Riemannian Geometry Spring 2019

Instructor: Danny Calegari

TuTh 12:30-1:50 Eckhart 206

Description of course:

This course is an introduction to Riemannian Geometry.

Cancellations:

None yet.

Homework/Midterm/Final

There will be a midterm and a final. There will also be weekly homework. Homework is posted to this website each Thursday and due at the start of class the following Thursday. Late homework will not be accepted.

• Homework 1, due April 11
• Homework 2, due April 18
• Homework 3, due April 25
• Homework 4, due May 2
• Midterm, due May 9
• Homework 5, due May 16
• Homework 6, due May 23
• Homework 7, due May 30

Syllabus:

This is an introduction to Riemannian geometry. Some topics will be covered very briefly. For more details see the notes.

• Riemannian metrics and Levi-Civita connection
• First variation formula; geodesics, Hopf-Rinow
• Curvature; Jacobi fields, Cartan-Hadamard
• Second variation formula; index theorems, Myers-Bonnet
• Lie groups and homogeneous spaces
• Hodge theory

Notes

Introductory notes on Differential Topology and Riemannian Geometry can be found here.

References:

The following references are just suggestions. There are no required texts for the course.

• A. Besse Einstein manifolds
• J. Cheeger and D. Ebin Comparison theorems in Riemannian Geometry
• T. Colding and W. Minicozzi A course in minimal surfaces
• S. Kobayashi and K. Nomizu Foundations of Differential Geometry
• J. Milnor Morse Theory
• F. Warner Foundations of Differentiable Manifolds and Lie Groups