Riemannian Metrics — Mathematics Knowledge System

Overview

A Riemannian metric on a smooth manifold M is a smooth assignment of an inner product to each tangent space. It provides the notions of length, angle, area, volume, and distance intrinsic to M, without reference to an ambient Euclidean space. The pair (M, g) is called a Riemannian manifold. Riemannian geometry, founded by Riemann's 1854 habilitation lecture, generalises Euclidean and non-Euclidean geometry and is the mathematical framework for Einstein's general theory of relativity.

Intuition

A Riemannian metric is like a local ruler and protractor at every point of your manifold. It tells you the length of a vector and the angle between two vectors. Once you have this, you can measure the length of any curve by integrating the speed, define the distance between two points as the length of the shortest connecting curve (a geodesic), and compute volumes and areas. The metric can vary from point to point — this variation encodes curvature.

Formal Definition

Definition

A Riemannian metric on a smooth n-manifold M is a (0,2) tensor field g that is symmetric and positive definite at every point. In local coordinates, g is represented by the matrix (gᵢⱼ).

g_p: T_pM \times T_pM \to \mathbb{R}, \quad g_p(u, v) = g_p(v, u), \quad g_p(v, v) > 0 \text{ for } v \neq 0

Symmetric positive-definite inner product at each point

g = g_{ij}\, dx^i \otimes dx^j

Local expression in coordinates (Einstein summation)

\|v\|_g = \sqrt{g_p(v,v)}, \quad L(\gamma) = \int_a^b \|\gamma'(t)\|_g\, dt

Length of a vector and length of a curve

d(p, q) = \inf_{\gamma} L(\gamma)

Infimum over smooth curves γ from p to q

Riemannian distance

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

Riemannian volume form

Properties

Metric determines a volume form

\mathrm{d}V_g = \sqrt{\det(g_{ij})}\, dx^1 \wedge \cdots \wedge dx^n

Musical isomorphisms

g \text{ induces } \flat: TM \to T^*M,\; v^\flat(w) = g(v,w), \text{ and its inverse } \sharp: T^*M \to TM

Positive definiteness implies non-degeneracy

\det(g_{ij}) > 0 \text{ everywhere}

Theorems

Theorem 1: Existence of Riemannian Metrics

\text{Every smooth manifold admits a Riemannian metric.}

Theorem 2: Nash Embedding Theorem

\text{Every Riemannian manifold } (M, g) \text{ isometrically embeds into } (\mathbb{R}^N, g_{\mathrm{Eucl}}) \text{ for sufficiently large } N.

Theorem 3: Hopf–Rinow Theorem

\text{A Riemannian manifold is geodesically complete iff it is complete as a metric space, and any two points can be joined by a minimizing geodesic.}

Worked Examples

1
The curve γ has velocity:
$\gamma'(t) = (-\sin t, \cos t, 1)$
2
The speed in the Euclidean metric on ℝ³ is:
$\|\gamma'(t)\| = \sqrt{\sin^2 t + \cos^2 t + 1} = \sqrt{2}$
3
The arc length is:
$L = \int_0^{2\pi} \sqrt{2}\, dt = 2\pi\sqrt{2}$

✓ Answer

The helix on the cylinder has length 2π√2 — longer than the circle of length 2π by a factor of √2 because of the vertical rise.

Practice Problems

Mediumfree response

Compute the distance between two antipodal points on S² with the round metric of radius r.

Mediumfree response

What is the volume of the Poincaré upper half-plane H = {(x,y) : y > 0} with metric g = (dx² + dy²)/y² inside the region 0 < x < 1, 1 < y < 2?

Common Mistakes

Common Mistake

Confusing a Riemannian metric with a pseudo-Riemannian (Lorentzian) metric

A Riemannian metric is positive definite; a pseudo-Riemannian metric (as in general relativity) has mixed signature (+−−−) and allows null vectors.

Common Mistake

Thinking the metric determines a unique coordinate system

The metric is a geometric object that exists independently of coordinates. Many different coordinate charts express the same metric.

Quiz

Historical Background

Bernhard Riemann's 1854 habilitation lecture 'Über die Hypothesen, welche der Geometrie zu Grunde liegen' introduced the concept of a curved n-dimensional space and the metric tensor as its fundamental invariant. Riemann proposed that space has intrinsic curvature, laying the groundwork for general relativity. Christoffel, Ricci, and Levi-Civita developed the tensor calculus needed to handle Riemannian geometry computationally. Einstein adopted this framework in 1915 for general relativity, while Marcel Grossmann guided him to the mathematical tools.

1854
Riemann's habilitation lecture introduces Riemannian geometry and the metric tensor
Bernhard Riemann
1869
Christoffel introduces what become the Christoffel symbols for covariant differentiation
Elwin Bruno Christoffel
1900
Ricci and Levi-Civita publish Méthodes de calcul différentiel absolu, the tensor calculus
Gregorio Ricci-Curbastro, Tullio Levi-Civita
1915
Einstein uses Riemannian geometry as the geometric foundation of general relativity
Albert Einstein

Summary

A Riemannian metric g is a smooth, symmetric, positive-definite (0,2) tensor field on M.
It provides notions of length, angle, volume, and distance intrinsic to M.
Every smooth manifold admits a Riemannian metric (proved via partition of unity).
The Hopf–Rinow theorem equates geodesic completeness with metric space completeness.
Nash's theorem guarantees every Riemannian manifold isometrically embeds in some Euclidean space.

References

Bookdo Carmo, M. P. — Riemannian Geometry, Birkhäuser, 1992
BookLee, J. M. — Riemannian Manifolds: An Introduction to Curvature, Springer, 1997
WebsiteWikipedia — Riemannian manifold
WebsiteWolfram MathWorld — Riemannian Metric