### Course info

**Text:** *Riemannian Geometry*, Manfredo do Carmo (beware of the title! It is NOT "Differential geometry", by the same author)

**Syllabus and course policy** here.

**Take home exam: here**

### HOMEWORKS

Homework 1 (due Feb 2, by noon) - Solutions

Homework 2 (due Feb 9, by noon) - Solutions

Homework 3 (due Feb 16, by noon) - Solutions

Homework 4 (due Feb 23, by noon) - Solutions

Homework 5 (due Mar 2, by noon) - Solutions

Homework 6 (due Mar 9, by noon) - Solutions

Homework 7 (due Mar 23, by noon) - Solutions

Homework 8 (due Apr 6, by noon) - Solutions

Homework 9 (due Apr 20, by noon)

Homework 10 (due Apr 28) **Last homework!**

### Lecture Notes

Class 01 - Review of manifolds and smooth maps (01/17/18)

Class 02 - Review of tangent spaces and orientability (01/19/18)

Class 03 - Vector bundles I; tangent and cotangent bundles (01/22/18)

Class 04 - Tensor bundles and sections (01/24/18)

Class 05 - Important sections; Riemannian metrics (01/29/18)

Class 06 - Riemannian manifolds (01/31/18)

Class 07 - Connections on Vector bundles (02/05/18)

Class 08 - Riemannian connections (02/07/18)

Class 09 - Geodesics (02/12/18)

Class 10 - The exponential map (02/14/18)

Class 11 - Geodesics locally minimize length, I (02/19/18)

Class 12 - Geodesics locally minimize length, II (02/21/18)

Class 13 - Riemannian manifolds as metric spaces, completeness and Hopf-Rinow Theorem (02/26/18)

Class 14 - Riemann curvature operator (02/28/18)

Class 15 - Curvature (03/05/18)

Class 16 - Jacobi fields, I (03/07/18)

Class 17 - Jacobi fields, II (03/21/18)

Class 18 - Cartan-Hadamard Theorem (03/23/18)

Class 19 - Conjugate points (03/26/18)

Class 20 - First Variation of energy (03/28/18)

Class 21 - Second variation of energy, and Bonnet-Myers Theorem (04/09/18)

Class 22 - More applications of variations: Synge-Weinstein Theorem (04/11/18)

Class 23 - Isometric immersions (04/13/18)

Class 24 - The second fundamental form. Totally geodesic and minimal submanifolds (04/16/18)

Class 25 - The fundamental equations of isometric immersions (04/18/18)

Class 26 - Distance between submanifolds, and Fraenkel's Theorem (04/23/18)

Class 27 - Riemannian submersions (04/25/18)

Class 28 - Curvature of Riemannian submersions (04/30/18)

Class 29 - The End (05/02/18)