Surfaces with specific types of second fundamental form

Question

Given a three dimensional Riemannian manifold $(M,g)$ and a surface $\Sigma \subset M$ can one categorize surfaces where the second fundamental form of $\Sigma$ is a scalar multiple of the induced metric on $\Sigma$? I mean do these surfaces exist and are they well studied? i.e. $ h_{ij} = f {\bar g}_{ij}$ where $f \in C^\infty (M)$ and $\bar{g}$ is the induced metric on $\Sigma$. This is clearly a conformally invariant condition btw... Thanks and sorry for being vague.

Answer 1

It's not a vague question. Such surfaces are known as umbilic surfaces. The generic $3$-dimensional Riemannian manifold $(M,g)$ has no umbilic surfaces whatsoever, and when they do exist, they form a finite dimensional family, in fact, of dimension at most $4$, since, if two umbilic surfaces are tangent at a point and have the same mean curvature vector there, then they are equal in a neighborhood.