Riemannian Geometry

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 $(\theta, \phi)$: $g = d\theta^2 + \cos^2\theta\, d\phi^2$.

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.

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 $(g_{k\ell})$.

A geodesic is a curve $\gamma : I \to M$ with zero acceleration:

$\nabla_{\dot\gamma} \dot\gamma = 0$

In local coordinates, this gives the geodesic equation:

$\ddot\gamma^k + \Gamma^k_{ij} \dot\gamma^i \dot\gamma^j = 0$

Proposition 5.2. For any $p \in M$ and $v \in T_p M$, there exists a unique geodesic $\gamma_v : I_v \to M$ with $\gamma_v(0) = p$ and $\dot\gamma_v(0) = v$.

Example 3. On $\mathbb{R}^n$ with the Euclidean metric, $\Gamma^k_{ij} = 0$, so geodesics are straight lines: $\gamma(t) = p + tv$.

Example 4. On $S^2$ with the round metric, geodesics are great circles (arcs of circles centered at the sphere’s center). In spherical coordinates $(\theta, \phi)$ with metric $g = d\theta^2 + \cos^2\theta\, d\phi^2$, the non-zero Christoffel symbols are $\Gamma^\theta_{\phi\phi} = \tan\theta$ and $\Gamma^\phi_{\theta\phi} = \Gamma^\phi_{\phi\theta} = \sec^2\theta$.

The Riemann curvature tensor $R : \mathfrak{X}(M) \times \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M)$ is defined by:

$R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$

Proposition 5.3 (Symmetries). For any vector fields $X, Y, Z, W$:

1. $R(X, Y)Z = -R(Y, X)Z$ (anti-symmetry in first two arguments)
2. $\langle R(X, Y)Z, W\rangle = -\langle R(X, Y)W, Z\rangle$ (anti-symmetry in last two arguments)
3. $R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0$ (first Bianchi identity)
4. $\langle R(X, Y)Z, W\rangle = \langle R(Z, W)X, Y\rangle$ (pair symmetry)
5. $\nabla_X R(Y, Z)W + \nabla_Y R(Z, X)W + \nabla_Z R(X, Y)W = 0$ (second Bianchi identity)

In local coordinates, the components are:

$R^\ell_{ijk} = \partial_i \Gamma^\ell_{jk} - \partial_j \Gamma^\ell_{ik} + \Gamma^m_{jk} \Gamma^\ell_{im} - \Gamma^m_{ik} \Gamma^\ell_{jm}$

Let $\Pi \subseteq T_p M$ be a 2-dimensional subspace spanned by $v, w \in T_p M$. The sectional curvature is:

$K(\Pi) = \frac{\langle R(v, w)w, v\rangle}{|v|^2|w|^2 - \langle v, w\rangle^2}$

Proposition 5.4. Sectional curvature determines the full Riemann curvature tensor.

The Ricci curvature is the trace of the Riemann tensor:

$\mathrm{Ric}(X, Y) = \sum_{i=1}^n \langle R(X, e_i)Y, e_i\rangle$

where $\{e_i\}$ is an orthonormal basis. In components: $R_{ij} = R^k_{ikj}$.

The scalar curvature is the trace of the Ricci tensor: $S = \sum_i \mathrm{Ric}(e_i, e_i) = g^{ij} R_{ij}$.

Example 5. For a sphere $S^n$ with radius $r$: sectional curvature $K = 1/r^2$, Ricci curvature $\mathrm{Ric} = (n-1)/r^2 \cdot g$, scalar curvature $S = n(n-1)/r^2$.

Example 6. For hyperbolic space $\mathbb{H}^n$: $K = -1$, $\mathrm{Ric} = -(n-1)g$, $S = -n(n-1)$.

Problem 1. Compute the Christoffel symbols for the Poincaré half-plane $\mathbb{H}^2 = \{(x, y) \in \mathbb{R}^2 : y > 0\}$ with metric $g = (dx^2 + dy^2)/y^2$.

Solution. The metric components are $g_{xx} = g_{yy} = 1/y^2$, $g_{xy} = 0$. The inverse metric is $g^{xx} = g^{yy} = y^2$, $g^{xy} = 0$. Using $\Gamma^k_{ij} = \frac{1}{2}g^{k\ell}(\partial_i g_{j\ell} + \partial_j g_{i\ell} - \partial_\ell g_{ij})$:

$\Gamma^x_{xy} = \Gamma^x_{yx} = -\frac{1}{y}, \quad \Gamma^y_{xx} = \frac{1}{y}, \quad \Gamma^y_{yy} = -\frac{1}{y}$

All other Christoffel symbols vanish. The geodesic equation gives: curves that are semicircles centered on the $x$-axis or vertical lines. $\blacksquare$

Problem 2. Show that $S^2$ with the round metric has constant sectional curvature $K = 1$.

Solution. The round metric $g = d\theta^2 + \sin^2\theta\, d\phi^2$ has non-zero Christoffel symbols $\Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta$, $\Gamma^\phi_{\theta\phi} = \Gamma^\phi_{\phi\theta} = \cot\theta$. Computing $R^\theta_{\phi\theta\phi}$ gives $-\sin^2\theta$, so $K = R^\theta_{\phi\theta\phi} / g_{\theta\theta}g_{\phi\phi} = 1$. $\blacksquare$