The Part III differential geometry examination will consist of questions from material covered in lecture and the example sheets.

The following may be used as a general guide to what was covered in MT 2011.

Definition of smooth manifolds, smooth functions and smooth maps and diffeomorphisms between manifolds. Tangent space at a point, the derivative of a smooth map and the Chain rule. Smooth curves. The tangent bundle of a manifold. The derivative of a smooth map, as a bundle map and in local coordinates.

Smooth vector fields, given as sections of the tangent bundle and as derivatives of the space of smooth functions. Partitions of unity (existence only examinable under assumption the manifold is compact). The Lie bracket of two vector fields, and the space of vector fields being a Lie algebra. Existence and uniqueness of integral curves (results from theory of ODE to be assumed). The flow defined by a vector field, and connection with one-parameter groups of diffeomorphisms. The Lie derivative.

Lie groups. Isomorphism between the lie algebra and the space of left invariant vector fields. Examples coming from matrix groups and their Lie algebra.

Submanifolds, examples arising from level sets of smooth maps, the inverse function theorem for open sets in Euclidean space to be assumed)

Smooth and involutive distributions. Frobenius Theorem

Smooth vector bundles, transition functions, smooth sections, frames and local trivialisations. Construction of vector bundles and examples of constructing new bundles from old (e.g. direct sum, tensor products, and dual bundles). Tensors, tensor bundles.

Exterior algebra, exterior bundles. Differential forms and exterior differentiation. Volume forms and orientability. De Rham cohomology, Poincare Lemma. The de-Rham's Theorem (statement only). Integration of top degree forms. Manifolds with boundary, induced orientation on boundary and Stoke's Theorem.

Connections on vector bundles, covariant derivatives. Curvature, and connection between curvature and covariant derivatives. The connection matrix and curvature matrix, and the Bianchi identity

Torsion. Existence of Levi-Cvita connection (proof non examinable). The curvature of a Riemannian metric and symmetries of its curvature. Definition of sectional curvature (non examinable: covariant derivative along a curve, geodesics, second fundamental form)