Chern-kuiper’s inequalities

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