each point can be reduced to a number (scalar), with the surfaces of constant positive or negative curvature being models of the <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometries</a>. </p><p>The Riemann metric is a collection of numbers at every point in space (i.e., a <a href="/wiki/Tensor" title="Tensor">tensor</a>) which allows measurements of speed in any trajectory, whose integral gives the distance between the trajectory's endpoints. For example, Riemann found that in four spatial dimensions, one needs ten numbers at each point to describe distances and curvatures on a <a href="/wiki/Manifold" title="Manifold">manifold</a>, no matter how distorted it is. </p> <div class="mw-heading mw-heading2"><h2 id="Complex_analysis">Complex analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=7" title="Edit section: Complex analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In his dissertation, he established a geometric foundation for <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> through <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>, through which multi-valued functions like the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> (with infinitely many sheets) or the <a href="/wiki/Square_root" title="Square root">square root</a> (with two sheets) could become <a href="/wiki/One-to-one_function" class="mw-redirect" title="One-to-one function">one-to-one functions</a>. Complex functions are <a href="/wiki/Harmonic_functions" class="mw-redirect" title="Harmonic functions">harmonic functions</a><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2021)">citation needed</span></a></i>]</sup> (that is, they satisfy <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a> and thus the <a href="/wiki/Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a>) on these surfaces and are described by the location of their singularities and the topology of the surfaces. The topological "genus" of the Riemann surfaces is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=w/2-n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=w/2-n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4428b3c9ec1f796a287308156be304db1bc2ec60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.441ex; height:2.843ex;" alt="{\displaystyle g=w/2-n+1}"></span>, where the surface has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> leaves coming together at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> branch points. For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3aa256fb29a830d501693b50832d0e09f65557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g>1}"></span> the Riemann surface has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3g-3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>g</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3g-3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17fbcba839c967e2e280e1ce29f4901ffca77764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.091ex; height:2.843ex;" alt="{\displaystyle (3g-3)}"></span> parameters (the "<a href="/wiki/Moduli_of_algebraic_curves" title="Moduli of algebraic curves">moduli</a>"). </p><p>His contributions to this area are numerous. The famous <a href="/wiki/Riemann_mapping_theorem" title="Riemann mapping theorem">Riemann mapping theorem</a> says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection between them that is holomorphic with a holomorphic inverse) to either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> or to the interior of the unit circle. The generalization of the theorem to Riemann surfaces is the famous <a href="/wiki/Uniformization_theorem" title="Uniformization theorem">uniformization theorem</a>, which was proved in the 19th century by <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> and <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>. Here, too, rigorous proofs were first given after the development of richer mathematical tools (in this case, topology). For the proof of the existence of functions on Riemann surfaces, he used a minimality condition, which he called the <a href="/wiki/Dirichlet_principle" class="mw-redirect" title="Dirichlet principle">Dirichlet principle</a>. <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> found a gap in the proof: Riemann had not noticed that his working assumption (that the minimum existed) might not work; the function space might not be complete, and therefore the existence of a minimum was not guaranteed. Through the work of <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> in the Calculus of Variations, the Dirichlet principle was finally established. Otherwise, Weierstrass was very impressed with Riemann, especially with his theory of <a href="/wiki/Abelian_function" class="mw-redirect" title="Abelian function">abelian functions</a>. When Riemann's work appeared, Weierstrass withdrew his paper from <i><a href="/wiki/Crelle%27s_Journal" title="Crelle's Journal">Crelle's Journal</a></i> and did not publish it. They had a good understanding when Riemann