This course is taken in sequence, part 1 in the fall, and part 2 in the spring.

### MODERN GEOMETRY I

I Differential Manifolds

- Smooth manifolds
- Smooth maps (submersions, immersions, embeddings)
- Tangent spaces and tangent bundles
- Vector bundles
- Vector fields (flows, Frobenius theorem)
- Lie groups and homogeneous spaces
- Covering maps and fibrations
- Tensors, differential forms, Stokes’ theorem

II Riemannian Geometry

- Riemannian metrics and connections
- Metrics and connections on vector bundles
- Geodesics
- Curvature
- Jacobi fields
- Isometric immersions (second fundamental form)

### MODERN GEOMETRY II

I Riemannian Geometry (continued)

- Complete manifolds
- Hopf-Rinow and Cartan-Hadamard theorems
- Manifolds of constant curvature
- First and second variations of energy
- Bonnet-Myers theorem, Synge’s theorem

II Principal Bundles

- Principal bundles and associated bundles
- Connections and curvatures on principal bundles
- Induced connections and curvatures on associated vector bundles
- Parallel transport and holonomy
- Characteristic classes

III Witten’s proof of the Positive Energy Theorem

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