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Modern Geometry

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


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)


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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