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<!DOCTYPE html><html><head><title>Manifolds and Differential Geometry - Jeffrey Marc Lee - Google Books</title><link rel="stylesheet" href="/books/css/_4f54b7ece244bea45b37a7418cb9cf10/kl_about_this_book_kennedy_full_bundle.css" type="text/css" /><link rel="stylesheet"href="https://fonts.googleapis.com/css2?family=Product+Sans:wght@400"><script src="/books/javascript/atb_4f54b7ece244bea45b37a7418cb9cf10__en.js"></script><link rel="canonical" href="https://books.google.com/books/about/Manifolds_and_Differential_Geometry.html?id=QqHdHy9WsEoC"/><meta property="og:url" content="https://books.google.com/books/about/Manifolds_and_Differential_Geometry.html?id=QqHdHy9WsEoC"/><meta name="title" content="Manifolds and Differential Geometry"/><meta name="description" content=""Differential geometry began as the study of curves and surfaces using the methods of calculus. 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In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature. At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle. This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory. The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hypersurfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations. The first chapters of the book are suitable for a one-semester course on manifolds. There is more than enough material for a year-long course on manifolds and geometry."--Publisher's website.</div></div></div><div class="search_box_wrapper"><form action=/books id=search_form style="margin:0px;padding:0px;" method=get> <input type=hidden name="id" value="QqHdHy9WsEoC"><table cellpadding=0 cellspacing=0 class="swv-table"><tr><td class="swv-td-search"><span><input id=search_form_input type=text maxlength=1024 class="text_flat swv-input-search" aria-label="Search in this book" name=q value="" title="Search inside" accesskey=i></span></td><td class="swv-td-space"><div> </div></td><td><input type=submit value="Search inside"></td></tr></table><script type="text/javascript">if (window['_OC_autoDir']) {_OC_autoDir('search_form_input');}</script></form><div id="preview-link"><a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&printsec=frontcover" class="primary"><span dir=ltr>Preview this book</span> »</a></div></div></td> </tr></table><div id="summary-second-column"></div></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="selected_pages_anchor"></a>Selected pages</h3><div id=selected_pages class=about_content><div id=selected_pages_v><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&pg=PR13&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/content?id=QqHdHy9WsEoC&pg=PR13&img=1&zoom=1&sig=ACfU3U2VG06uAJrkEFOfbq4GOnf1MP7n1w" alt="Page xiii" title="Page xiii" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&pg=PR13&source=gbs_selected_pages&cad=1" >Page xiii</a></div><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&pg=PA12&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/content?id=QqHdHy9WsEoC&pg=PA12&img=1&zoom=1&sig=ACfU3U0ACyWi4ZiH4bODrI-oCrq55dpCMQ" alt="Page 12" title="Page 12" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&pg=PA12&source=gbs_selected_pages&cad=1" >Page 12</a></div><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&pg=PR5&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/content?id=QqHdHy9WsEoC&pg=PR5&img=1&zoom=1&sig=ACfU3U0aJ6mufOp5EX0kn8RF9xQy9RINhA" alt="Table of Contents" title="Table of Contents" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&pg=PR5&source=gbs_selected_pages&cad=1" >Table of Contents</a></div><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&pg=PA667&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/content?id=QqHdHy9WsEoC&pg=PA667&img=1&zoom=1&sig=ACfU3U22rtOAOEK2TLi70D6cIwF2vVNXCA" alt="Index" title="Index" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&pg=PA667&source=gbs_selected_pages&cad=1" >Index</a></div><div style="clear:both;"></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="toc_anchor"></a>Contents</h3><div id=toc class=about_content><div id=toc_v><div class="first_toc_column"><div class="first_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The Tangent Structure </span></span></div></td><td class="toc_number" align=right>55</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Immersion and Submersion </span></span></div></td><td class="toc_number" align=right>127</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Curves and Hypersurfaces in Euclidean Space </span></span></div></td><td class="toc_number" align=right>143</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Lie Groups </span></span></div></td><td class="toc_number" align=right>189</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Fiber Bundles </span></span></div></td><td class="toc_number" align=right>257</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Tensors </span></span></div></td><td class="toc_number" align=right>307</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Differential Forms </span></span></div></td><td class="toc_number" align=right>345</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Integration and Stokes Theorem </span></span></div></td><td class="toc_number" align=right>391</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><div class="second_toc_column"><div class="second_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Distributions and Frobenius Theorem </span></span></div></td><td class="toc_number" align=right>467</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Connections and Covariant Derivatives </span></span></div></td><td class="toc_number" align=right>501</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Riemannian and SemiRiemannian Geometry </span></span></div></td><td class="toc_number" align=right>547</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Appendix A The Language of Category Theory </span></span></div></td><td class="toc_number" align=right>637</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Appendix B Topology </span></span></div></td><td class="toc_number" align=right>643</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Modules and Multilinearity </span></span></div></td><td class="toc_number" align=right>649</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Bibliography </span></span></div></td><td class="toc_number" align=right>663</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><a class="primary" href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&printsec=copyright" ><span title="Copyright" style="white-space:nowrap"><span dir=ltr>Copyright</span></span></a></div></td><td class="toc_number" align=right></td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><br style="clear:both;"/></div><span onclick="_OC_setListSectionVisible('toc_h', 1)" class=morelesslink id=toc_hc0 style="display:none"><br>More</span><div id=toc_hd1><div class="first_toc_column"><div class="first_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>De Rham Cohomology </span></span></div></td><td class="toc_number" align=right>441</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><div class="second_toc_column"><div class="second_toc_pad"><table></table></div></div><br style="clear:both;"/><span onclick="_OC_setListSectionVisible('toc_h', 0)" class=morelesslink id=toc_hc1 style="display:none"><br>Less</span></div><script type="text/javascript">if (window['_OC_setListSectionVisible']) {_OC_setListSectionVisible('toc_h', 0);}</script></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="book_other_versions_anchor"></a>Other editions - <a href='https://books.google.com.sg/books?q=editions:ISBN0821848151&id=QqHdHy9WsEoC'>View all</a></h3><div id=book_other_versions class=about_content><div id=book_other_versions_v><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=QKFnEAAAQBAJ&source=gbs_book_other_versions_r&cad=3" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=QKFnEAAAQBAJ&printsec=frontcover&img=1&zoom=5&edge=curl" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=QKFnEAAAQBAJ&printsec=frontcover&source=gbs_book_other_versions_r&cad=3"><span dir=ltr>Manifolds and Differential Geometry</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Jeffrey+M.