Part 2: Riemannian Geometry
Riemannian Geometry
Metric tensor, Levi-Civita connection, parallel transport, and Christoffel symbols
Connections Curvature
Riemann curvature tensor, Ricci tensor and scalar, Bianchi identities, and sectional curvature
Geodesics
Geodesic equation, exponential map, Jacobi fields, conjugate points, and completeness
Gauss Bonnet
Gaussian curvature, Euler characteristic, and the generalized Gauss-Bonnet theorem