Riemannian metric, tensors, vector bundles, connections and curvature on a vector bundle, the Levi-Civita connection. The positivity of curvature and the topology of a manifold. Geodesics,
Lie derivative, the exponential map, completeness. Characteristic classes, the Chern-Weil theory. Bianchi identities, the Gauss-Bonnet theorem. Introduction to Lie groups.
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