visited him in Berlin in 1859. Weierstrass encouraged his student <a href="/wiki/Hermann_Amandus_Schwarz" class="mw-redirect" title="Hermann Amandus Schwarz">Hermann Amandus Schwarz</a> to find alternatives to the Dirichlet principle in complex analysis, in which he was successful. An anecdote from <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> shows the difficulties which contemporary mathematicians had with Riemann's new ideas. In 1870, Weierstrass had taken Riemann's dissertation with him on a holiday to Rigi and complained that it was hard to understand. The physicist <a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a> assisted him in the work overnight and returned with the comment that it was "natural" and "very understandable". </p><p>Other highlights include his work on abelian functions and <a href="/wiki/Theta_functions" class="mw-redirect" title="Theta functions">theta functions</a> on Riemann surfaces. Riemann had been in a competition with Weierstrass since 1857 to solve the Jacobian inverse problems for abelian integrals, a generalization of <a href="/wiki/Elliptic_integrals" class="mw-redirect" title="Elliptic integrals">elliptic integrals</a>. Riemann used theta functions in several variables and reduced the problem to the determination of the zeros of these theta functions. Riemann also investigated period matrices and characterized them through the "Riemannian period relations" (symmetric, real part negative). By <a href="/wiki/Ferdinand_Georg_Frobenius" title="Ferdinand Georg Frobenius">Ferdinand Georg Frobenius</a> and <a href="/wiki/Solomon_Lefschetz" title="Solomon Lefschetz">Solomon Lefschetz</a> the validity of this relation is equivalent with the embedding of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}/\Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}/\Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a9a930759fcb2a93fe8355db46ef3284ef704e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.737ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} ^{n}/\Omega }"></span> (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> is the lattice of the period matrix) in a projective space by means of theta functions. For certain values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, this is the <a href="/wiki/Jacobian_variety" title="Jacobian variety">Jacobian variety</a> of the Riemann surface, an example of an abelian manifold. </p><p>Many mathematicians such as <a href="/wiki/Alfred_Clebsch" title="Alfred Clebsch">Alfred Clebsch</a> furthered Riemann's work on algebraic curves. These theories depended on the properties of a function defined on Riemann surfaces. For example, the <a href="/wiki/Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a> (Roch was a student of Riemann) says something about the number of linearly independent differentials (with known conditions on the zeros and poles) of a Riemann surface. </p><p>According to <a href="/wiki/Detlef_Laugwitz" title="Detlef Laugwitz">Detlef Laugwitz</a>,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Automorphic_function" title="Automorphic function">automorphic functions</a> appeared for the first time in an essay about the Laplace equation on electrically charged cylinders. Riemann however used such functions for conformal maps (such as mapping topological triangles to the circle) in his 1859 lecture on hypergeometric functions or in his treatise on <a href="/wiki/Minimal_surface" title="Minimal surface">minimal surfaces</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Real_analysis">Real analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=8" title="Edit section: Real analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the field of <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>, he discovered the <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a> in his <a href="/wiki/Habilitation" title="Habilitation">habilitation</a>. Among other things, he showed that every piecewise continuous function is integrable. Similarly, the <a href="/wiki/Stieltjes_integral" class="mw-redirect" title="Stieltjes integral">Stieltjes integral</a> goes back to the Göttinger mathematician, and so they are named together the <a href="/wiki/Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a>. </p><p>In his habilitation work on <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>, where he followed the work of his teacher Dirichlet, he showed that Riemann-integrable functions are "representable" by Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an example of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the <a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a>: if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large <i>n</i>. </p><p>Riemann's essay was also the starting point for <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>'s work with Fourier series, which was the impetus for <a href="/wiki/Set_theory" title="Set theory">set theory</a>. </p><p>He also worked with <a