extremal points on a manifold intrinsiclly

Question

I am wondering if there is a geometric object for real analytic manifolds that characterizes extremal points of the manifold intrinsically. For instance, suppose I live in the manifold, can I calculate a geometric quantity on a point by point basis to determine the nature of the points (extremal or not)? For example, in my minds eye, I imagine this quantity would vanish when evaluated for extremal points. I would appreciated any words on something like this. Thanks.