Announcements: Homework 4 is now available. Also, a correction has been made to Homework 3, problem 3.
Homework:
Instructor: Ben Weinkove.
Email address: weinkove@math.
Lectures: In AP&M 5402, MWF 9-9.50am.
Office hours: Wednesdays 11am-12pm in AP&M 6240.
Course Description: This is a year-long graduate course on differential geometry. This course is aimed primarily at graduate students in mathematics who are interested in studying geometry or related fields. However, the course may also be of interest to advanced undergraduates in mathematics or students in physics.
Differential geometry is a cornerstone of the modern study of geometry. It takes the ideas of calculus and extends them to geometric spaces. These objects of study are called differentiable manifolds and may have many dimensions and be endowed with a number of different structures. A basic knowledge of differential geometry is essential to graduate students interested in studying any of the many fields of geometry, including geometric analysis, topology, algebraic geometry, symplectic geometry and Lie groups.
Differential geometry is also the language of modern physics. One of the best known examples is Einstein's general relativity, where the three dimensions of space together with time are regarded as a four-dimensional differentiable manifold. The presence of matter gives rise to curvature of this manifold and we experience this as gravity.
The Fall 2009 quarter (250A) will be an introduction to differentiable manifolds. The Winter 2010 quarter (250B) will be a course on Riemannian geometry and the Spring 2010 quarter (250C) will be a course on complex geometry. 250B and 250C will be largely independent of each other, but will both assume knowledge of material from 250A (or a similar course). For more details, see below.
250A - Fall 2009. This class is an introduction to differentiable manifolds and should be accessible to any first year graduate student in mathematics or advanced undergraduate who has completed rigorous courses in analytic topology (topological spaces, continuous maps of topological spaces, the Hausdorff property, compactness etc) and calculus of several variables. We will use the text:
An Introduction to Differentiable Manifolds and Riemannian Geometry, Second Edition, by William M. Boothby.
We will aim to cover: definition of a smooth manifold, examples, maps between smooth manifolds, submanifolds, tangent spaces, vector fields, tensors, differential forms, the Lie derivative, integration, Stokes' theorem for manifolds, and other topics as time allows (e.g. Lie Groups, actions on manifolds, covering spaces.)
250B - Winter 2010. This is a course on Riemannian geometry. A knowledge of 250A or a similar course is assumed. We will use the text:
Riemannian Geometry - A Modern Introduction, by Isaac Chavel, Second Edition.
We will aim to cover: Riemannian metrics, the Levi-Civita connection, curvature, first and second variation of arc length, geodesics, the Jacobi equation, Bonnet-Myers theorem, comparison theorems and other topics as time allows.
250C - Spring 2010. This is a course on complex geometry. A knowledge of 250A or a similar course is assumed. We will use the text:
TBA
We will aim to cover: almost complex structures, holomorphic coordinates, examples of complex manifolds, Kahler metrics, curvature, holomorphic line bundles, the Kodaira embedding theorem and other topics as time allows.