href="/wiki/Hypergeometric_differential_equations" class="mw-redirect" title="Hypergeometric differential equations">hypergeometric differential equations</a> in 1857 using complex analytical methods and presented the solutions through the behaviour of closed paths about singularities (described by the <a href="/wiki/Monodromy_matrix" title="Monodromy matrix">monodromy matrix</a>). The proof of the existence of such differential equations by previously known monodromy matrices is one of the Hilbert problems. </p> <div class="mw-heading mw-heading2"><h2 id="Number_theory">Number theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=9" title="Edit section: Number theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Riemann made some famous contributions to modern <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>. In <a href="/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude" title="On the Number of Primes Less Than a Given Magnitude">a single short paper</a>, the only one he published on the subject of number theory, he investigated the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">zeta function</a> that now bears his name, establishing its importance for understanding the distribution of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>. The <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> was one of a series of conjectures he made about the function's properties. </p><p>In Riemann's work, there are many more interesting developments. He proved the functional equation for the zeta function (already known to <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>), behind which a theta function lies. Through the summation of this approximation function over the non-trivial zeros on the line with real portion 1/2, he gave an exact, "explicit formula" for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a62d9bf7b2ba03b30a95e7059362cc4b3865123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.471ex; height:2.843ex;" alt="{\displaystyle \pi (x)}"></span>. </p><p>Riemann knew of <a href="/wiki/Pafnuty_Chebyshev" title="Pafnuty Chebyshev">Pafnuty Chebyshev</a>'s work on the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">Prime Number Theorem</a>. He had visited Dirichlet in 1852. </p> <div class="mw-heading mw-heading2"><h2 id="Writings">Writings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=10" title="Edit section: Writings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Riemann's works include: </p> <ul><li>1851 – <i><a href="/wiki/List_of_important_publications_in_mathematics#Grundlagen_für_eine_allgemeine_Theorie_der_Functionen_einer_veränderlichen_complexen_Grösse" title="List of important publications in mathematics">Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse</a></i>, Inaugural dissertation, Göttingen, 1851.</li> <li>1857 – <i><a href="/wiki/List_of_important_publications_in_mathematics#Theorie_der_Abelschen_Functionen" title="List of important publications in mathematics">Theorie der Abelschen Functionen</a></i>, Journal für die reine und angewandte Mathematik, Bd. 54. S. 101–155.</li> <li>1859 – <i>Über die Anzahl der Primzahlen unter einer gegebenen Größe</i>, in: <i>Monatsberichte der Preußischen Akademie der Wissenschaften.</i> Berlin, November 1859, S. 671ff. With Riemann's conjecture. <i><a href="https://en.wikisource.org/wiki/%C3%9Cber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6%C3%9Fe" class="extiw" title="s:Über die Anzahl der Primzahlen unter einer gegebenen Größe">Über die Anzahl der Primzahlen unter einer gegebenen Grösse.</a></i> (Wikisource), <a rel="nofollow" class="external text" href="http://www.claymath.org/sites/default/files/riemann1859.pdf">Facsimile of the manuscript</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303224135/http://www.claymath.org/sites/default/files/riemann1859.pdf">Archived</a> 2016-03-03 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> with Clay Mathematics.</li> <li>1861 – <i><a rel="nofollow" class="external text" href="https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Paris/">Commentatio mathematica, qua respondere tentatur quaestioni ab Illma Academia Parisiensi propositae</a></i>, submitted to the Paris Academy for a prize competition</li> <li>1867 – <i><a href="/wiki/List_of_important_publications_in_mathematics#Über_die_Darstellbarkeit_einer_Function_durch_eine_trigonometrische_Reihe" title="List of important publications in mathematics">Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe</a></i>, Aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen.</li> <li>1868 – <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002019213"><i>Über die Hypothesen, welche der Geometrie zugrunde liegen</i>.