Mathematics 268: Topics in Differential Geometry (Symmetric Spaces)

Mathematics 268: Topics in Differential Geometry (Symmetric Spaces) (Spring 2002)

Instructor

Description

Depending on the background of the students, we will begin with a review of the basic theory of Lie groups and complex semisimple Lie algebras (chapters 2 and 3 of Helgason's book). We will then cover the material in chapters 4 - 10 of Helgason's book, though we will not always follow Helgason directly. Specifically, after establishing the relation between symmetric spaces (defined differential geometrically) and certain real Lie algebras with involution, we show that any symmetric space may be decomposed into irreducible symmetric spaces. There are three types of irreducible symmetric spaces: Euclidean, noncompact, and compact. We study the latter two types in detail and the duality between them. If there is time we will consider the special case of Hermitian symmetric spaces (and their relation with bounded symmetric domains) as well as the classification of symmetric spaces (which is intimately tied with the classification of semisimple real Lie algebras).

Symmetric spaces (and locally symmetric spaces) play crucial roles in Algebraic Geometry, Differential Geometry, Mathematical Physics, Number Theory, and Representation Theory. They arise as moduli spaces (parameter spaces) for variations of geometric and arithmetic objects.

Prerequisites

Mathematics 267 (Differential Geometry)
Mathematics 253 (Lie Algebras and Representation Theory, formerly 254)

Specifically, students should have had a graduate differential geometry course (say on the level of do Carmo's Riemannian Geometry) and be familiar with the basic theory of Lie groups. They also should be familiar with the structure theory of semisimple Lie algebras over the complex numbers (Cartan subalgebras, roots, Dynkin diagrams), however we will cover the more subtle situation over the real numbers in the class.

Course times

Tuesdays and Thursdays, 10:55 AM - 12:10 PM.

Text(s)

Differential Geometry, Lie Groups, and Symmetric Spaces, by Sigurdur Helgason (American Mathematical Society, 2001)

(Note that American Mathematical Society members may purchase the book from the AMS directly at the member price which may be less than charged at the bookstore.)

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Last modified: 17 October 2001