Differential Geometry II/Differentialgeometrie II

Summer term 2024

Prof. Bernd Ammann, Office M 119

Current Announcements

Language/Sprache

Recommended previous knowledge

• Analysis I, II and IV
• Linear Algebra I and II
• Differential geometry I (or comparable)

Content of the Lecture

The lecture builds on the lecture Differential Geometry I, where we have started to treat the first such relations. The main focus of the lecture is to understand

• Lie groups: these are smooth manifolds with a group structure, such as the orthogonal group O(n), the unitary group U(n) and many more. They are important to decribe symmetries and to construct new examples of Riemannian manifolds
• Some examples of Lorentzian manifolds that are important in general relativity: flat spacetime (=Minkowski space); Schwarzschild spacetime to describe black holes; cosmological aspects
• A deeper understanding of the consequences of positive or negative curvature, for various notions of curvature (sectional, Ricci, scalar). A list of examples in the focus of the lecture, including some relations partially treated at the end of Differential Geometry I are:
  • 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

Literature

• 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

Exercise group

The first exercise group will take place on Thursday, April 18th, and several notions of basic algebraic toplogy such as, fundamental groups, coverings (Überlagerungen), universal coverings, normal coverings, connections between subgroups of the fundamental groups and coverings, deck transformations, etc. will be recalled and discussed.

Exercise Sheets

• Exercise Sheet no. 1,
• Exercise Sheet no. 2,

Links to related web pages

• Link to the G.R.I.P.S. page
• This lecture in the KVV (= Commented list of courses)

Examination/Prüfung

Formal criteria/Kriterien für Leistungsnachweise