TITLE: Submanifolds whose second fundamental form has constant rank in every direction
QUESTION [5 upvotes]: 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)?

REPLY [5 votes]: 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 $\mathbb{R^4}=\mathbb{C}^2$.