• Manifolds
  • Definitions; elementary facts about Lie groups;
  • Tensor algebra, Grassman algebra, exterior differentiation.

• A guiding example: Riemannian geometry of embedded surfaces in R³
  • Equivalent definitions of curvature;
  • Simple implications of the sign of the curvature;

• Affine connections
  • Covariant derivatives, parallelism;
  • Geodesics;

• Riemannian metrics
  • Sectional, Ricci and Scalar curvature;
  • Volume form: integration on Riemannian manifolds.
  • Examples: left-invariant metrics on Lie groups.

• Complete Riemannian manifolds
  • Hopf-Rinow theorem.

• Riemannian manifolds of negative curvature
• Totally geodesic submanifolds
• Riemannian functionals
  • The total scalar curvature: Einstein metrics.
  • Conformal deformation of metrics: surfaces with negative curvature.

Homework: classes will be filled with exercises for you to work on. I encourage you to work these out and submit written solutions with regularity. On occasions, we will discuss some of these problems in class.

Grading: your grade will be based upon your work on the assignments and class participation.

Class notes: You may find some brief class notes in here. Use these as an indication of where we are in the course. They are not intended to be substitutes for the relevant literature on the subject.

If you have a physical, psychological, medical or learning disability that my impact on your ability to carry out assigned course work, please contact the staff in the Disabled Student Service Office, Room 133, Humanities, 632-6784/TDD. DSS will review your concerns and determine with you what accommodations are necessary and appropriate. All information and documentation of disability is confidential.

The State University of New York at Stony Brook is an affirmative action/equal employment opportunity educator and employer.