- Analysis I, II and IV
- Linear Algebra I and II

The main topic are Riemannian metrics on manifolds. The simplest examples are surfaces in Euclidean space ℝ

The goal is to understand not only surfaces, but similar curvature quantities in arbitrary dimensions and codimensions, a generalization going back to work of Bernhard Riemann. Very similar structures were later used by Einstein and others in order to get a mathematical framework to describe general relativity. The theory is still a very active area in mathematics and theoretical physics. The lecture will be continued in the summer term.

- C. Bär, Differential Geometry (Unpublished Lecture Notes), Summer Term 2023, (German version from a previous lecture; Old version from Summer Term 2013), (German version from a previous lecture)
- B. Ammann, Lineare Algebra I, WS 2007/08
- B. Ammann, Analysis I+II, 2018/19
- B. Ammann, Analysis III, WS 2019/20
- B. Ammann, Analysis IV, SS 2020
- B. Ammann, Kurzskript zu Tensoren (German language)
- N. Ginoux (former member of my group), Kurzskript zur Topologie (German language)

- M. do Carmo, Riemannian Geometry, Birkhäuser
- Cheeger, Ebin, Comparison theorems in Riemannian Geometry
- B. O'Neill, Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press
- 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

- Zero is not a natural number in this lecture
- The empty set is connected and path-connected
- A topological space is quasi-compact, if and only if each open covering admits a finite subcovering. Compact spaces are defined as quasi-compact Hausdorff spaces.
- We use the term "paracompact" as explained on this Wikipedia page. However, it will be rarely used. A topological manifold is defined as a locally euclidean Hausdorff space, whose connected components have a countable basis of topology. There is no restriction on the number of connected components, we also allow uncountably many. A smooth manifold is a topological manifold with a smooth atlas. This is equivalent to having a paracompact locally euclidean Hausdorff space with a smooth atlas.

- Thursday 14-16, M009
- Friday 12-14, M102

- Link to the G.R.I.P.S. page
- This lecture in the KVV (= Commented list of courses)
- This lecture in the Campusportal of the University of Regensburg
- This lecture in the Campusportal of the University of Regensburg
- The exercises for this lecture in the Campusportal of the University of Regensburg

51107: Übungen

Bernd Ammann, 27.09.2023

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