Chern-kuiper’s inequalities

Given a Euclidean submanifold \(g:M^{n}\rightarrow {\mathbb {R}}^{n+p}\), 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

$$\begin{aligned} \nu _g\le \mu \le \nu _g+p. \end{aligned}$$(1)

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 \(\nu _g\le n-p-1\).