Math 250C-- differential geometry
The geometry studies the metric properties. The metric tensor (a Minkowski metric with signature (+, -, -, -)) is used by Einstein to model the gravity. The magnitude of the curvature at each point characterizes the strength of the gravitational force at that point (cf. Modern Geometry by Dubrovin, Fomenko and Novikov, Vol 1, Section 39). Riemannian geometry studies the properties of positive definite metrics. The relevant curvature is the one related to the canonical Levi-Civita connection. The metric/geometric properties of the space are of fundamental importance to the physical events since every event takes place in the space. The study of the metric properties is also supported by quantum mechanics since the atomic laws fix an absolute length (cf. page 129 of Symmetry, Chapter 4, by H. Weyl), where he also proclaimed that (on page 132) `objectivity means invariance with respect to the group of automorphisms'.
The concept of a group is used to understand the symmetry. There are finite, discrete and continuous groups. They all arise in the study of geometry. The main purpose of this course is to illustrate some interactions between the curvature and groups. We shall prove some nontrivial theorems when introducing some basic materials.
For this quarter, all lectures and office hours shall be online. I plan to schedule a zoom meeting at the time (MWF 12-12:50) when the class is scheduled (not necessarily recorded). The meeting can be joined via the Canvas. Office hours shall be handled similarly.
The complete course schedule will be available and updated weekly.
There will be no final exam.
||AP&M 5250 (Via Zoom meeting)
Course Time and Location
||MWF 12:00-12:50 am
||APM 7421 (Zoom meeting via Canvas)
Text Recommended by the Instructor:
(i) Riemannian Geometry by M. P. do Carmo, Birkhauser;
(ii) Morse Theory by J. Milnor, Princeton Press;
(iii) Riemannian geometry by T. Sakai, AMS translation series;
(iv) Modern Geometry by Dubrovin, Fomenko and Novikov, Vol 1 and 2, Springer
There will be no exam.
Partial Reading Materials
A quick introduction to tensor algebra, smooth manifolds and vector bundles
A proof of Stokes theorem
My lecture notes on connection and curvature
Last modified: Wed March 11, 14:49:08 PST 2020