Intrinsicality of second fundamentals of hypersurfaces in space form

TL;DR

This paper proves that the normalized second fundamental form of certain hypersurfaces in space forms is intrinsic under specific conditions, advancing understanding of geometric properties related to curvature and hypersurface theory.

Contribution

It establishes the intrinsic nature of the normalized second fundamental form for hypersurfaces in space forms under conditions involving higher order curvature functions.

Findings

Normalized second fundamental form is intrinsic when k+1(A)  for some k.

Intrinsicity holds for hypersurfaces that are boundaries of domains in space forms.

Results apply to hypersurfaces in space forms with dimension n3.

Abstract

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