</a> Abh. Kgl. Ges. Wiss., Göttingen 1868. Translation <a rel="nofollow" class="external text" href="http://www.emis.de/classics/Riemann/Geom.pdf">EMIS, pdf</a> <i>On the hypotheses which lie at the foundation of geometry</i>, translated by <a href="/wiki/William_Kingdon_Clifford" title="William Kingdon Clifford">W.K.Clifford</a>, Nature 8 1873 183 – reprinted in Clifford's Collected Mathematical Papers, London 1882 (MacMillan); New York 1968 (Chelsea) <a rel="nofollow" class="external free" href="http://www.emis.de/classics/Riemann/">http://www.emis.de/classics/Riemann/</a>. Also in Ewald, William B., ed., 1996 "From Kant to Hilbert: A Source Book in the Foundations of Mathematics", 2 vols. Oxford Uni. Press: 652–61.</li> <li>1876 – <i>Bernhard Riemann's Gesammelte Mathematische Werke und wissenschaftlicher Nachlass. herausgegeben von Heinrich Weber unter Mitwirkung von Richard Dedekind</i>, Leipzig, B. G. Teubner 1876, 2. Auflage 1892, Nachdruck bei Dover 1953 (with contributions by Max Noether and Wilhelm Wirtinger, Teubner 1902). Later editions <i>The collected Works of Bernhard Riemann: The Complete German Texts.</i> Eds. Heinrich Weber; Richard Dedekind; M Noether; Wilhelm Wirtinger; Hans Lewy. Mineola, New York: Dover Publications, Inc., 1953, 1981, 2017</li> <li>1876 – <i>Schwere, Elektrizität und Magnetismus</i>, Hannover: Karl Hattendorff.</li> <li>1882 – <i>Vorlesungen über Partielle Differentialgleichungen</i> 3. Auflage. Braunschweig 1882.</li> <li>1901 – <i>Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen</i>. <a href="https://commons.wikimedia.org/wiki/Bernard_Riemann_%26_Heinrich_Weber_%E2%80%93_Die_partiellen_Differential-Gleichungen_der_mathematischen_Physik_Nach_Riemann%27s_Vorlesungen,_Zweiter_Band,_1901.pdf" class="extiw" title="c:Bernard Riemann & Heinrich Weber – Die partiellen Differential-Gleichungen der mathematischen Physik Nach Riemann's Vorlesungen, Zweiter Band, 1901.pdf">PDF on Wikimedia Commons</a>. On archive.org: <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRiemann1901" class="citation web cs1">Riemann, Bernhard (1901). Weber, Heinrich Martin (ed.). <a rel="nofollow" class="external text" href="https://archive.org/details/diepartiellendi02webegoog">"Die partiellen differential-gleichungen der mathematischen physik nach Riemann's Vorlesungen"</a>. <i>archive.org</i>. Friedrich Vieweg und Sohn<span class="reference-accessdate">. Retrieved <span class="nowrap">1 June</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=archive.org&rft.atitle=Die+partiellen+differential-gleichungen+der+mathematischen+physik+nach+Riemann%27s+Vorlesungen&rft.date=1901&rft.aulast=Riemann&rft.aufirst=Bernhard&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdiepartiellendi02webegoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></li> <li>2004 – <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiemann2004" class="citation cs2">Riemann, Bernhard (2004), <i>Collected papers</i>, Kendrick Press, Heber City, UT, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9740427-2-5" title="Special:BookSources/978-0-9740427-2-5"><bdi>978-0-9740427-2-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2121437">2121437</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Collected+papers&rft.pub=Kendrick+Press%2C+Heber+City%2C+UT&rft.date=2004&rft.isbn=978-0-9740427-2-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2121437%23id-name%3DMR&rft.aulast=Riemann&rft.aufirst=Bernhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_things_named_after_Bernhard_Riemann" title="List of things named after Bernhard Riemann">List of things named after Bernhard Riemann</a></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a></li> <li><a href="/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude" title="On the Number of Primes Less Than a Given Magnitude">On the Number of Primes Less Than a Given Magnitude</a>, Riemann's 1859 paper introducing the complex zeta function</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDudenredaktionKleinerKnöbl2015" class="citation book cs1 cs1-prop-foreign-lang-source">Dudenredaktion; Kleiner, Stefan; Knöbl, Ralf (2015) [First published 1962]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=T6vWCgAAQBAJ"><i>Das Aussprachewörterbuch</i></a> [<i>The Pronunciation Dictionary</i>] (in German) (7th ed.). Berlin: Dudenverlag. pp. 229, 381, 398, 735. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-411-04067-4" title="Special:BookSources/978-3-411-04067-4"><bdi>978-3-411-04067-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Das+Aussprachew%C3%B6rterbuch&rft.place=Berlin&rft.pages=229%2C+381%2C+398%2C+735&rft.edition=7th&rft.pub=Dudenverlag&rft.date=2015&rft.isbn=978-3-411-04067-4&rft.au=Dudenredaktion&rft.au=Kleiner%2C+Stefan&rft.au=Kn%C3%B6bl%2C+Ralf&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DT6vWCgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrechStockHirschfeldAnders2009" class="citation book cs1 cs1-prop-foreign-lang-source">Krech, Eva-Maria; Stock, Eberhard; Hirschfeld, Ursula; Anders, Lutz Christian (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=E-1tr_oVkW4C&q=deutsches+ausspracheworterbuch"><i>Deutsches Aussprachewörterbuch</i></a> [<i>German Pronunciation Dictionary</i>] (in German). Berlin: Walter de Gruyter. pp. 366, 520, 536, 875. