D. Kotschick: Geometry of Manifolds II

(Differentialgeometrie II, in englischer Sprache)

• Time and place: Tu 11-13 room B 047, Th 11-13 room 252
• Recitation classes: We 16-18, room B 252
  
• Lecture notes
• Contents: This is the second half of a full-year course on differentiable manifolds. We shall discuss mostly Riemannian geometry, including notions of curvature, geodesics, and relations between curvature and topology. Sample topics include: Lie groups and homogeneous spaces; spaces of constant curvature; the Bochner technique and some applications; structure theorems for manifolds with sectional or Ricci curvatures of a fixed sign (Myers, Synge and Cheeger-Gromoll theorems for positive curvature, Cartan-Hadamard, Cartan and Preissmann theorems for negative curvature).
• Intended audience: This course is obligatory for all master's degree students wishing to take more advanced courses and seminars in geometry during their second year. It is also suitable for those who do not want to specialize in this area, but want to be examined in geometry to cover the pure mathematics requirement for the master's degree.
  Diplom- und Lehramts-Studenten die eine Einführung in die Differentialgeometrie hören wollen, sollten diese Vorlesung besuchen. (Bei Bedarf werden sowohl deutsche als auch englische Übungsgruppen angeboten.) Für Lehramtstudenten eignet sich diese Vorlesung für das Prüfungsgebiet Geometrie im Staatsexamen.
• Prerequisites: We shall assume only a basic knowledge of differentiable manifolds. It is not necessary to have attended Geometry of manifolds I, which covered more than enough background material for this course.
• Main text: P. Pedersen: Riemannian Geometry. Springer Verlag 1998.
  Other recommended reading: R. L. Bishop and R. J. Crittenden: Geometry of Manifolds. 1964, reprinted 2001 by AMS Chelsea Publishing.
  F. Warner: Foundations of Differentiable Manifolds and Lie Groups. Springer Verlag 1983.
  S. Lang: Fundamentals of Differential Geometry. Springer Verlag 1999.
  L. Conlon: Differentiable Manifolds  --- A first course. Birkhäuser Verlag 1993. M. H. Freedman and F. Luo: Selected Applications of Geometry to Low-Dimensional Topology. Amer. Math. Soc.
  B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry --- Methods and Applications, Vol. II, Springer Verlag 1990.
  F. Warner: Foundations of Differentiable Manifolds and Lie Groups. Springer Verlag 1983.