A nice topic to read about is Chern-Weil theory. This is the generalisation of Gauss-Bonnet to higher dimensions and to vector bundles other than the tangent bundle. Put very briefly, topological invariants of a vector bundle over a manifold (its characteristic classes - certain classes in the cohomology of the base) can be computed using the curvature tensor of any choice of connection in the bundle.
The prototype is Gauss-Bonnet in which, as you know, the Euler characteristic of a (compact orientable) surface is equal to a fixed constant times the integral of the scalar curvature of any Riemannian metric on the surface.
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