• Basics of smooth manifolds: Inverse function theorem, implicit function theorem, submanifolds, integration on manifolds
  • Basics of matrix Lie groups over R and C: The definitions of Gl(n), SU(n), SO(n), U(n), their manifold structures, Lie algebras, right and left invariant vector fields and differential forms, the exponential map.
  • Definition of real and complex vector bundles, tangent and cotangent bundles, basic operations on bundles such as dual bundle, tensor products, exterior products, direct sums, pull-back bundles.
  • Definition of differential forms, exterior product, exterior derivative, de Rham cohomology, behavior under pull-back.
  • Metrics on vector bundles.
  • Riemannian metrics, definition of a geodesic, existence and uniqueness of geodesics.
  • Definition of a principal Lie group bundle for matrix groups.
  • Associated vector bundles: Relation between principal bundles and vector bundles
  • Definition of covariant derivative for a vector bundle and connection on a principal bundle. Relations between the two.
  • Definition of curvature, flat connections, parallel transport.
  • Definition of Levi-Civita connection and properties of the Riemann curvature tensor.