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)?



Submanifolds whose second fundamental form has constant rank in every direction
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 $\mathbb{R^4}=\mathbb{C}^2$.


