Zeros of second fundamental form

Question

Let $M$ be a submanifold of $N$ and suppose that $M$ and $N$ are equipped with arbitrary connections $\nabla^M$ and $\nabla^N.$ Let $A = \{ X \in TM : \mathrm{I\!I}(X,X)=0 \},$ where $\mathrm{I\!I}$ is the second fundamental form, i.e. $\mathrm{I\!I}(X,X)=\nabla^N_XX-\nabla^M_XX$. Are there any geometric significances of the set $A$? Or can such set even exist?

Answer 1

Here is a somewhat classical implication.

When $M$ is embedded into Euclidean space, the connection (on vector fields) can be thought of as the orthogonal projection of the directional derivative of a vector field onto $TM$.

The vanishing of the second fundamental form is then equivalent to this orthogonal projection of a directional derivative being the same as differentiation by the connection. Specifically, if we consider the derivative of one vector field $e_j$ along another $e_i$ (which provide a local basis), we expand it as

$$\frac{\partial}{\partial x^i} (\frac{\partial}{\partial x^j}) = \Gamma^{k}_{ij}\frac{\partial}{\partial x^k} + \vec{n}$$

The coefficients of the Second Fundamental Form are, on the other hand, defined as the projections (by the ambient dot product) onto this normal vector $\vec{n}$ (found by a normalized cross product of the basis of the tangent space as usual).

Hence, when the second fundemental form vanishes, such basis vector derivatives are completely within the tangent space.