### Comparison Geometry (Summer Semester 2015)

- Lecturer: PD Dr. Fernando Galaz-Garcia
- Classes: Lecture (01540800), Problem class (0154810)
- Weekly hours: 2+1

**Tutor**: Mr. Jan-Bernhard Kordaß (jan-bernhard.kordass at student.kit.edu).

**News**

**No lecture on 18.5.2015.**

Schedule | ||
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Lecture: | Thursday 11:30-13:00 | 20.30 SR 2.67 |

Problem class: | Wednesday 14:00-15:30 | 20.30 SR 2.59 |

Lecturers | ||
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Lecturer, Problem classes | PD Dr. Fernando Galaz-Garcia | |

Office hours: By appointment. | ||

Room 1.016 Kollegiengebäude Mathematik (20.30) | ||

Email: galazgarcia@kit.edu |

**Description**

The course provides a thorough introduction to comparison theory in Riemannian geometry:

What can be said about a complete Riemannian manifold when (mainly lower) bounds for the sectional or Ricci curvature are given? Starting from the comparison theory for the Riccati ODE which describes the evolution of the principal curvatures of equidistant hypersurfaces, we discuss the global estimates for volume and length given by Bishop-Gromov and Toponogov. An application is Gromov’s estimate of the number of generators of the fundamental group and the Betti numbers when lower curvature bounds are given. Using convexity arguments, we prove the ”soul theorem” of Cheeger and Gromoll and the sphere theorem of Berger and Klingenberg for nonnegative curvature. If lower Ricci curvature bounds are given we exploit subharmonicity instead of convexity and show the rigidity theorems of Myers-Cheng and the splitting theorem of Cheeger and Gromoll. The Bishop-Gromov inequality shows polynomial growth of finitely generated subgroups of the fundamental group of a space with nonnegative Ricci curvature (Milnor). We also discuss briefly Bochner’s method.

**Homework sets**

- Homework set 1
- Homework set 2
- Homework set 3
- Homework set 4
- Homework set 5
- Homework set 6
- Homework set 7
- Homework set 8
- Homework set 9
- Homework set 10
- Homework set 11

# References

- do Carmo, Manfredo Perdigão. Riemannian geometry. Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992.
- J.-H. Eschenburg, "Comparison theorems in Riemannian Geometry" (figures here)