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-018202-6" title="Special:BookSources/978-3-11-018202-6"><bdi>978-3-11-018202-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Deutsches+Aussprachew%C3%B6rterbuch&rft.place=Berlin&rft.pages=366%2C+520%2C+536%2C+875&rft.pub=Walter+de+Gruyter&rft.date=2009&rft.isbn=978-3-11-018202-6&rft.aulast=Krech&rft.aufirst=Eva-Maria&rft.au=Stock%2C+Eberhard&rft.au=Hirschfeld%2C+Ursula&rft.au=Anders%2C+Lutz+Christian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DE-1tr_oVkW4C%26q%3Ddeutsches%2Bausspracheworterbuch&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWendorf2020" class="citation web cs1">Wendorf, Marcia (2020-09-23). <a rel="nofollow" class="external text" href="https://interestingengineering.com/science/bernhard-riemann-the-mind-who-laid-the-foundations-for-einsteins-theory-of-relativity">"Bernhard Riemann Laid the Foundations for Einstein's Theory of Relativity"</a>. <i>interestingengineering.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-10-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=interestingengineering.com&rft.atitle=Bernhard+Riemann+Laid+the+Foundations+for+Einstein%27s+Theory+of+Relativity&rft.date=2020-09-23&rft.aulast=Wendorf&rft.aufirst=Marcia&rft_id=https%3A%2F%2Finterestingengineering.com%2Fscience%2Fbernhard-riemann-the-mind-who-laid-the-foundations-for-einsteins-theory-of-relativity&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFJiPapadopoulosYamada2017">Ji, Papadopoulos & Yamada 2017</a>, p. 614</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMccleary" class="citation book cs1">Mccleary, John. <i>Geometry from a Differentiable Viewpoint</i>. Cambridge University Press. p. 282.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+from+a+Differentiable+Viewpoint&rft.pages=282&rft.pub=Cambridge+University+Press&rft.aulast=Mccleary&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></span> </li> <li id="cite_note-Hawking2005-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hawking2005_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephen_Hawking2005" class="citation book cs1">Stephen Hawking (4 October 2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3zdFSOS3f4AC"><i>God Created The Integers</i></a>. Running Press. pp. 814–815. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7624-1922-7" title="Special:BookSources/978-0-7624-1922-7"><bdi>978-0-7624-1922-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=God+Created+The+Integers&rft.pages=814-815&rft.pub=Running+Press&rft.date=2005-10-04&rft.isbn=978-0-7624-1922-7&rft.au=Stephen+Hawking&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3zdFSOS3f4AC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></span> </li> <li id="cite_note-:0-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWendorf2020" class="citation web cs1">Wendorf, Marcia (2020-09-23). <a rel="nofollow" class="external text" href="https://interestingengineering.com/science/bernhard-riemann-the-mind-who-laid-the-foundations-for-einsteins-theory-of-relativity">"Bernhard Riemann Laid the Foundations for Einstein's Theory of Relativity"</a>. <i>interestingengineering.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-04-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=interestingengineering.com&rft.atitle=Bernhard+Riemann+Laid+the+Foundations+for+Einstein%27s+Theory+of+Relativity&rft.date=2020-09-23&rft.aulast=Wendorf&rft.aufirst=Marcia&rft_id=https%3A%2F%2Finterestingengineering.com%2Fscience%2Fbernhard-riemann-the-mind-who-laid-the-foundations-for-einsteins-theory-of-relativity&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Werke, p. 268, edition of 1876, cited in <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.bams/1183493815">Pierpont, Non-Euclidean Geometry, A Retrospect</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.geni.com/people/Ida-Schilling/6000000025101232998">"Ida Schilling"</a>. 22 December 1862.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Ida+Schilling&rft.date=1862-12-22&rft_id=https%3A%2F%2Fwww.geni.com%2Fpeople%2FIda-Schilling%2F6000000025101232998&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></span> </li> <li id="cite_note-duSautoy2003-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-duSautoy2003_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-duSautoy2003_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFdu_Sautoy2003" class="citation book cs1"><a href="/wiki/Marcus_du_Sautoy" title="Marcus du Sautoy">du Sautoy, Marcus</a> (2003). <a href="/wiki/The_Music_of_the_Primes" title="The Music of the Primes"><i>The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics</i></a>. HarperCollins. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-06-621070-4" title="Special:BookSources/978-0-06-621070-4"><bdi>978-0-06-621070-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Music+of+the+Primes%3A+Searching+to+Solve+the+Greatest+Mystery+in+Mathematics&rft.pub=HarperCollins&rft.date=2003&rft.isbn=978-0-06-621070-4&rft.aulast=du+Sautoy&rft.aufirst=Marcus&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://godandmath.com/2012/04/24/christian-mathematicians-riemann/">"Christian Mathematician – Riemann"</a>. 