Chern-Kuiper's inequalities

Abstract

Given a Euclidean submanifold $\map{g}{M}{n}{\R^{n+p}}$, Chern and Kuiper provided inequalities between $\mu$ and ${\nu }_g$, the ranks of the nullity of $M^n$ and the relative nullity of $g$ respectively. Namely, they prove that $${\nu }_g\le \mu \le {\nu }_g+p$$. In this work, we study the submanifolds with ${\nu }_g\ne \mu$. More precisely, we characterize locally the ones with $0\ne (\mu -{\nu }_g)\in \{p,p-1,p-2\}$ under the hypothesis of ${\nu }_g\le n-p-1$.