Riemannian Geometry

Math 250B. Riemannian Geometry - Winter 05 - Hans Lindblad

In the early 1800s Gauss asked how much of the geometry of a surface is independent of how it bends in space. Riemannian geometry is designed to describe the universe of creatures who live on a curved surface and who are unaware of space outside and can only measure distances and areas on the surface. This lead to the modern notion of a manifold independent of a surrounding space. 250B is about manifolds with a notion of a distance, how this can be used to define curvatures and how the curvatures can characterize a manifold. Einstein realized that this theory could be used to describe how space curves under the influence of gravity which lead to the general theory of relativity.

The core topics are:

Curvature of curves, surfaces and submanifolds of Euclidean space.(week 1-2)
Riemannian metric on a manifold. Riemannian connection. Geodesics. (week 3-4)
Curvature, Sectional curvature. Bianchi identities. Jacobi fields.(week 5-6)
Submanifolds, immersions. The second fundamental form. Codazzi equation.(week 7)

Additional topics that we might cover:

Moving frames and the structural equations.(week 8)
General relativity.(week 9)

I plan to cover basic Riemannian geometry from ch 1-6 in Do Carmo (or ch 3-5 in O'Neill or ch 2-3(5) in Gallot et al) and some Semi-Riemannian geometry and general relativity from O'Neill:

Do Carmo Riemannian Geometry Birkhauser (Springer) 1992
O'Neill Semi-Riemannian Geometry with applications to relativity Academic Press 1983
Gallot, Hulin & Lafontaine Riemannian Geometry Springer 2004

We will however start by reviewing curvature of curves and surfaces in three dimensional space, in particular Gauss Theorema Egregium, see references for surfaces. 250B is essentially independent of 250A, apart from that it will be assumed that you know the introductory chapter about manifold theory in the textbook (Ch 0 in Do Carmo) I am not sure how much relativity we will have time to cover but I hope to also teach a course on general relativity next Fall and 250B will be good preparation for it. (More references: General Relativity, Riemannian Geometry and Lecture notes on the web.)

I hope to make my lecture notes available. There will be no exams but some homework problems. The lectures are MWF 3-4 in APM6218. Office hour/problem session W 4-5 in 5829.

    Lecture Notes: L 1, L 2, L 3, L 4, L 5, L 6, L 7,
    L 8, L 9, L 10, L 11, L 12, L 13, L 14, L 15, L 16,
    L 17, L 18, L 19, L 20, L 21,
    Problems: P 1,   P2.