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, de Rham cohomolgy.

The second part is "Riemannian geometry": Riemannian metric, parallel transport, connections, geodesics, curvature.

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.
• Hitchin's lecture notes on "Differentiable Manifolds", 2014.