Whitney embedding theorem

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class="vector-toc-numb">1.3</span> <span>Eventual consequences of the Whitney trick</span> </div> </a> <ul id="toc-Eventual_consequences_of_the_Whitney_trick-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sharper_results" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sharper_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Sharper results</span> </div> </a> <button aria-controls="toc-Sharper_results-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Sharper results subsection</span> </button> <ul id="toc-Sharper_results-sublist" class="vector-toc-list"> <li id="toc-Restrictions_on_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Restrictions_on_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Restrictions on manifolds</span> </div> </a> <ul id="toc-Restrictions_on_manifolds-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Isotopy_versions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Isotopy_versions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Isotopy versions</span> </div> </a> <ul id="toc-Isotopy_versions-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 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<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">Any smooth real m-dimensional manifold can be smoothly embedded in real 2m-space</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, particularly in <a href="/wiki/Differential_topology" title="Differential topology">differential topology</a>, there are two Whitney embedding theorems, named after <a href="/wiki/Hassler_Whitney" title="Hassler Whitney">Hassler Whitney</a>: </p> <ul><li>The <b>strong Whitney embedding theorem</b> states that any <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">smooth</a> <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real</a> <span class="texhtml mvar" style="font-style:italic;">m</span>-<a href="/wiki/Dimension_(mathematics)" class="mw-redirect" title="Dimension (mathematics)">dimensional</a> <a href="/wiki/Manifold" title="Manifold">manifold</a> (required also to be <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> and <a href="/wiki/Second-countable" class="mw-redirect" title="Second-countable">second-countable</a>) can be <a href="/wiki/Smooth_map" class="mw-redirect" title="Smooth map">smoothly</a> <a href="/wiki/Embedding" title="Embedding">embedded</a> in the <a href="/wiki/Real_coordinate_space" title="Real coordinate space">real <span class="texhtml">2<i>m</i></span>-space</a>, <span class="nowrap">⁠<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 {R} ^{2m},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2m},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36261130e07c5a8abf3ec05c2dc9fe745fb7583e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.822ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{2m},}"></span>⁠</span> if <span class="texhtml"><i>m</i> > 0</span>. This is the best linear bound on the smallest-dimensional Euclidean space that all <span class="texhtml mvar" style="font-style:italic;">m</span>-dimensional manifolds embed in, as the <a href="/wiki/Real_projective_space" title="Real projective space">real projective spaces</a> of dimension <span class="texhtml mvar" style="font-style:italic;">m</span> cannot be embedded into real <span class="texhtml">(2<i>m</i> − 1)</span>-space if <span class="texhtml mvar" style="font-style:italic;">m</span> is a <a href="/wiki/Power_of_two" title="Power of two">power of two</a> (as can be seen from a <a href="/wiki/Characteristic_class" title="Characteristic class">characteristic class</a> argument, also due to Whitney).</li> <li>The <b>weak Whitney embedding theorem</b> states that any continuous function from an <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional manifold to an <span class="texhtml mvar" style="font-style:italic;">m</span>-dimensional manifold may be approximated by a smooth embedding provided <span class="texhtml"><i>m</i> > 2<i>n</i></span>. Whitney similarly proved that such a map could be approximated by an <a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">immersion</a> provided <span class="texhtml"><i>m</i> > 2<i>n</i> − 1</span>. This last result is sometimes called the <b><a href="/wiki/Whitney_immersion_theorem" title="Whitney immersion theorem">Whitney immersion theorem</a></b>.</li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="About_the_proof">About the proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Whitney_embedding_theorem&action=edit&section=1" title="Edit section: About the proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Weak_embedding_theorem">Weak embedding theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Whitney_embedding_theorem&action=edit&section=2" title="Edit section: Weak embedding theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The weak Whitney embedding is proved through a projection argument. </p><p>When the manifold is <i>compact</i>, one can first use a covering by finitely many local charts and then reduce the dimension with suitable projections.