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<p>Murphy and the second author showed that a generic closed Riemannian manifold has no totally geodesic submanifolds, provided the ambient space is at least four dimensional. Lytchak and Petrunin established a similar result in dimension 3. For the higher dimensional result, the “generic set” is open and dense in the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9998_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{q}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>–topology for any <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9998_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(q\ge 2.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≥</mo> <mn>2</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In Lytchak and Petrunin’s work, the “generic set” is a dense <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9998_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(G_{\delta }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mi>δ</mi> </msub> </math></EquationSource> </InlineEquation> in the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9998_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{q}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>–topology for any <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9998_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(q\ge 2.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≥</mo> <mn>2</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Here we show that the set of such metrics on a compact 3–manifold actually contains a set that is that is open and dense set in the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9998_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{q}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>–topology, provided <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9998_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(q\ge 3.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≥</mo> <mn>3</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Random 3-manifolds have no totally geodesic submanifolds

  • 期刊论文

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  • 发表日期：

    2025年5月15日

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  • Hasan M. El-Hasan,
  • Frederick Wilhelm

摘要

Murphy and the second author showed that a generic closed Riemannian manifold has no totally geodesic submanifolds, provided the ambient space is at least four dimensional. Lytchak and Petrunin established a similar result in dimension 3. For the higher dimensional result, the “generic set” is open and dense in the \(C^{q}\) C q –topology for any \(q\ge 2.\) q ≥ 2 . In Lytchak and Petrunin’s work, the “generic set” is a dense \(G_{\delta }\) G δ in the \(C^{q}\) C q –topology for any \(q\ge 2.\) q ≥ 2 . Here we show that the set of such metrics on a compact 3–manifold actually contains a set that is that is open and dense set in the \(C^{q}\) C q –topology, provided \(q\ge 3.\) q ≥ 3 .