Math 445 Fall

Fall: Riemannian Geometry

Text: Cheeger and Ebin (or similar)

- Levi-Civita connection
- geodesics, existence
- completeness
- geodesics as flows on the tangent bundle
- geodesics as Hamiltonian flows on the cotangent bundle
- Jacobi fields and curvature
- Riemann normal coordinates
- Riemann curvature tensor and its symmetries
- sectional curvature, scalar curvature
- spaces of constant curvature, geodesics
- volume of balls, Taylor expansion, Ricci curvature
- exponential map, normal coordinates
- Theorems of Cartan-Hadamard and Synge
- The sphere theorem

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