<sup id="cite_ref-Hirsch_1-0" class="reference"><a href="#cite_note-Hirsch-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Ch. 1 §3">: Ch. 1 §3 </span></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><sup class="reference nowrap"><span title="Page / location: Ch. 6">: Ch. 6 </span></sup><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><sup class="reference nowrap"><span title="Page / location: Ch. 5 §3">: Ch. 5 §3 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Strong_embedding_theorem">Strong embedding theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Whitney_embedding_theorem&action=edit&section=3" title="Edit section: Strong embedding theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> The general outline of the proof is to start with an immersion <span class="nowrap">⁠<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 f:M\to \mathbb {R} ^{2m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:M\to \mathbb {R} ^{2m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793c0b4dc9fe37b9507dc74cb2cb55b78fb9422e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.447ex; height:3.009ex;" alt="{\displaystyle f:M\to \mathbb {R} ^{2m}}"></span>⁠</span> with <a href="/wiki/Transversality_(mathematics)" title="Transversality (mathematics)">transverse</a> self-intersections. These are known to exist from Whitney's earlier work on <b>the weak immersion theorem</b>. Transversality of the double points follows from a general-position argument. The idea is to then somehow remove all the self-intersections. If <span class="texhtml mvar" style="font-style:italic;">M</span> has boundary, one can remove the self-intersections simply by isotoping <span class="texhtml mvar" style="font-style:italic;">M</span> into itself (the isotopy being in the domain of <span class="texhtml mvar" style="font-style:italic;">f</span>), to a submanifold of <span class="texhtml mvar" style="font-style:italic;">M</span> that does not contain the double-points. Thus, we are quickly led to the case where <span class="texhtml mvar" style="font-style:italic;">M</span> has no boundary. Sometimes it is impossible to remove the double-points via an isotopy—consider for example the figure-8 immersion of the circle in the plane. In this case, one needs to introduce a local double point. </p><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Whitneytrickstep1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Whitneytrickstep1.svg/350px-Whitneytrickstep1.svg.png" decoding="async" width="350" height="53" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Whitneytrickstep1.svg/525px-Whitneytrickstep1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Whitneytrickstep1.svg/700px-Whitneytrickstep1.svg.png 2x" data-file-width="900" data-file-height="135" /></a><figcaption>Introducing double-point.</figcaption></figure><p> Once one has two opposite double points, one constructs a closed loop connecting the two, giving a closed path in <span class="nowrap">⁠<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 {R} ^{2m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2m}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b771ab83305f7396c76b1b1ab27d0d9a0f439ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.822ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2m}.}"></span>⁠</span> Since <span class="nowrap">⁠<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 {R} ^{2m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f793c7e7d89f1189bd4b5c365b4ee385375f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.175ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2m}}"></span>⁠</span> is <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>, one can assume this path bounds a disc, and provided <span class="texhtml">2<i>m</i> > 4</span> one can further assume (by the <b>weak Whitney embedding theorem</b>) that the disc is embedded in <span class="nowrap">⁠<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 {R} ^{2m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f793c7e7d89f1189bd4b5c365b4ee385375f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.175ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2m}}"></span>⁠</span> such that it intersects the image of <span class="texhtml mvar" style="font-style:italic;">M</span> only in its boundary. Whitney then uses the disc to create a <a href="/wiki/Homotopy" title="Homotopy">1-parameter family</a> of immersions, in effect pushing <span class="texhtml mvar" style="font-style:italic;">M</span> across the disc, removing the two double points in the process. In the case of the figure-8 immersion with its introduced double-point, the push across move is quite simple (pictured).