- 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- C. Bär, Differential Geometry (Unpublished Lecture Notes), Summer Term 2023, (German version from a previous lecture; Old version from Summer Term 2013),
- 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.
- A topological manifold is defined as a locally Euclidean Hausdorff space with a countable basis of topology.

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

- Presence Exercise Sheet,
- Exercise Sheet no. 1,
- Exercise Sheet no. 2,
- Exercise Sheet no. 3,
- Exercise Sheet no. 4,
- Exercise Sheet no. 5,
- Exercise Sheet no. 6,
- Exercise Sheet no. 7,
- Exercise Sheet no. 8,
- Exercise Sheet no. 9,
- Exercise Sheet no. 10,
- Exercise Sheet no. 11,
- Exercise Sheet no. 12,
- Exercise Sheet no. 13,
- Exercise Sheet no. 14,

- 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

we will offer several alternatives and you also ma suggest other dates.

The examination can take place during the week Feb 5th to Feb 9th (last week of the teaching period), or after February 26th.

51107: Übungen

Bernd Ammann, 31.01.2024

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