+Lee%22" class="secondary"><span dir=ltr>Jeffrey M. Lee</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2022</span><br/></span></div></td><td align=right></td></tr></table></div></div><script>(function () {var fn = window['_OC_WSBookList'] || window['_OC_BookList'];fn && fn('book_other_versions', [{"title":"Manifolds and Differential Geometry","authors":"Jeffrey M. Lee","bib_key":"ISBN:9781470469825","pub_date":"8 Mar 2022","snippet":"Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated ...","subject":"Mathematics","info_url":"https://books.google.com.sg/books?id=QKFnEAAAQBAJ\u0026source=gbs_book_other_versions","preview_url":"https://books.google.com.sg/books?id=QKFnEAAAQBAJ\u0026printsec=frontcover\u0026source=gbs_book_other_versions","thumbnail_url":"https://books.google.com.sg/books/content?id=QKFnEAAAQBAJ\u0026printsec=frontcover\u0026img=1\u0026zoom=1\u0026edge=curl","num_pages":671,"viewability":2,"preview":"partial","embeddable":true,"my_ebooks_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fas_coll%3D7\u0026hl=en","has_scanned_text":true,"can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false}]);})();</script></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="word_cloud_anchor"></a>Common terms and phrases</h3><div id=word_cloud class=about_content><div id=word_cloud_v><style type="text/css">.cloud9 {color: #7777cc;font-size: 10px;}.cloud8 {color: #6963CC;font-size: 10.5px;}.cloud7 {color: #6057CC;font-size: 11px;}.cloud6 {color: #574BCC;font-size: 11.5px;}.cloud5 {color: #4E3DCC;font-size: 12px;}.cloud4 {color: #4632CC;font-size: 14px;}.cloud3 {color: #3D26CC;font-size: 16px;}.cloud2 {color: #341ACC;font-size: 18px;}.cloud1 {color: #2B0DCC;font-size: 20px;}.cloud0 {color: #2200CC;font-size: 22px;}.cloud {margin-top: 4px;line-height: 24px;}.cloud a {margin-right: 6px;text-decoration: none;}.cloud a:hover {text-decoration: underline;}</style><div class=cloud><a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=action&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>action</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=atlas&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>atlas</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=basis&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>basis</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=called&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>called</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=chart&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>chart</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=cohomology&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>cohomology</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=commutes&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>commutes</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=compact&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>compact</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=components&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>components</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=connected&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>connected</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=consider&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>consider</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=coordinate&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>coordinate</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Corollary&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>Corollary</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=countable&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>countable</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=covariant+derivative&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>covariant derivative</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=curvature&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>curvature</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=defined&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>defined</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Definition&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>Definition</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=denote&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>denote</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=diffeomorphism&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>diffeomorphism</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=differential&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>differential</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=domain&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>domain</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=element&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>element</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=equation&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>equation</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=equivalence&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>equivalence</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=example&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>example</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Exercise&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>Exercise</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=expp&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>expp</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=%C6%8Fxi&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>Əxi</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=finite&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>finite</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=formula&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>formula</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=frame+field&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>frame field</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=geodesic&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>geodesic</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=geometry&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>geometry</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=given&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>given</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=GL(n&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>GL(n</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=GL(V&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>GL(V</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=global&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>global</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=homomorphism&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>homomorphism</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=integral+curve&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>integral curve</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=isometry&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>isometry</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=isomorphism&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>isomorphism</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Lemma&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>Lemma</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Lie+algebra&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>Lie algebra</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Lie+group&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>Lie group</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=linear+map&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>linear