Homepage of M392C, 57755: Introduction to Riemannian Geometry; Fall 2003

Motto

"Out of nothing I created a strange new world." (János Bolyai after discovering hyperbolic geometry)

Course

  • See Survey papers in Riemannian Geometry, written by students of this class.
  • Syllabus online, printable version here.
  • Instructor: Tamás Hausel, Office Hours: TTh 2-3pm, Office: RLM 11.168, Phone: 471-7169, Email: hausel@math.utexas.edu
  • Time: TTh 11am-12.30pm, Room: RLM 11.176

    Description of course

    This course gives an introduction to Riemannian geometry. I will concentrate on various curvatures of the Riemannian manifold, how they arise, what properties they have. Emphasis will be put on examples, partly arising in physics, like Einstein's equations and black holes and in low dimensions like curves in the plane and space, and surfaces in the space. The course then will lead to study of topological obstructions of existence of Riemannian manifolds with certain properties, like constant or nonpositive sectional curvature, positive Ricci curvature and such. If time permits we will close with studying holonomy of Riemannian manifolds, and its relation to curvature, with some examples arising in string theory recently.

    Prerequisites

    The basic prerequisite is the prelim course in topology, in other words the notion of smooth manifolds, vector fields and differential forms on such. However the course will be self-contained in that in the first few lectures will consist of an overview of these notions. So a motivated student can overcome the missing prerequisite with some extra efforts in the beginning of the course. Also a look at the first Chapter of the main book (see below) can give an idea about this basic material. If you are not sure if this course is for you contact me for more details!

    Text books

    All textbooks below are available at the University Co-op.

    We will roughly follow

  • Gallot, S.; Hulin, D.; Lafontaine, J Riemannian geometry. Universitext. Springer-Verlag, Berlin, 1987. xii+248 pp.

    some parts of the course maybe taken from

  • Spivak, Michael A comprehensive introduction to differential geometry. Vol. II Second edition. Publish or Perish, Inc., Wilmington, Del., 1979.

    and towards the end we may study parts of

  • Joyce, D. Compact manifolds with special holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000.

    Links

  • Links to the biographies of Levi-Civita and Christoffel.
  • For quick introduction to some basic notions in point-set topology, differentiable manifolds, Lie groups and the like check out the handouts of the Warmup Programme of the University of Chicago.
  • Links to the biographies of Euclid, Bolyai, Gauss , Lobachevsky, Riemann, Einstein and Superstring theory.
  • Check out these two pages, where survey papers in Riemannian Geometry by graduate students are posted.

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