Intrinsicality of second fundamentals of hypersurfaces in space form

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$.