</p><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Whitneytrickstep2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Whitneytrickstep2.svg/450px-Whitneytrickstep2.svg.png" decoding="async" width="450" height="62" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Whitneytrickstep2.svg/675px-Whitneytrickstep2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Whitneytrickstep2.svg/900px-Whitneytrickstep2.svg.png 2x" data-file-width="1450" data-file-height="200" /></a><figcaption>Cancelling opposite double-points.</figcaption></figure><p> This process of eliminating <b>opposite sign</b> double-points by pushing the manifold along a disc is called the <b>Whitney Trick</b>. </p><p>To introduce a local double point, Whitney created immersions <span class="nowrap">⁠<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 \alpha _{m}:\mathbb {R} ^{m}\to \mathbb {R} ^{2m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{m}:\mathbb {R} ^{m}\to \mathbb {R} ^{2m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58e4863ea2c7753bc87fac9e2f8bda7971db994a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.242ex; height:3.009ex;" alt="{\displaystyle \alpha _{m}:\mathbb {R} ^{m}\to \mathbb {R} ^{2m}}"></span>⁠</span> which are approximately linear outside of the unit ball, but containing a single double point. For <span class="texhtml"><i>m</i> = 1</span> such an immersion is given by </p> <dl><dd><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 {\begin{cases}\alpha :\mathbb {R} ^{1}\to \mathbb {R} ^{2}\\\alpha (t)=\left({\frac {1}{1+t^{2}}},\ t-{\frac {2t}{1+t^{2}}}\right)\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>α</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\alpha :\mathbb {R} ^{1}\to \mathbb {R} ^{2}\\\alpha (t)=\left({\frac {1}{1+t^{2}}},\ t-{\frac {2t}{1+t^{2}}}\right)\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ee7bb35bf0fe78c4275d91fdfe4252e2cac175" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26.654ex; height:7.676ex;" alt="{\displaystyle {\begin{cases}\alpha :\mathbb {R} ^{1}\to \mathbb {R} ^{2}\\\alpha (t)=\left({\frac {1}{1+t^{2}}},\ t-{\frac {2t}{1+t^{2}}}\right)\end{cases}}}"></span></dd></dl> <p>Notice that if <span class="texhtml">α</span> is considered as a map to <span class="nowrap">⁠<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 {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>⁠</span> like so: </p> <dl><dd><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 \alpha (t)=\left({\frac {1}{1+t^{2}}},\ t-{\frac {2t}{1+t^{2}}},0\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (t)=\left({\frac {1}{1+t^{2}}},\ t-{\frac {2t}{1+t^{2}}},0\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d688b1bf7d522ded9ea9779da2366060706813a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.613ex; height:6.176ex;" alt="{\displaystyle \alpha (t)=\left({\frac {1}{1+t^{2}}},\ t-{\frac {2t}{1+t^{2}}},0\right)}"></span></dd></dl> <p>then the double point can be resolved to an embedding: </p> <dl><dd><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 \beta (t,a)=\left({\frac {1}{(1+t^{2})(1+a^{2})}},\ t-{\frac {2t}{(1+t^{2})(1+a^{2})}},\ {\frac {ta}{(1+t^{2})(1+a^{2})}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mi>a</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta (t,a)=\left({\frac {1}{(1+t^{2})(1+a^{2})}},\ t-{\frac {2t}{(1+t^{2})(1+a^{2})}},\ {\frac {ta}{(1+t^{2})(1+a^{2})}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d46016b299bc3ee11c8abeba46f067c41b4cf39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:70.622ex; height:6.343ex;" alt="{\displaystyle \beta (t,a)=\left({\frac {1}{(1+t^{2})(1+a^{2})}},\ t-{\frac {2t}{(1+t^{2})(1+a^{2})}},\ {\frac {ta}{(1+t^{2})(1+a^{2})}}\right).}"></span></dd></dl> <p>Notice <span class="texhtml">β(<i>t</i>, 0) = α(<i>t</i>)</span> and for <span class="texhtml"><i>a</i> ≠ 0</span> then as a function of <span class="texhtml mvar" style="font-style:italic;">t</span>, <span class="texhtml">β(<i>t</i>, <i>a</i>)</span> is an embedding. </p><p>For higher dimensions <span class="texhtml mvar" style="font-style:italic;">m</span>, there are <span class="texhtml">α<sub><i>m</i></sub></span> that can be similarly resolved in <span class="nowrap">⁠<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 {R} ^{2m+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2m+1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b814a38e8a66c08b5d4e2e782e76f5cfc17dca4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.922ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2m+1}.}"></span>⁠</span> For an embedding into <span class="nowrap">⁠<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 {R} ^{5},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{5},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b308a3661817394a7e6119de25ad557bebe09929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{5},}"></span>⁠</span> for example, define </p> <dl><dd><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 \alpha _{2}(t_{1},t_{2})=\left(\beta (t_{1},t_{2}),\ t_{2}\right)=\left({\frac {1}{(1+t_{1}^{2})(1+t_{2}^{2})}},\ t_{1}-{\frac {2t_{1}}{(1+t_{1}^{2})(1+t_{2}^{2})}},\ {\frac {t_{1}t_{2}}{(1+t_{1}^{2})(1+t_{2}^{2})}},\ t_{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{2}(t_{1},t_{2})=\left(\beta (t_{1},t_{2}),\ t_{2}\right)=\left({\frac {1}{(1+t_{1}^{2})(1+t_{2}^{2})}},\ t_{1}-{\frac {2t_{1}}{(1+t_{1}^{2})(1+t_{2}^{2})}},\ {\frac {t_{1}t_{2}}{(1+t_{1}^{2})(1+t_{2}^{2})}},\ t_{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66a07eafc839c8b6bfc65688b42d62fb936ebdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:93.582ex; height:7.509ex;" alt="{\displaystyle \alpha _{2}(t_{1},t_{2})=\left(\beta (t_{1},t_{2}),\ t_{2}\right)=\left({\frac {1}{(1+t_{1}^{2})(1+t_{2}^{2})}},\ t_{1}-{\frac {2t_{1}}{(1+t_{1}^{2})(1+t_{2}^{2})}},\ {\frac {t_{1}t_{2}}{(1+t_{1}^{2})(1+t_{2}^{2})}},\ t_{2}\right).}"></span></dd></dl> <p>This process ultimately leads one to the definition: </p> <dl><dd><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 \alpha _{m}(t_{1},t_{2},\cdots ,t_{m})=\left({\frac {1}{u}},t_{1}-{\frac {2t_{1}}{u}},{\frac {t_{1}t_{2}}{u}},t_{2},{\frac {t_{1}t_{3}}{u}},t_{3},\cdots ,{\frac {t_{1}t_{m}}{u}},t_{m}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mrow> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mi>u</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>u</mi> </mfrac> </mrow> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mi>u</mi> </mfrac> </mrow> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mi>u</mi> </mfrac> </mrow> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{m}(t_{1},t_{2},\cdots ,t_{m})=\left({\frac {1}{u}},t_{1}-{\frac {2t_{1}}{u}},{\frac {t_{1}t_{2}}{u}},t_{2},{\frac {t_{1}t_{3}}{u}},t_{3},\cdots ,{\frac {t_{1}t_{m}}{u}},t_{m}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b656162c8519525aad0e0df29dd4c62a368c1d95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:68.009ex; height:6.176ex;" alt="{\displaystyle \alpha _{m}(t_{1},t_{2},\cdots ,t_{m})=\left({\frac {1}{u}},t_{1}-{\frac {2t_{1}}{u}},{\frac {t_{1}t_{2}}{u}},t_{2},{\frac {t_{1}t_{3}}{u}},t_{3},\cdots ,{\frac {t_{1}t_{m}}{u}},t_{m}\right),}"></span></dd></dl> <p>where </p> <dl><dd><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 u=(1+t_{1}^{2})(1+t_{2}^{2})\cdots (1+t_{m}^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=(1+t_{1}^{2})(1+t_{2}^{2})\cdots (1+t_{m}^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0045e2f0a809521e8896d85798945bf9ae3a77ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.311ex; height:3.176ex;" alt="{\displaystyle u=(1+t_{1}^{2})(1+t_{2}^{2})\cdots (1+t_{m}^{2}).}"></span></dd></dl> <p>The key properties of <span class="texhtml">α<sub><i>m</i></sub></span> is that it is an embedding except for the double-point <span class="texhtml">α<sub><i>m</i></sub>(1, 0, ... , 0) = α<sub><i>m</i></sub>(−1, 0, ... , 0)</span>. Moreover, for <span class="texhtml">|(<i>t</i><sub>1</sub>, ... , <i>t<sub>m</sub></i>)|</span> large, it is approximately the linear embedding <span class="texhtml">(0, <i>t</i><sub>1</sub>, 0, <i>t</i><sub>2</sub>, ... , 0, <i>t<sub>m</sub></i>)</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Eventual_consequences_of_the_Whitney_trick">Eventual consequences of the Whitney trick</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Whitney_embedding_theorem&action=edit&section=4" title="Edit section: Eventual consequences of the Whitney trick"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Whitney trick was used by <a href="/wiki/Stephen_Smale" title="Stephen Smale">Stephen Smale</a> to prove the <a href="/wiki/H-cobordism_theorem" class="mw-redirect" title="H-cobordism theorem"><i>h</i>-cobordism theorem</a>; from which follows the <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a> in dimensions <span class="texhtml"><i>m</i> ≥ 5</span>, and the classification of <a href="/wiki/Smooth_structure" title="Smooth structure">smooth structures</a> on discs (also in dimensions 5 and up). This provides the foundation for <a href="/wiki/Surgery_theory" title="Surgery theory">surgery theory</a>, which classifies manifolds in dimension 5 and above. </p><p>Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension ≥ 5, one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Whitney_embedding_theorem&action=edit&section=5" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output 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">See also: <a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History of manifolds and varieties</a></div> <p>The occasion of the proof by <a href="/wiki/Hassler_Whitney" title="Hassler Whitney">Hassler Whitney</a> of the embedding theorem for smooth manifolds is said (rather surprisingly) to have been the first complete exposition of the <i>manifold concept</i> precisely because it brought together and unified the differing concepts of manifolds at the time: no longer was there any confusion as to whether abstract manifolds, intrinsically defined via charts, were any more or less general than manifolds extrinsically defined as submanifolds of Euclidean space. See also the <a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">history of manifolds and varieties</a> for context. </p> <div class="mw-heading mw-heading2"><h2 id="Sharper_results">Sharper results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Whitney_embedding_theorem&action=edit&section=6" title="Edit section: Sharper results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although every <span class="texhtml mvar" style="font-style:italic;">n</span>-manifold embeds in <span class="nowrap">⁠<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 {R} ^{2n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/244fdd1e23c4d2f60a8da8ff339e14af86a0382d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.365ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{2n},}"></span>⁠</span> one can frequently do better. Let <span class="texhtml"><i>e</i>(<i>n</i>)</span> denote the smallest integer so that all compact connected <span class="texhtml mvar" style="font-style:italic;">n</span>-manifolds embed in <span class="nowrap">⁠<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 {R} ^{e(n)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{e(n)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5dcfbe54dbf40d11a8197f2896ea70cba3e3033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.589ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} ^{e(n)}.}"></span>⁠</span> Whitney's strong embedding theorem states that <span class="texhtml"><i>e</i>(<i>n</i>) ≤ 2<i>n</i></span>. For <span class="texhtml"><i>n</i> = 1, 2</span> we have <span class="texhtml"><i>e</i>(<i>n</i>) = 2<i>n</i></span>, as the <a href="/wiki/Circle" title="Circle">circle</a> and the <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a> show. More generally, for <span class="texhtml"><i>n</i> = 2<sup><i>k</i></sup></span> we have <span class="texhtml"><i>e</i>(<i>n</i>) = 2<i>n</i></span>, as the <span class="texhtml">2<sup><i>k</i></sup></span>-dimensional <a href="/wiki/Real_projective_space" title="Real projective space">real projective space</a> show. Whitney's result can be improved to <span class="texhtml"><i>e</i>(<i>n</i>) ≤ 2<i>n</i> − 1</span> unless <span class="texhtml mvar" style="font-style:italic;">n</span> is a power of 2. This is a result of <a href="/wiki/Andr%C3%A9_Haefliger" title="André Haefliger">André Haefliger</a> and <a href="/wiki/Morris_Hirsch" title="Morris Hirsch">Morris Hirsch</a> (for <span class="texhtml"><i>n</i> > 4</span>) and <a href="/wiki/C._T._C._Wall" title="C. T. C. Wall">C. T. C. Wall</a> (for <span class="texhtml"><i>n</i> = 3</span>); these authors used important preliminary results and particular cases proved by Hirsch, <a href="/wiki/William_S._Massey" title="William S. Massey">William S. Massey</a>, <a href="/wiki/Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Sergey Novikov</a> and <a href="/wiki/Vladimir_Rokhlin_(Soviet_mathematician)" class="mw-redirect" title="Vladimir Rokhlin (Soviet mathematician)">Vladimir Rokhlin</a>.<sup id="cite_ref-skopenkov2_4-0" class="reference"><a href="#cite_note-skopenkov2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> At present the function <span class="texhtml mvar" style="font-style:italic;">e</span> is not known in closed-form for all integers (compare to the <a href="/wiki/Whitney_immersion_theorem" title="Whitney immersion theorem">Whitney immersion theorem</a>, where the analogous number is known). </p> <div class="mw-heading mw-heading3"><h3 id="Restrictions_on_manifolds">Restrictions on manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Whitney_embedding_theorem&action=edit&section=7" title="Edit section: Restrictions on manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One can strengthen the results by putting additional restrictions on the manifold. For example, the <a href="/wiki/N-sphere" title="N-sphere"><span class="texhtml mvar" style="font-style:italic;">n</span>-sphere</a> always embeds in <span class="nowrap">⁠<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 {R} ^{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccea3976e1f8a1bb853c8ca00e52d518a3a4fe07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.997ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n+1}}"></span>⁠</span> – which is the best possible (closed <span class="texhtml mvar" style="font-style:italic;">n</span>-manifolds cannot embed in <span class="nowrap">⁠<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 {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>⁠</span>). Any compact <i>orientable</i> surface and any compact surface <i>with non-empty boundary</i> embeds in <span class="nowrap">⁠<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 {R} ^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb17c1074c77de2cf88d45bcd6d7a795b0f5d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{3},}"></span>⁠</span> though any <i>closed non-orientable</i> surface needs <span class="nowrap">⁠<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 {R} ^{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d430c4fb99f020f9bab89c75847092a44842c7c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}.