24 April 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">13 October</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Christian+Mathematician+%E2%80%93+Riemann&rft.date=2012-04-24&rft_id=http%3A%2F%2Fgodandmath.com%2F2012%2F04%2F24%2Fchristian-mathematicians-riemann%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://mihai-caragiu-maths.blogspot.co.uk/2009/09/bernhard-riemann.html">"Riemann's Tomb"</a>. 18 September 2009<span class="reference-accessdate">. Retrieved <span class="nowrap">13 October</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Riemann%27s+Tomb&rft.date=2009-09-18&rft_id=http%3A%2F%2Fmihai-caragiu-maths.blogspot.co.uk%2F2009%2F09%2Fbernhard-riemann.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.deutschestextarchiv.de/book/view/riemann_hypothesen_1867?p=7">Riemann, Bernhard: Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. In: Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1868), S. 133-150.</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.emis.de/classics/Riemann/WKCGeom.pdf"><i>On the Hypotheses which lie at the Bases of Geometry</i>. Bernhard Riemann. Translated by William Kingdon Clifford [Nature, Vol. VIII. Nos. 183, 184, pp. 14–17, 36, 37.]</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiemannJost2016" class="citation book cs1">Riemann, Bernhard; Jost, Jürgen (2016). <i>On the Hypotheses Which Lie at the Bases of Geometry</i>. Classic Texts in the Sciences (1st ed. 2016 ed.). Cham: Springer International Publishing : Imprint: Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-26042-6" title="Special:BookSources/978-3-319-26042-6"><bdi>978-3-319-26042-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+the+Hypotheses+Which+Lie+at+the+Bases+of+Geometry&rft.place=Cham&rft.series=Classic+Texts+in+the+Sciences&rft.edition=1st+ed.+2016&rft.pub=Springer+International+Publishing+%3A+Imprint%3A+Birkh%C3%A4user&rft.date=2016&rft.isbn=978-3-319-26042-6&rft.aulast=Riemann&rft.aufirst=Bernhard&rft.au=Jost%2C+J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a>, „<a href="/wiki/Vorlesungen_%C3%BCber_theoretische_Physik" class="mw-redirect" title="Vorlesungen über theoretische Physik">Vorlesungen über theoretische Physik</a>“, Bd.2 (Mechanik deformierbarer Medien), Harri Deutsch, S.124. Sommerfeld heard the story from Aachener Professor of Experimental Physics <a href="/wiki/Adolf_W%C3%BCllner" title="Adolf Wüllner">Adolf Wüllner</a>.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="/wiki/Detlef_Laugwitz" title="Detlef Laugwitz">Detlef Laugwitz</a>: <i>Bernhard Riemann 1826–1866</i>. Birkhäuser, Basel 1996, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-5189-2" title="Special:BookSources/978-3-7643-5189-2">978-3-7643-5189-2</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDerbyshire2003" class="citation cs2"><a href="/wiki/John_Derbyshire" title="John Derbyshire">Derbyshire, John</a> (2003), <i><a href="/wiki/Prime_Obsession" title="Prime Obsession">Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics</a></i>, Washington, DC: John Henry Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-309-08549-7" title="Special:BookSources/0-309-08549-7"><bdi>0-309-08549-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prime+Obsession%3A+Bernhard+Riemann+and+the+Greatest+Unsolved+Problem+in+Mathematics&rft.place=Washington%2C+DC&rft.pub=John+Henry+Press&rft.date=2003&rft.isbn=0-309-08549-7&rft.aulast=Derbyshire&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMonastyrsky1999" class="citation cs2">Monastyrsky, Michael (1999), <i>Riemann, Topology and Physics</i>, Boston, MA: Birkhäuser, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-3789-3" title="Special:BookSources/0-8176-3789-3"><bdi>0-8176-3789-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Riemann%2C+Topology+and+Physics&rft.place=Boston%2C+MA&rft.pub=Birkh%C3%A4user&rft.date=1999&rft.isbn=0-8176-3789-3&rft.aulast=Monastyrsky&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJiPapadopoulosYamada2017" class="citation book cs1">Ji, Lizhen; Papadopoulos, Athanese; Yamada, Sumio, eds. (2017). <i>From Riemann to Differential Geometry and Relativity</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783319600390" title="Special:BookSources/9783319600390"><bdi>9783319600390</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Riemann+to+Differential+Geometry+and+Relativity&rft.pub=Springer&rft.date=2017&rft.isbn=9783319600390&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernhard_Riemann&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></span></span></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Bernhard_Riemann" class="extiw" title="q:Special:Search/Bernhard Riemann">Bernhard Riemann</a></b></i>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Georg_Friedrich_Bernhard_Riemann" class="extiw" title="commons:Georg Friedrich Bernhard Riemann"><span style="font-style:italic; 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<a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Riemann.html">"Bernhard Riemann"</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bernhard+Riemann&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FRiemann.