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Modern Geometry

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

II Riemannian Geometry

  • Riemannian metrics and connections
  • Geodesics
  • Curvature
  • Jacobi fields
  • Isometric immersions (second fundamental form)
  • Hopf-Rinow and Cartan-Hadamard theorems
  • Manifolds of constant curvature
  • Bonnet-Myers theorem

Modern Geometry II

I Differential Topology

  • Transversality
  • Tubular neighborhoods
  • Intersection theory (mod 2 and oriented)
  • Degrees
  • Poincare-Hopf index theorem
  • Lefschetz fixed-point theorem
  • de Rham cohomology
  • Poincare duality

II Vector Bundles and Principal Bundles

  • Real and complex vector bundles
  • Metrics, connections, and curvature on vector bundles
  • Chern, Pontryagin, and Euler classes
  • Principal bundles
  • Connections and curvature on principal bundles
  • Parallel transport and holonomy
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