Chern-Kuiper's inequalities

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 $$\nu_g\leq\mu\leq\nu_g+p$$. In this work, we study the submanifolds with $\nu_g\neq\mu$. More precisely, we characterize locally the ones with $0\neq(\mu-\nu_g)\in\{p,p-1,p-2\}$ under the hypothesis of $\nu_g\leq n-p-1$.