I'm interested in how the concept of saddle point, easily defined for graphs of functions, can be generalized for two-dimensional surfaces embedded in arbitrary semi-Riemannian manifolds.
I think the answer to that is something related to the second fundamental form. Let $\Sigma\subset M$ be an embedded surface in $M$. Let $N\Sigma$ be its normal bundle and let $\Pi_N:TM|_{\Sigma} \to N\Sigma$ be the projection onto the orthogonal component to the surface. Define the second fundamental form to be $$\mathrm{II}(v,w)=\Pi_N(\nabla_v w),$$
where $\nabla$ is the Levi-Civita connection of $M$. Given this definition of the second fundamental form, can it be used to define a saddle point in $\Sigma$? If not, what is the correct way of defining saddle points for arbitrary surfaces?