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20050903035028/http://www.fh-lueneburg.de/u1/gym03/englpage/chronik/riemann/riemann.htm">Bernhard Riemann – one of the most important mathematicians</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160318034045/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/">Bernhard Riemann's inaugural lecture</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_Wolfgang_Weisstein" class="mw-redirect" title="Eric Wolfgang Weisstein">Weisstein, Eric Wolfgang</a> (ed.). <a rel="nofollow" class="external text" href="http://scienceworld.wolfram.com/biography/Riemann.html">"Riemann, Bernhard (1826–1866)"</a>. <i><a href="/wiki/ScienceWorld" class="mw-redirect" title="ScienceWorld">ScienceWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ScienceWorld&rft.atitle=Riemann%2C+Bernhard+%281826%26ndash%3B1866%29&rft_id=http%3A%2F%2Fscienceworld.wolfram.com%2Fbiography%2FRiemann.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernhard+Riemann" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Leben/Leben.pdf">Richard Dedekind (1892), Transcripted by D. R. 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class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Bernhard_Riemann" title="Template:Bernhard Riemann"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Bernhard_Riemann" title="Template talk:Bernhard Riemann"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Bernhard_Riemann" title="Special:EditPage/Template:Bernhard Riemann"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Bernhard_Riemann" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Bernhard Riemann</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a></li> <li><a href="/wiki/Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">Generalized Riemann hypothesis</a></li> <li><a href="/wiki/Grand_Riemann_hypothesis" title="Grand Riemann hypothesis">Grand Riemann hypothesis</a></li> <li><a href="/wiki/Grothendieck%E2%80%93Riemann%E2%80%93Roch_theorem" title="Grothendieck–Riemann–Roch theorem">Grothendieck–Hirzebruch–Riemann–Roch theorem</a></li> <li><a href="/wiki/Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem" title="Hirzebruch–Riemann–Roch theorem">Hirzebruch–Riemann–Roch theorem</a></li> <li><a href="/wiki/Local_zeta_function" title="Local zeta function">Local zeta function</a></li> <li><a href="/wiki/Measurable_Riemann_mapping_theorem" title="Measurable Riemann mapping theorem">Measurable Riemann mapping theorem</a></li> <li><a href="/wiki/Riemann_(crater)" title="Riemann (crater)">Riemann (crater)</a></li> 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title="Riemann sphere">Riemann sphere</a></li> <li><a href="/wiki/Riemann_sum" title="Riemann sum">Riemann sum</a></li> <li><a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a></li> <li><a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a></li> <li><a href="/wiki/Riemann%27s_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li> <li><a href="/wiki/Riemann%27s_minimal_surface" title="Riemann's minimal surface">Riemann's minimal surface</a></li> <li><a href="/wiki/Riemannian_circle" class="mw-redirect" title="Riemannian circle">Riemannian circle</a></li> <li><a href="/wiki/Riemannian_connection_on_a_surface" title="Riemannian connection on a surface">Riemannian connection on a surface</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li> <li><a href="/wiki/Riemann%E2%80%93Hilbert_correspondence" title="Riemann–Hilbert correspondence">Riemann–Hilbert correspondence</a></li> <li><a href="/wiki/Riemann%E2%80%93Hilbert_problem" title="Riemann–Hilbert problem">Riemann–Hilbert problems</a></li> <li><a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a></li> <li><a href="/wiki/Riemann%E2%80%93Liouville_integral" title="Riemann–Liouville integral">Riemann–Liouville integral</a></li> <li><a href="/wiki/Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a></li> <li><a href="/wiki/Riemann%E2%80%93Roch_theorem_for_smooth_manifolds" title="Riemann–Roch theorem for smooth manifolds">Riemann–Roch theorem for smooth manifolds</a></li> <li><a href="/wiki/Riemann%E2%80%93Siegel_formula" title="Riemann–Siegel formula">Riemann–Siegel formula</a></li> <li><a href="/wiki/Riemann%E2%80%93Siegel_theta_function" title="Riemann–Siegel theta function">Riemann–Siegel theta function</a></li> <li><a href="/wiki/Riemann%E2%80%93Silberstein_vector" title="Riemann–Silberstein vector">Riemann–Silberstein vector</a></li> <li><a href="/wiki/Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a></li> <li><a href="/wiki/Riemann%E2%80%93von_Mangoldt_formula" title="Riemann–von Mangoldt formula">Riemann–von Mangoldt formula</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a 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