Differential Geometry

A rigorous graduate-level course on differential geometry—from smooth manifolds and tangent spaces through Riemannian geometry, fiber bundles, characteristic classes, and applications to modern theoretical physics.

Course Overview

Differential geometry provides the mathematical language of modern physics, from general relativity's curved spacetimes to gauge theories in particle physics. This course develops the complete framework of manifolds, connections, curvature, and topology needed to understand the geometric foundations of contemporary theoretical physics.

What You'll Learn

  • • Smooth manifolds, charts, and atlases
  • • Tangent spaces, vector fields, and differential forms
  • • Integration on manifolds and Stokes' theorem
  • • Riemannian metrics, connections, and curvature
  • • Geodesics and the Gauss-Bonnet theorem
  • • Fiber bundles and gauge connections
  • • Characteristic classes and Chern-Weil theory
  • • Homotopy, homology, cohomology, and index theorems

Prerequisites

  • • Linear algebra and multivariable calculus
  • • Real analysis and point-set topology
  • • Abstract algebra basics (groups, rings)
  • • Ordinary differential equations
  • • Classical mechanics (helpful)

References

  • • M. Nakahara, Geometry, Topology and Physics (2nd ed.)
  • • T. Frankel, The Geometry of Physics (3rd ed.)
  • • M. P. do Carmo, Riemannian Geometry
  • • J. Baez & J. P. Muniain, Gauge Fields, Knots and Gravity

Course Structure

🌐

Part I: Manifolds

Manifolds and charts, tangent spaces, differential forms, and integration on manifolds.

📐

Part II: Riemannian Geometry

Riemannian geometry, connections and curvature, geodesics, and the Gauss-Bonnet theorem.

🔗

Part III: Fiber Bundles

Fiber bundles, gauge connections, characteristic classes, and Chern-Weil theory.

🔮

Part IV: Topology & Physics

Topology in physics, homotopy groups, homology and cohomology, and index theorems.

Key Equations

Metric Tensor

$$ds^2 = g_{\mu\nu} \, dx^\mu \, dx^\nu$$

The line element defining distances on a Riemannian manifold

Christoffel Symbols

$$\Gamma^\lambda_{\mu\nu} = \frac{1}{2} g^{\lambda\sigma} \left( \partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu} \right)$$

Connection coefficients for the Levi-Civita connection

Riemann Curvature Tensor

$$R^\rho{}_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}$$

Measures the intrinsic curvature of a manifold

Gauss-Bonnet Theorem

$$\int_M K \, dA = 2\pi \chi(M)$$

Relates total Gaussian curvature to the Euler characteristic

Chern Number

$$c_1 = \frac{1}{2\pi} \int_M \text{Tr}(F)$$

First Chern number as integral of the curvature 2-form

Atiyah-Singer Index Theorem

$$\text{ind}(D) = \int_M \hat{A}(M) \wedge \text{ch}(E)$$

Relates analytical index of an elliptic operator to topological invariants