Connection in a vector bundle

Christoffel symbols; connection matrix (of 1-forms)

Induced connections on tensor products and dual spaces. Vector valued differential forms.

Covariant derivative; Differential Bianchi identity.

Parallel transport and holonomy group.

Ambrose-Singer' theorem; holonomy principle. Linear connections - torsion tensor. Ricci identities.

Bianchi identities. Bianchi and Ricci identities in the language of forms-Cartan's structure equations.

Local calculations. Geodesics of a linear connection.

Examples of computations of geodesics. The geodesic vector field (spray) on TM.

Examples. The exponential map.

Normal coordinates and their properties. Convex normal neighborhoods-Whitehead's theorem.

Finish the proof of Whitehead's theorem. Variations of a (geodesic) curve. Jacobi's equation and fields.

Conjugate points. Riemannian metric and the Levi-Civita connection.

Riemannian vector bundles and their holonomy. Equivalence of the metric and manifold topologies.

The Heisenberg group and Lie groups - left invariant vector fields and the Lie algebra,

left-invariant metrics and their Levi-Civita connection.

Computations of the connection and curvature using forms and Cartan's structure equations-

hyperbolic space example. Properties of the curvature tensor. Spaces of constant curvature.

The musical isomorphism. Sectional curvature. Proof that the sectional curvatures determine

the curvature.

Ricci and scalar curvatures. Kulkarni-Nomizu product and the decomposition of the curvature tensor.

Einstein manifolds. Schur's teorem. Contructed differential Bianchi identity.

Riemannian sub-manifolds - Gauss' equation and the second fundamental form, the induced

connection on the normal bundle.

Weingarten's equation. Gauss' equation for curvature. The Gauss and Weingarten maps for

a hypersurface in Euclidean space; principle curavtures. Gauss' lemma and Hopf-Rinow's theorem.