Course Identification

Familiarity with multivariate calculus (say, the divergence threorem) and point-set topology (say, a Hausdorff topological space).

This course consists of two parts.

The first part is "Analysis on Manifolds", starting from the definition a differentiable manifold, vectors fields, differential forms, integration, Stokes theorem.

The second part is "Curvature" and it focuses on Riemannian manifolds and submanifolds of Euclidean space: Riemannian metric, parallel transport, connections, geodesics, curvature, isoperimetric inequalities.

Upon successful completion of this course students will be able to:

• Demonstrate intuition and knowledge of the basic theorems in differential geometry.
• Apply the language and tools of differential geometry in other areas of mathematics and science.
• Work with differential forms.
• Understand the intuition behind curvature (at least in two dimensions) and its effects.
• Appreciate the interplay between analysis and geometry.

• Lee, J. M, Introduction to Smooth Manifolds, 2003.
• DoCarmo, Riemannian Geometry, 1992.
• A short appendix by Gromov, "Isoperimetric Inequalities in Riemannian Manifolds" in the book by Milman and Schechtman, "Asymptotic Theory of Finite Dimensional Normed Spaces", 2001.