Submanifolds whose second fundamental form has constant rank in every direction Let $M$ be a submanifold of a Riemannian manifold $\widetilde{M}$. Let $A$ be the second fundamental form of $M$. 
Suppose that, for all $p \in M$, the linear map $A(v, \cdot)\: \colon T_{p}M \to N_{p}M$ has rank $k$ for every nonzero $v \in T_{p}M$.
Does this condition define some well-known class of submanifolds (if any)?
 A: Sometimes, this is an open condition, so that it does not impose any differential equations on the submanifold.  For example, consider the case of a surface $\Sigma^2$ in a $4$-manifold $M^4$.  The condition that the rank of $A(v,\cdot):T_p\Sigma\to N_p\Sigma$ be equal to $2$ for every nonzero $v\in T_p\Sigma$ is an open condition on the submanifold $\Sigma$, so once one has such a submanifold, any $C^2$-nearby submanifold will also have this property.  Probably, someone has a name for this class of surfaces, but I don't think that it is well-known.
There are many examples of such surfaces in $\mathbb{R}^4$.  For example, let $\Sigma\subset\mathbb{C}^2$ be a holomorphic curve without flexes regarded as a smooth surface in $\mathbb{R^4}=\mathbb{C}^2$.
A: 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 $\nu_1=n-k$.
