Applications

9.1 General Relativity

In general relativity, the metric $g$ on a 4-dimensional manifold encodes gravity. Geodesics of the metric describe the trajectories of freely falling particles. The curvature of spacetime is determined by the distribution of matter and energy via the Einstein field equations.

Example (Schwarzschild Metric). The metric outside a spherically symmetric mass $M$ is:

$ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right)c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 d\Omega^2$

This metric describes non-rotating black holes and predicts phenomena such as gravitational redshift, perihelion precession, and gravitational lensing.

9.2 Gauge Theory

In Yang-Mills gauge theory, the connection (gauge field) on a principal $G$-bundle over spacetime plays a role analogous to the Levi-Civita connection in Riemannian geometry. The curvature of this connection is the field strength $F = dA + A \wedge A$.

The Chern-Weil theory relates the curvature of a connection to topological invariants (characteristic classes) of the bundle. For example, the first Chern class $c_1(E) \in H^2(M, \mathbb{R})$ is represented by the closed 2-form $\frac{i}{2\pi}\mathrm{tr}(F)$.