D. Kotschick: Riemannian Geometry

• Time and place: Mo 12-14 in HS B 006, Tu 12-14 in HS B 005
• Recitation classes: First class on Wednesday, April 16, 14-16, in B 004; second class on Friday, May 2, 8-10, in B 252. After that the class will always be on Fridays, same time and same place.
  
• Lecture notes
• Contents: This is a first course in Riemannian geometry, covering the following topics: geodesics, completeness, exponential map, Jacobi fields, isometries, spaces of constant curvature and other model spaces, relations between curvature and topology, for example the classical theorems of Bonnet-Myers and Cartan-Hadamard. If time permits, further topics will be covered.
• Topics not covered in this course: We shall not cover Hodge theory and the Bochner method in the lectures, since these form the subject of the parallel seminar. We will also not cover spectral geometry, which is discussed in a two-hour parallel course by Prof. Kokarev. This course is recommended as a supplement for students who are interested in studying more Riemannian geometry than we can cover in this course, and in particular its relations with functional analysis and with the spectral theory of differential operators.
• Intended audience: All students of mathematics and/or physics who are interested in geometry and have a working knowledge of the basic facts about smooth manifolds and vector bundles.
• Prerequisites: We shall assume only a basic knowledge of smooth manifolds. It is not necessary to have attended Differential Geometry/Differentiable Manifolds last semester. That course covered much more preliminary material than is needed to understand this course.
• Main references for the course:
  M. P. do Carmo: Riemannian Geometry, Birkhäuser Verlag 1992.
  P. Pedersen: Riemannian Geometry, Springer Verlag 1998.
  Other recommended reading:
  S. Gallot, D. Hulin and J. Lafontaine: Riemannian Geometry, Springer Verlag 1987, 1990.
  R. L. Bishop and R. J. Crittenden: Geometry of Manifolds, 1964, reprinted 2001 by AMS Chelsea Publishing.
  I. Chavel: Riemannian Geometry: A modern introduction, Cambridge University Press 1993.
  For the prerequisites:
  L. Conlon: Differentiable Manifolds  --- A first course, Birkhäuser Verlag 1993.
  F. Warner: Foundations of Differentiable Manifolds and Lie Groups, Springer Verlag 1983.
• Examination: Those who failed the final in July can take an oral makeup exam on October 1st.