map</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=map+f&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>map f</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=map+%C6%92&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>map ƒ</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=matrix&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>matrix</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=metric&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>metric</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=module&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>module</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=morphism&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>morphism</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=n-manifold&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>n-manifold</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=neighborhood&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>neighborhood</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=normal&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>normal</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=notation&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>notation</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=obtain&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>obtain</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=open+set&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>open set</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=open+subset&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>open subset</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=oriented&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>oriented</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=orthonormal&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>orthonormal</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=principal+bundle&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>principal bundle</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Proof&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>Proof</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Proposition&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>Proposition</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=pull-back&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>pull-back</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=regular+submanifold&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>regular submanifold</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=restriction&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>restriction</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Riemannian+manifold&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>Riemannian manifold</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=scalar+product&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>scalar product</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=semi-Riemannian+manifold&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>semi-Riemannian manifold</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Show&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>Show</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=smooth+function&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>smooth function</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=smooth+manifold&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>smooth manifold</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=smooth+map&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>smooth map</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=smooth+structure&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>smooth structure</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Suppose&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Suppose</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=tangent+space&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>tangent space</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=tensor+field&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>tensor field</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=Theorem&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>Theorem</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=timelike&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>timelike</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=topology&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>topology</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=unique&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>unique</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=V%E2%82%81&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>V₁</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=vector+bundle&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>vector bundle</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=vector+field&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>vector field</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=vector+space&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>vector space</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=zero&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>zero</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=%D3%98%D2%BB&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>Әһ</span></a> <a href="https://books.google.com.sg/books?id=QqHdHy9WsEoC&q=%E1%83%9B%E1%83%94&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>მე</span></a></div></div></div></div><div class=vertical_module_list_row><h3 class="about_title">Bibliographic information</h3><div class="about_content" id="metadata_content" style="padding-bottom:.3em"><div class="metadata_sectionwrap"><table id="metadata_content_table"><tr class="metadata_row"><td class="metadata_label">Title</td><td class="metadata_value"><span dir=ltr>Manifolds and Differential Geometry</span><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22Graduate+studies+in+mathematics%22&source=gbs_metadata_r&cad=5"><i><span dir=ltr>Volume 107 of Graduate studies in mathematics</span></i></a>, <span dir=ltr>ISSN</span> <span dir=ltr>1065-7339</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Author</span></td><td class="metadata_value"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Jeffrey+Marc+Lee%22&source=gbs_metadata_r&cad=5"><span dir=ltr>Jeffrey Marc Lee</span></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Publisher</span></td><td class="metadata_value"><span dir=ltr>American Mathematical Soc., 2009</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>ISBN</span></td><td class="metadata_value"><span dir=ltr>0821848151, 9780821848159</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Length</span></td><td class="metadata_value"><span dir=ltr>671 pages</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Subjects</span></td><td class="metadata_value"><div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics%22" itemprop="url" dir=ltr><span itemprop="title">Mathematics</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Geometry%22" itemprop="url" dir=ltr><span itemprop="title">Geometry</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Geometry+Differential%22" itemprop="url" dir=ltr><span itemprop="title">Differential</span></a></div><br><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Geometry+/+Differential%22&source=gbs_metadata_r&cad=5"><span dir=ltr>Mathematics / Geometry / Differential</span></a><br/><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Topology%22&source=gbs_metadata_r&cad=5"><span dir=ltr>Mathematics / Topology</span></a></td></tr><tr class="metadata_row"><td>&nbsp</td><td>&nbsp</td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Export Citation</span></td><td class="metadata_value"><a class="gb-button " 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