We prove that the normalized second fundamental form A of immersed <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2118_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$C^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> hypersurface in a space form <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2118_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$(N^{n+1}, {{\bar{g}}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mover accent="true"> <mrow> <mi>g</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is intrinsic provided <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2118_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma _{2k+1}(A)\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2118_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We also establish the intrinsicality of the normalized second fundamental form A of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2118_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$M^{n}=\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>M</mi> <mi>n</mi> </msup> <mo>=</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> for domain <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2118_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="128" /> </InlineMediaObject> <EquationSource Format="TEX">$ \Omega \subset N^{n+1}, \ n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mi>N</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mspace width="4pt" /> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.

Intrinsicality of Second Fundamentals of Hypersurfaces in Space Forms

期刊论文
发表日期：

2025年8月4日

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Pengfei Guan,
Yu Yuan

摘要

We prove that the normalized second fundamental form A of immersed $C^2$ $C^{2}$ hypersurface in a space form $(N^{n+1}, {{\bar{g}}})$ $(N^{n + 1}, \bar{g})$ is intrinsic provided $\sigma _{2k+1}(A)\ne 0$ $σ_{2 k + 1} (A) \neq 0$ for some $k\ge 1$ $k \geq 1$ . We also establish the intrinsicality of the normalized second fundamental form A of $M^{n}=\partial \Omega $ $M^{n} = \partial Ω$ for domain $ \Omega \subset N^{n+1}, \ n\ge 3$ $Ω \subset N^{n + 1}, n \geq 3$ .