MATHEMATICS 551 P Fall 2002

Elementary Differential Geometry

Time and location: Tue., Thu. 10.50am-12.05pm, room CC411 

Instructor: Dr. Lars Andersson

Office: Ungar 547

Phone: 284-3742

Email: larsa@math.miami.edu

Office hours: By appointment

Text: Spivak, A comprehensive introduction to differential geometry, vol I, II.

Examination: homework

MTH551 Geometrical structures on differentiable manifolds play a central role in moderna mathematics and physics, from the mathematical formulation of classical mechanics, via Morse theory, Hodge theory, Riemannian geometry, to the theory of partial differential operators.

This course aims to give an introduction to the subject. Some points which will be discussed are

· Manifolds, differential topology, Sards theorem, the Whitney imbedding theorem

· Differential forms, tensors, integration on manifolds, mapping degree theorem

· The Poincare’ Lemma, Stokes theorem

· Riemannian geometry: curves and surfaces in the plane and in space, Riemannian metric, covariant derivative, ‘moving frames’, the curvature tensor, the Laplace operator

The goal of the course is a proof of the Gauss-Bonnet theorem, which gives a relation between the integral of the Gauss curvature over a surface, and the Euler charateristic or genus of the surface.

Modern differential geometry is based on ideas which arose out of the work of Gauss and Riemann. As course book we will use the first two volumes of the fascinating text of Riemann (in 5 volumes). The first gives a modern presentation of the basic machinery of differential geometry, while the second volume contains historical material presented from a modern point of view.

I will use material from both volume I and II, and complete this with lecture notes, available at the course homepage, www.math.miami.edu/~larsa/MTH551

The prerequisites of the course are a good knowledge of linear algebra and multivariate calculus, and some basic notions of topology. I will give a quick introduction to the necessary concepts.

Welcome,

Lars Andersson