The lecture builds on the lecture Differential Geometry I, where we have started to treat the first such relations. The lecture shall lead to a good understand about the consequences of positive or negative curvature, for various notions of curvature (sectional, Ricci, scalar). We list some examples of relations in the focus of the lecture, including some relations partially treated at the end of Differential Geometry I. We will probably not have the time to cover all of them.

• The theorem of Cartan Hadamard: the universal covering of a complete manifold with non-positive sectional curvature is diffeomorphic to ℝⁿ
• Bonnet-Myers: the fundamental group of a closed manifold with positive ricci curvature is finite
• Cheeger splitting theorem: a complete manifold with non-negative Ricci-curvature, containing a line, splits as a Riemannian product with the line
• Existence of closed geodesics
• Synge's obstruction to positive sectional curvature
• Structure theorems for ricci-flat manifolds
• Exponential growth of the fundamental group for closed manifolds with negative sectional curvature (Theorem by Milnor)
• Polynomial growth of the fundamental group for closed manifolds with non-negative Ricci curvature (another Theorem by Milnor)
• Special holonomy

• M. do Carmo, Riemannian Geometry, Birkhäuser
• Cheeger, Ebin, Comparison theorems in Riemannian Geometry
• F. Warner, Foundations of differentiable manifolds and Lie groups, Springer
• T. Sakai, Riemannian Geometry, Transl. Math. Monogr., AMS
• W. Kühnel, Differentialgeometrie, Vieweg
• J. Lee, Introduction to topological manifolds, Springer
• J. Lee, Introduction to smooth manifolds, Springer
• J. Lee, Riemannian manifolds, Springer

• Analysis I, II and IV
• Linear Algebra I and II
• Differential geometry I

• Registration for the exercise classes: An exercise group is currently planned for Thursday
  10-12 (the time and day of week might change!). The registration will be in the first lecture.
• Please register on the GRIPS system. We will send out additional information and emails via
  GRIPS
• Registration for course work/examination/ECTS: FlexNow

• Successful participation in the exercise classes:

• Oral exam: Duration: 30 minutes , Date: individual arrangements, re-exam: Date: