Curvature of Riemannian Manifolds

Curvature measures how a Riemannian manifold deviates from flat Euclidean space. It comes in several flavors -- sectional, Ricci, and scalar -- each capturing different aspects of the geometry.

The Riemann Curvature Tensor

Definition6.6Riemann curvature tensor

The Riemann curvature tensor of a Riemannian manifold $(M, g)$ with Levi-Civita connection $\nabla$ is the $(1,3)$ -tensor

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

for vector fields $X, Y, Z$ . In components: $R^l{}_{ijk} = \partial_i \Gamma^l_{jk} - \partial_j \Gamma^l_{ik} + \Gamma^l_{im}\Gamma^m_{jk} - \Gamma^l_{jm}\Gamma^m_{ik}$ . The curvature measures the failure of parallel transport to be path-independent.

RemarkSymmetries of the curvature tensor

The Riemann tensor satisfies:

$R_{ijkl} = -R_{jikl}$ (skew-symmetry in first pair).
$R_{ijkl} = -R_{ijlk}$ (skew-symmetry in second pair).
$R_{ijkl} = R_{klij}$ (pair symmetry).
$R_{ijkl} + R_{iklj} + R_{iljk} = 0$ (first Bianchi identity).

These reduce the independent components of $R$ from $n^4$ to $\frac{n^2(n^2-1)}{12}$ in dimension $n$ .

Types of Curvature

Definition6.7Sectional curvature

For a 2-plane $\sigma = \operatorname{span}(u, v) \subset T_pM$ , the sectional curvature is

K(\sigma) = K(u, v) = \frac{g(R(u,v)v, u)}{g(u,u)g(v,v) - g(u,v)^2}.

This is the Gaussian curvature of the surface obtained by exponentiating $\sigma$ . The sectional curvatures determine the full curvature tensor.

Definition6.8Ricci and scalar curvature

The Ricci curvature is the trace of the curvature tensor: $\mathrm{Ric}(u, v) = \sum_{i=1}^n g(R(e_i, u)v, e_i)$ for an orthonormal basis $\{e_i\}$ . In components: $R_{ij} = R^k{}_{ikj}$ .

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

ExampleConstant curvature spaces

The three model geometries:

$\mathbb{R}^n$ (Euclidean): $K = 0$ everywhere. Flat space.
$S^n(r)$ (sphere of radius $r$ ): $K = 1/r^2$ everywhere. Positive curvature.
$\mathbb{H}^n$ (hyperbolic): $K = -1$ everywhere. Negative curvature.

By Schur's lemma, if the sectional curvature $K$ is pointwise constant on a connected manifold of dimension $\geq 3$ , then $K$ is globally constant.

Curvature and Topology

Theorem6.4Gauss-Bonnet theorem (dimension 2)

For a compact oriented Riemannian 2-manifold $(M, g)$ ,

\int_M K \, \mathrm{dvol}_g = 2\pi \chi(M),

where $K$ is the Gaussian curvature and $\chi(M)$ is the Euler characteristic. This remarkable formula connects local geometry (curvature) to global topology (Euler characteristic).

RemarkHigher-dimensional Gauss-Bonnet

The Chern-Gauss-Bonnet theorem generalizes this to even-dimensional manifolds: $\chi(M) = \int_M \mathrm{Pf}(\Omega)$ , where $\mathrm{Pf}(\Omega)$ is the Pfaffian of the curvature form. This is a prototype of index theorems connecting analysis and topology.