}"></span>⁠</span> </p><p>If <span class="texhtml mvar" style="font-style:italic;">N</span> is a compact orientable <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional manifold, then <span class="texhtml mvar" style="font-style:italic;">N</span> embeds in <span class="nowrap">⁠<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 {R} ^{2n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a05cbaac96ee725b237c62ab10faf0a0d28119b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.819ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2n-1}}"></span>⁠</span> (for <span class="texhtml mvar" style="font-style:italic;">n</span> not a power of 2 the orientability condition is superfluous). For <span class="texhtml mvar" style="font-style:italic;">n</span> a power of 2 this is a result of <a href="/wiki/Andr%C3%A9_Haefliger" title="André Haefliger">André Haefliger</a> and <a href="/wiki/Morris_Hirsch" title="Morris Hirsch">Morris Hirsch</a> (for <span class="texhtml"><i>n</i> > 4</span>), and Fuquan Fang (for <span class="texhtml"><i>n</i> = 4</span>); these authors used important preliminary results proved by Jacques Boéchat and Haefliger, <a href="/wiki/Simon_Donaldson" title="Simon Donaldson">Simon Donaldson</a>, Hirsch and <a href="/wiki/William_S._Massey" title="William S. Massey">William S. Massey</a>.<sup id="cite_ref-skopenkov2_4-1" class="reference"><a href="#cite_note-skopenkov2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Haefliger proved that if <span class="texhtml mvar" style="font-style:italic;">N</span> is a compact <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional <a href="/wiki/N-connected" class="mw-redirect" title="N-connected"><span class="texhtml mvar" style="font-style:italic;">k</span>-connected</a> manifold, then <span class="texhtml mvar" style="font-style:italic;">N</span> embeds in <span class="nowrap">⁠<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 {R} ^{2n-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2n-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7a1e40cc18e3688d2e7eb4560f4446beff0f415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.854ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2n-k}}"></span>⁠</span> provided <span class="texhtml">2<i>k</i> + 3 ≤ <i>n</i></span>.<sup id="cite_ref-skopenkov2_4-2" class="reference"><a href="#cite_note-skopenkov2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Isotopy_versions">Isotopy versions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Whitney_embedding_theorem&action=edit&section=8" title="Edit section: Isotopy versions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A relatively 'easy' result is to prove that any two embeddings of a 1-manifold into <span class="nowrap">⁠<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 {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}"></span>⁠</span> are isotopic (see <a href="/wiki/Knot_theory#Higher_dimensions" title="Knot theory">Knot theory#Higher dimensions</a>). This is proved using general position, which also allows to show that any two embeddings of an <span class="texhtml mvar" style="font-style:italic;">n</span>-manifold into <span class="nowrap">⁠<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 {R} ^{2n+2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2n+2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f379bc4bea2e8d31c33cfe13c0663c31195191b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.819ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2n+2}}"></span>⁠</span> are isotopic. This result is an isotopy version of the weak Whitney embedding theorem. </p><p>Wu proved that for <span class="texhtml"><i>n</i> ≥ 2</span>, any two embeddings of an <span class="texhtml mvar" style="font-style:italic;">n</span>-manifold into <span class="nowrap">⁠<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 {R} ^{2n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aac07172c3ef7c08f78b1b6aa513d909aa09870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.819ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2n+1}}"></span>⁠</span> are isotopic. This result is an isotopy version of the strong Whitney embedding theorem. </p><p>As an isotopy version of his embedding result, <a href="/wiki/Andr%C3%A9_Haefliger" title="André Haefliger">Haefliger</a> proved that if <span class="texhtml mvar" style="font-style:italic;">N</span> is a compact <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional <span class="texhtml mvar" style="font-style:italic;">k</span>-connected manifold, then any two embeddings of <span class="texhtml mvar" style="font-style:italic;">N</span> into <span class="nowrap">⁠<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 {R} ^{2n-k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2n-k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d004bea678cfc580877d2f357baf6624eb6b5f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.954ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2n-k+1}}"></span>⁠</span> are