Manfredo do Carmo, Riemannian Geometry, chapters 1-9
S. Kobayashi, K. Nomizu, Foundations of Differential Geometry
J. Milnor, J. Stasheff, Characteristic classes, Appendix C
J. Dupont, Fibre bundles and Chern-Weil theory
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L. I. Nicolaescu, Notes on Atiyah-Singer Index Theorem
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Topics covered:
Riemannian geometry: Hopf-Rinow theorem, curvature and second
fundamental form, Jacobi fields, the theorems of Gauss-Bonnet,
Cartan-Hadamard, and Bonnet-Myers, spaces of constant sectional curvature...
Connections and curvature on vector bundles, Spin structures, Spaces of
flat connections, Characteristic classes via Chern-Weil appoach