I am working on the frame of singular Riemannian foliations and I am interested on examples of foliations whose eigenvalues of the shape operator of a given leaf are all positive. One intermediate step to understand such examples consists in to understand the following:
Given a Riemannian manifold $(M,g)$ and an immersed manifold $N$. Let $A$ be the shape operator on $N$. In which conditions on $M$ and $N$, $A$ has positive eigenvalues for any point?
It follows, in particular, that if the shape operator is positive definite, then the claim follows.