Riemannian Metrics - Core Definitions

A Riemannian metric endows a smooth manifold with the structure needed to measure lengths, angles, areas, and volumes, transforming differential geometry from a purely topological subject into a geometric one.

Definition and Basic Properties

Definition6.1Riemannian metric

A Riemannian metric $g$ on a smooth manifold $M$ is a smooth assignment of an inner product $g_p: T_pM \times T_pM \to \mathbb{R}$ to each tangent space, meaning:

Symmetric: $g_p(u, v) = g_p(v, u)$ for all $u, v \in T_pM$ .
Positive definite: $g_p(v, v) > 0$ for all $v \neq 0$ .
Smooth: In any local coordinates $(x^1, \ldots, x^n)$ , the components $g_{ij}(x) = g(\partial/\partial x^i, \partial/\partial x^j)$ are smooth functions.

In local coordinates, $g = g_{ij}(x) \, dx^i \otimes dx^j$ (Einstein summation). A smooth manifold equipped with a Riemannian metric is a Riemannian manifold $(M, g)$ .

Theorem6.1Existence of Riemannian metrics

Every smooth manifold admits a Riemannian metric.

The proof uses partitions of unity: choose an arbitrary inner product on each chart domain, then average using a partition of unity. The convex combination of positive definite bilinear forms is positive definite.

Examples

ExampleStandard and classical metrics

Euclidean metric: On $\mathbb{R}^n$ , $g = dx^1 \otimes dx^1 + \cdots + dx^n \otimes dx^n = \delta_{ij} dx^i dx^j$ .
Round metric on $S^n$ : The restriction of the Euclidean metric to the sphere embedded in $\mathbb{R}^{n+1}$ .
Hyperbolic metric: On the upper half-plane $\mathbb{H}^2 = \{(x,y) : y > 0\}$ , $g = \frac{dx^2 + dy^2}{y^2}$ . This has constant curvature $-1$ .
Flat torus: $T^2 = \mathbb{R}^2 / \mathbb{Z}^2$ with the metric inherited from $\mathbb{R}^2$ . This is flat ( $R = 0$ ).
Product metric: If $(M, g_M)$ and $(N, g_N)$ are Riemannian, then $M \times N$ has the product metric $g_M \oplus g_N$ .

Lengths, Distances, and Volumes

Definition6.2Length and distance

The length of a piecewise smooth curve $\gamma: [a,b] \to M$ is

L(\gamma) = \int_a^b \sqrt{g_{\gamma(t)}(\gamma'(t), \gamma'(t))} \, dt = \int_a^b |\gamma'(t)|_g \, dt.

The Riemannian distance between points $p, q \in M$ is

d(p, q) = \inf\{L(\gamma) : \gamma \text{ is a piecewise smooth curve from } p \text{ to } q\}.

This makes $(M, d)$ a metric space whose topology agrees with the manifold topology.

Definition6.3Riemannian volume form

On an oriented Riemannian $n$ -manifold, the volume form is

\mathrm{dvol}_g = \sqrt{\det(g_{ij})} \, dx^1 \wedge \cdots \wedge dx^n.

The volume of a measurable region $U \subset M$ is $\operatorname{Vol}(U) = \int_U \mathrm{dvol}_g$ .

RemarkMusical isomorphisms

A Riemannian metric provides canonical isomorphisms between tangent and cotangent spaces: the flat map $\flat: TM \to T^*M$ sends $v \mapsto g(v, \cdot)$ , and the sharp map $\sharp = \flat^{-1}$ is its inverse. These "raise and lower indices" in tensor calculus: $v_i = g_{ij} v^j$ and $\alpha^i = g^{ij} \alpha_j$ .