Cohomogeneity one Einstein manifolds

Topics:

1. Riemannian homogeneous spaces: Introduction, examples and geometric properties.
2. Cohomogeneity one manifolds (generalized surfaces of revolution): Introduction, examples and geometric properties.
3. The Einstein ODE: Riccati equation, Gauß equation and second Bianchi identity.
4. Structure results: Qualitative methods applied to the Einstein ODE.
5. Examples.

Relevance

Pre-requisites

Knowledge concerning the following topics would be helpful:

• Basic Riemannian geometry (Riemannian metric, Levi-Civita connection, Riemannian curvature tensor –> do Carmo, “Riemannian geometry”)
• Homogeneous spaces (Cheeger-Ebin, “Comparison Theorems in Riemannian Geometry”, important for section 3 of this unit)
• Ordinary differential equations (Perko, “Differential equations and dynamical systems)

Pre-Reading

• M. Do Carmo, Riemannian Geometry 
  https://www.scribd.com/document/123292244/Manfredo-Do-Carmo-Riemannian-Geometry
• J. Cheeger, D. Ebin, Comparison Theorems in Riemannian Geometry
• L. Perko, Differential equations and dynamical systems
  https://www.scribd.com/doc/236334149/Differential-Equations-and-Dynamical-Systems-Lawrence-Perko