I want to know
Does 2-dimensional Gauss-Bonnet theorem applicable (any topological or geometrical obstruction) in higher dimensions?
My idea is that one can consider 2-dimensional embedded sub-manifolds of $(M^n,g)$ and then applying Gauss-Bonnet theorem to all of such sub-manifolds then gluing them together somehow and finding a topological or geometrical property. Is that possible at all?