isotopic provided <span class="texhtml">2<i>k</i> + 2 ≤ <i>n</i></span>. The dimension restriction <span class="texhtml">2<i>k</i> + 2 ≤ <i>n</i></span> is sharp: Haefliger went on to give examples of non-trivially embedded 3-spheres in <span class="nowrap">⁠<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 {R} ^{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/547484e275658bac48b1ad8f5407446612d4a65c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{6}}"></span>⁠</span> (and, more generally, <span class="texhtml">(2<i>d</i> − 1)</span>-spheres in <span class="nowrap">⁠<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 {R} ^{3d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b9daa57ef354eae155ee2a9ed8d2559dae15d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.592ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3d}}"></span>⁠</span>). See <a rel="nofollow" class="external text" href="http://www.map.mpim-bonn.mpg.de/High_codimension_embeddings:_classification">further generalizations</a>. </p> <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=Whitney_embedding_theorem&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Representation_theorem" title="Representation theorem">Representation theorem</a></li> <li><a href="/wiki/Whitney_immersion_theorem" title="Whitney immersion theorem">Whitney immersion theorem</a></li> <li><a href="/wiki/Nash_embedding_theorem" class="mw-redirect" title="Nash embedding theorem">Nash embedding theorem</a></li> <li><a href="/wiki/Takens%27s_theorem" title="Takens's theorem">Takens's theorem</a></li> <li><a href="/wiki/Nonlinear_dimensionality_reduction" title="Nonlinear dimensionality reduction">Nonlinear dimensionality reduction</a></li> <li><a href="/wiki/Universal_space" title="Universal space">Universal space</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Whitney_embedding_theorem&action=edit&section=10" title="Edit section: Notes"><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"><ol class="references"> <li id="cite_note-Hirsch-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hirsch_1-0">^</a></b></span> <span class="reference-text"><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="CITEREFHirsch1976" class="citation book cs1"><a href="/wiki/Morris_Hirsch" title="Morris Hirsch">Hirsch, Morris W.</a> (1976). <i>Differential topology</i>. Graduate texts in mathematics. New York Heidelberg Berlin: <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4684-9449-5" title="Special:BookSources/978-1-4684-9449-5"><bdi>978-1-4684-9449-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+topology&rft.place=New+York+Heidelberg+Berlin&rft.series=Graduate+texts+in+mathematics&rft.pub=Springer&rft.date=1976&rft.isbn=978-1-4684-9449-5&rft.aulast=Hirsch&rft.aufirst=Morris+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWhitney+embedding+theorem" 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="CITEREFLee2013" class="citation book cs1"><a href="/wiki/John_M._Lee" title="John M. Lee">Lee, John M.</a> (2013). <a rel="nofollow" class="external text" href="https://www.worldcat.org/title/800646950"><i>Introduction to smooth manifolds</i></a>. Graduate texts in mathematics (2nd ed.). New York; London: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-9981-8" title="Special:BookSources/978-1-4419-9981-8"><bdi>978-1-4419-9981-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/800646950">800646950</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+smooth+manifolds&rft.place=New+York%3B+London&rft.series=Graduate+texts+in+mathematics&rft.edition=2nd&rft.pub=Springer&rft.date=2013&rft_id=info%3Aoclcnum%2F800646950&rft.isbn=978-1-4419-9981-8&rft.aulast=Lee&rft.aufirst=John+M.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Ftitle%2F800646950&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWhitney+embedding+theorem" 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="CITEREFPrasolov2006" class="citation book cs1"><a href="/wiki/Prasolov" title="Prasolov">Prasolov, Victor V.</a> (2006). <i>Elements of Combinatorial and Differential Topology</i>. Providence: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-1153-4" title="Special:BookSources/978-1-4704-1153-4"><bdi>978-1-4704-1153-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Combinatorial+and+Differential+Topology&rft.place=Providence&rft.pub=American+Mathematical+Society&rft.date=2006&rft.isbn=978-1-4704-1153-4&rft.aulast=Prasolov&rft.aufirst=Victor+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWhitney+embedding+theorem" class="Z3988"></span></span> </li> <li id="cite_note-skopenkov2-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-skopenkov2_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-skopenkov2_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-skopenkov2_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">See