Riemannian Geometry IV

Epiphany 2020

The Michaelmas 2019 webpage

Textbooks:

• S. Gudmundsson, An Introduction to Riemannian Geometry, Lecture Notes, Lund University (2014).
• J.Lee, Riemannian Manifolds, An Introduction to curvature, Springer (1997)
• M.P. Do Carmo, Riemannian Geometry, Birkhäuser (1992)

The content of the course can also be found in any standard textbook on Riemannian Geometry, e.g.

• F. Morgan, Riemannian Geometry.
• T. Sakai, Riemannian Geometry.
• S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry.
• P. Petersen, Riemannian Geometry.

Further (recommended) reading:

• J. Gallier, Notes On Group Actions, Manifolds, Lie Groups and Lie Algebra (2005)
• J. Gallier, Differential Geometry and Lie Groups (2019)
• M. Gromov, Sign and geometric meaning of curvature (1990).

Preliminary course content (subject to change): Hopf -- Rinow theorem; introduction to Lie groups; Riemannian curvature tensor, Ricci curvature, sectional curvature and scalar curvature; manifolds of positive curvature and Bonnet-Myers theorem; Jacobi fields and conjugate points, manifolds of nonpositive curvature and Cartan-Hadamard theorem; comparison theorems.

Schedule: