Solutions

Topics

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.