In this last lecture we prove the Hopf-Rinow Theorem, the Cartan-Hadamard Theorem, and the Bonnet-Myers Theorem.

Read MoreWe show how a Riemannian manifold can be given the structure of a metric space, and prove geodesics are locally length-minimising with respect to this metric.

Read MoreWe prove that geodesics are critical points of the energy functional, and that the null-space of the Hessian is spanned by Jacobi fields vanishing at the endpoints.

Read MoreWe define Jacobi fields and prove that the Jacobi field equation is the linearisation of the geodesic equation. We prove the Gauss Lemma.

Read MoreWe state the Killing-Hopf Theorem. We then define the Ricci curvature and the scalar curvature, and introduce Einstein metrics.

Read MoreWe define the divergence of a vector field. We define the gradient and the Hessian of a function, and give three equivalent ways of defining the Laplacian.

Read MoreWe define the sectional curvature of a Riemannian manifold, which in dimension 2 reduces to the Gaussian curvature. We prove Schur’s Theorem.

Read MoreWe define isometric maps, and investigate how the Levi-Civita connection behaves with respect to isometric maps.

Read MoreWe prove the so-called “Fundamental Theorem of Riemannian Geometry” on the existence and uniqueness of the Levi-Civita connection.

Read MoreWe define the torsion tensor of a connection, and prove that a connection is uniquely determined by its geodesics and its torsion tensor.

Read MoreWe define the exponential map of a spray and prove it is a diffeomorphism near the zero section. We then prove the Ambrose-Palais-Singer Spray Theorem.

Read MoreWe define geodesics, and prove geodesics with prescribed initial conditions always exist. We define sprays, and use this to define the geodesic flow.

Read MoreWe define the holonomy of a connection on a principal bundle. We prove the principal bundle version of the Ambrose-Singer Holonomy Theorem.

Read MoreWe prove Cartan’s Structure Equation and discuss how curvature on a principal bundle is related to curvature on any associated bundle.

Read MoreWe define the connection form and the curvature form of a connection on a principal bundle, and state Cartan's Structure Equation.

Read MoreWe define connections on principal bundles. We prove that a connection on a principal bundle induces a connection on every associated vector bundle.

Read MoreWe state and prove the Chern-Weil Theorem. We define the Pontryagin classes of a vector bundle, and give an application to the embedding problem.

Read MoreWe state and prove the Bianchi Identity, and explain how this can be used to construct characteristic classes of vector bundles.

Read MoreWe give a sheaf-theoretic definition of a connection, and introduce the exterior covariant differential associated to a connection.

Read MoreWe define the holonomy algebra of a connection and prove it defines a subbundle of the endomorphism bundle. We state the Ambrose-Singer Holonomy Theorem.

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