This condition means that the 1-nullity $\nu_1=n-k$.

Let $U,\ V$ and $W$ be real vector spaces of finite dimension, and let $\beta:V\times U\to W$ be a bilinear form. The *nullity* subspace $\mathcal{N}\left(\beta\right)\subset U$ of $\beta$ is defined by

$$\mathcal{N}\left(\beta\right)=\left\{Y\in U:\beta\left(X,Y\right)=0\text{ for all }X\in V\right\},$$

and its *image* subspace $\mathcal{S}\left(\beta\right)\subset W$ by

$$\mathcal{S}\left(\beta\right)=\text{span}\left\{\beta\left(X,Y\right):X\in V\text{ and }Y\in U\right\}.$$

Assume that $W$ has a positive definite inner product and that $\beta:V\times V\to W$ is a symmetric bilinear form. For an $s$-dimensional subspace $U^s\subset W$, we denote by $\beta_{U^s}:V\times V\to U^s$ the map given by

$$\beta_{U^s}\left(X,Y\right)=\pi_{U^s}\circ\beta\left(X,Y\right),$$

where $\pi_{U^s}$ stands for the orthogonal projection $\pi_{U^s}:W\to U^s$. The $s$*-nullity* $\nu_s$ of the bilinear form $\beta$ is defined by

$$\nu_s=\max_{U^s\subset W}\left\{\dim\mathcal{N}\left(\beta_{U^s}\right)\right\}$$

for each integer $1\leq s\leq\dim W$.

For an isometric immersion $f:M^n\to\tilde M^m$, the $s$*-nullity* $\nu_s\left(x\right)$ at $x\in M^n$ is defined as the $s$-nullity of its second fundamental form $\alpha$ at $x$.

If the subspaces $U^s$ in the definition of $\nu_s\left(x\right)$ are restricted to subspaces of $N_1\left(x\right)$, then one obtains the $s$*-nullity of $f$ on the first normal space*, which we denote by $\nu_s^*\left(x\right)$. Notice that when $k=\dim N_1\left(x\right)$ then $\nu_k^*\left(x\right)$ is the usual index of relative nullity.

The concept of $s$-nullities plays a key role in the study of rigidity aspects of submanifolds. Roughly speaking, the $s$-nullities measure the degeneracy of the extrinsic geometry of a submanifold. As a general rule of thumb, the higher the $s$-nullities, the more deformable the submanifold. Indeed, do Carmo-Dajczer rigidity result states that in low codimension submanifolds with low $s$-nullities are rigid.

**Theorem.** *An isometric immersion $f:M^n\to\mathbb{Q}_c^{n+p},\ p\leq5,$ with $s$-nullities satisfying $\nu_s\leq n-2s-1$ for all $1\leq s\leq p$ at any point is rigid.*

Moreover, the above assumption on the $s$-nullities also implies infinitesimal rigidity.

Since rigid submanifolds tend not to use more codimension than strictly necessary to immerse their second fundamental form, it is not surprising that the notion of $s$-nullities plays also a role in the study of reduction of codimension. Indeed, the next result ensures the parallelism of the first normal bundle for submanifolds with low $s$-nullities in low codimension.

**Proposition.** *Let $f:M^n\to\mathbb{Q}_c^m$ be a 1-regular isometric immersion such that* $\text{rank}\,N_1=q\leq n-1$. *If $\nu_s^*\left(x\right)<n-s$ for all $1\leq s\leq q$ at any point $x\in M^n$, then the first normal bundle $N_1$ is parallel.*

Since a generic set of vectors is as linearly independent as possible, it follows that generic submanifolds will have as high $s$-nullities as possible. In the particular case of 1-nullity, if $\nu_1=n-k$ everywhere, then the index of relative nullity $\nu=n-k$. Thus every shape operator $A_\xi|_{\mathcal{N}^\perp}$ is an isomorphism. The maximal dimension of a subspace of symmetric isomorphisms of a $k$-dimensional real vector space is given by

$$p\left(k\right)=\rho\left(\frac{k}{2}\right)+1,$$

where $\rho\left(k\right)$ is the Hurwitz-Radon number. Thus, in codimension $p\leq p\left(k\right)$, a generic $n$-dimensional submanifold with constant index of relative nullity $\nu=n-k$ will have $\nu_1=n-k$.

nonzero$v\in T_pM$. Otherwise, taking $v=0$, you will have to have $k=0$, and then you'll only get the totally geodesic submanifolds. $\endgroup$ – Robert Bryant Sep 16 '18 at 11:53