Lecture Notes Werner Ballmann

Kurz und bündig

1. Notizen zur Vorlesung Lineare Algebra. Skript zum ersten Teil meiner Vorlesung für Informatiker. (2002)
2. Überlagerungen und Fundamentalgruppe. Ergänzendes Skript zu meinen Vorlesungen über Differentialgeometrie. (2003)
3. Einführung in die Geometrie und Topologie. Mathematik Kompakt. Birkhäuser, 2015. x+162 pp. Zweite Auflage: Birkhäuser, 2018. xii+163 pp. (Link)

Lectures on Differential Geometry

1. Introduction to Geometry and Topology. Translated by Walker Stern. Compact Textbooks in Mathematics. Birkhäuser, 2018. x+169 pp. (Link)
2. Connections and Geodesics. Connections on manifolds, geodesics, exponential map. (2002)
3. Vector Bundles and Connections. A short and elementary exposition of vector bundles and connections on vector bundles. (2002).
4. Semi-Riemannian Metrics. Semi-Riemannian metrics, Levi-Civita connection, curvature. (2003)
5. Riemannian Immersions and Submersions. Riemannian immersions and submersions, Gauss and Codazzi equation, O'Neill's formula, projective spaces, Hopf map, Fubini-Study metric. (2003)
6. Variational Theory of Geodesics. In these notes, I discuss first and second variation of length and energy and boundary conditions on path spaces. In the later version, I also discuss the theorem of Birkhoff / Lusternik-Fet and the Morse index theorem. (2003, 2015)
7. Global Riemannian Geometry. Riemannian distance, theorems of Hopf-Rinow, Bonnet-Myers, Hadamard-Cartan. (2003)
8. Riccati Equation and Volume Estimates. Distance functions and Riccati equation, comparison theory for scalar Riccati equations, Bishop-Gromov inequality and applications, Heintze-Karcher inequalities. (2016)
9. Lectures on Spaces of Nonpositive Curvature. I discuss the geometry of metric spaces, spaces of nonpositive curvature, rank one spaces, and rank rigidity. In an appendix, Misha Brin proves Anosov's theorem on the ergodicity of geodesic flows on closed manifolds of negative curvature. (Published by Birkhäuser as DMV Seminar 25, 1995) (1995)
10. Lectures on Kähler Manifolds. ESI Lectures in Mathematics and Physics. EMS Publishing House, 2006. x+172 pp. These notes contain basics on Kähler geometry, cohomology of closed Kähler manifolds, Yau's proof of the Calabi conjecture, Gromov's Kähler hyperbolic spaces, and the Kodaira embedding theorem. (Link)

Topics in Differential Geometry

1. On the Geometry of Metric Spaces. Length and geodesic spaces, length metrics on simplicial complexes, Theorem of Hopf-Rinow for geodesic spaces, upper curvature bounds in the sense of Alexandrov, barycenters, filling discs, cones, tangent cones, spherical joins, Tits buildings, short homotopies, Theorem of Hadamard-Cartan. (2004)
2. Automorphism Groups. I show that the automorphism groups of certain natural geometric structures are Lie groups with respect to the compact-open topology. As a particular application I get that the isometry group of a Riemannian or semi-Riemannian manifold is a Lie group with respect to the compact-open topology. (2000)
3. Geometric Structures. I discuss geometric structures with the main aim of proving a theorem of Singer on the local homogeneity of Riemannian manifolds and Gromov's Open Orbit Theorem. Recall that the latter theorem is crucial in the celebrated work of Benoist, Foulon and Labourie on the regularity of stable foliations of contact Anosov flows. This work was one of my main motivations for discussing geometric structures. (2000)
4. Homogeneous Structures. In these notes I discuss the theorem of Ambrose and Hicks on parallel translation of torsion and curvature and the Lie theoretic description of affine manifolds with parallel torsion and curvature of Nomizu. (2000)
5. Lectures on the Blaschke Conjecture. In a joint lecture with Karsten Grove, we discuss Wiedersehen manifolds, Zoll surfaces, Blaschke manifolds, and harmonic spaces. (2016)
6. Integrability of Hamiltonian Systems. I explain complete integrability of geodesic flows and other Hamiltonian systems after the method developed by Anton Thimm. (2015)
7. Symmetric Spaces. Unfinished notes on symmetric spaces. (1999)