section 2 of Skopenkov (2008)</span> </li> </ol></div></div> <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=Whitney_embedding_theorem&action=edit&section=11" title="Edit section: References"><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="CITEREFWhitney1992" class="citation cs2"><a href="/wiki/Hassler_Whitney" title="Hassler Whitney">Whitney, Hassler</a> (1992), <a href="/wiki/James_Eells" title="James Eells">Eells, James</a>; Toledo, Domingo (eds.), <i>Collected Papers</i>, Boston: Birkhäuser, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-3560-2" title="Special:BookSources/0-8176-3560-2"><bdi>0-8176-3560-2</bdi></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.place=Boston&rft.pub=Birkh%C3%A4user&rft.date=1992&rft.isbn=0-8176-3560-2&rft.aulast=Whitney&rft.aufirst=Hassler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWhitney+embedding+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilnor1965" class="citation cs2"><a href="/wiki/John_Milnor" title="John Milnor">Milnor, John</a> (1965), <i>Lectures on the </i>h<i>-cobordism theorem</i>, Princeton University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+the+h-cobordism+theorem&rft.pub=Princeton+University+Press&rft.date=1965&rft.aulast=Milnor&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWhitney+embedding+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdachi1993" class="citation cs2">Adachi, Masahisa (1993), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JcMwHWSBSB4C"><i>Embeddings and Immersions</i></a>, translated by Hudson, Kiki, American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-4612-4" title="Special:BookSources/0-8218-4612-4"><bdi>0-8218-4612-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Embeddings+and+Immersions&rft.pub=American+Mathematical+Society&rft.date=1993&rft.isbn=0-8218-4612-4&rft.aulast=Adachi&rft.aufirst=Masahisa&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJcMwHWSBSB4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWhitney+embedding+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSkopenkov2008" class="citation cs2">Skopenkov, Arkadiy (2008), "Embedding and knotting of manifolds in Euclidean spaces", in Nicholas Young; Yemon Choi (eds.), <i>Surveys in Contemporary Mathematics</i>, London Math. Soc. Lect. Notes., vol. 347, Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, pp. 248–342, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0604045">math/0604045</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006math......4045S">2006math......4045S</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=2388495">2388495</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Embedding+and+knotting+of+manifolds+in+Euclidean+spaces&rft.btitle=Surveys+in+Contemporary+Mathematics&rft.place=Cambridge&rft.series=London+Math.+Soc.+Lect.+Notes.&rft.pages=248-342&rft.pub=Cambridge+University+Press&rft.date=2008&rft_id=info%3Aarxiv%2Fmath%2F0604045&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2388495%23id-name%3DMR&rft_id=info%3Abibcode%2F2006math......4045S&rft.aulast=Skopenkov&rft.aufirst=Arkadiy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWhitney+embedding+theorem" 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=Whitney_embedding_theorem&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.map.mpim-bonn.mpg.de/Embeddings_in_Euclidean_space:_an_introduction_to_their_classification">Classification of embeddings</a></li></ul> <div class="navbox-styles"><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 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.navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Manifolds_(Glossary)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><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:Manifolds" title="Template:Manifolds"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manifolds" title="Template talk:Manifolds"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Manifold" title="Manifold">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Topological_manifold" title="Topological manifold">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size:85%;"><a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a class="mw-selflink selflink">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />manifolds</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_manifold" title="Closed manifold">Closed</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="/wiki/Complex_manifold" title="Complex manifold">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler</a></li> <li><a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Whitney_embedding_theorem&oldid=1248529519">https://en.wikipedia.org/w/index.php?title=Whitney_embedding_theorem&oldid=1248529519</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Theorems_in_differential_topology" title="Category:Theorems in differential topology">Theorems in differential 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Whitney embedding theorem - Wikipedia