Riemannian Geometry

5.1 Riemannian Metrics

A Riemannian metric on a smooth manifold $M$ is a smooth family of inner products $g_p : T_p M \times T_p M \to \mathbb{R}$, varying smoothly with $p$. In local coordinates:

$g = g_{ij}(x)\, dx^i \otimes dx^j$

where $g_{ij} = g\left(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}\right)$ forms a symmetric positive-definite matrix.

Example 1. The Euclidean metric on $\mathbb{R}^n$: $g = \sum dx^i \otimes dx^i$, so $g_{ij} = \delta_{ij}$.

Example 2. The standard metric on $S^2 \subseteq \mathbb{R}^3$: induced by the embedding. In spherical coordinates $(\theta, \phi)$: $g = d\theta^2 + \cos^2\theta\, d\phi^2$.

5.2 The Levi-Civita Connection

A connection on $M$ is a map $\nabla : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M)$ satisfying linearity and the Leibniz rule. A connection is Riemannian if it is compatible with the metric ($\nabla g = 0$) and torsion-free ($\nabla_X Y - \nabla_Y X = [X, Y]$).

Theorem 5.1 (Levi-Civita). On any Riemannian manifold, there exists a unique Riemannian (metric-compatible, torsion-free) connection, called the Levi-Civita connection.

5.3 Christoffel Symbols

In local coordinates, the Levi-Civita connection is determined by the Christoffel symbols $\Gamma^k_{ij}$:

$\nabla_{\frac{\partial}{\partial x^i}} \frac{\partial}{\partial x^j} = \Gamma^k_{ij} \frac{\partial}{\partial x^k}$

These are given by:

$\Gamma^k_{ij} = \frac{1}{2} g^{k\ell}\left(\frac{\partial g_{j\ell}}{\partial x^i} + \frac{\partial g_{i\ell}}{\partial x^j} - \frac{\partial g_{ij}}{\partial x^\ell}\right)$

where $(g^{k\ell})$ is the inverse matrix of $(g_{k\ell})$.