Course Outline for Math 240A (Fall 2014)

Differential Geometry

MWF 10 am, Girvetz 1119

Instructor: John Douglas Moore

The first quarter of the differential geometry course will be devoted to global calculus of several variables, also known as calculus on manifolds.

The simplest examples of manifolds are surfaces in Euclidean space, and in fact manifolds can be thought of as n-dimensional generalizations of such surfaces. Manifolds are necessary for the formulation of global problems in geometry, analysis and physics. They form the foundation for three great theories of modern mathematics: differential equations, differential topology and differential geometry.

Topics to be treated in 240A include tangent vectors, vector bundles, submanifolds, tensors, differential forms and the manifold version of Stokes’ Theorem. This will lead to de Rham’s beautiful cohomology theory, which measures the number of holes of various dimensions within a manifold.

The second and third quarters of the continuing course (240BC) will treat Riemannian geometry, Lie groups and the theory of connections.

We plan to provide brief lecture notes for the course, which will include homework problems.

The recommended text for the course is: John M. Lee, Introduction to Smooth Manifolds, Springer, second edition, 2012. Since I will not follow the text very closely it is not necessary to buy the text. This book is available from www.amazon.com for $82.40 at the time this is written. (The first edition is also acceptable.)