Skip to content

Riemannian Metric

Summary

A Riemannian metric is the additional structure that endows a smooth manifold with lengths, angles and volumes. It is the fundamental object that Hamilton's Ricci flow deforms in time — without a metric there is no curvature, and without curvature there is no Ricci flow.

1. From Topology to Geometry

In Act 1 (Article 02) manifolds were introduced as spaces that locally look like \(\mathbb{R}^n\). A smooth manifold knows only its topological and differentiable structure: which functions are smooth, which vector fields exist. It does not know how long a vector is or what angle two vectors enclose.

This is exactly what a Riemannian metric supplies. It turns a smooth manifold into a Riemannian manifold \((M, g)\) and is the smallest extra structure with which one can do classical geometry.

2. Definition

Let \(M\) be a smooth \(n\)-manifold. A Riemannian metric \(g\) assigns to every point \(p \in M\) an inner product \(g_p \colon T_pM \times T_pM \to \mathbb{R}\) on the tangent space that is

  • bilinear in both arguments,
  • symmetric (\(g_p(v,w) = g_p(w,v)\)),
  • positive definite (\(g_p(v,v) \ge 0\) with equality only for \(v = 0\)),

and depends smoothly on \(p\). In local coordinates \((x^1,\dots,x^n)\) one writes

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

with \(\bigl(g_{ij}(p)\bigr)\) symmetric and positive definite. Einstein's summation convention is used silently throughout.

"A Riemannian metric on a smooth manifold \(M\) is a smooth, symmetric, positive-definite 2-tensor field on \(M\)." — John M. Lee, Introduction to Riemannian Manifolds (2018), p. 11

3. What the Metric Buys Us

Once \(g\) is in place, all classical geometric quantities are defined.

Length of a vector. For \(v \in T_pM\): \(\lvert v\rvert_g = \sqrt{g_p(v,v)}\).

Angle. Between \(v, w \in T_pM \setminus\{0\}\):

\[\cos\theta = \frac{g_p(v,w)}{\lvert v\rvert_g\,\lvert w\rvert_g}.\]

Arc length of a curve \(\gamma\colon [a,b]\to M\):

\[L(\gamma) = \int_a^b \sqrt{g_{\gamma(t)}\bigl(\dot\gamma(t),\dot\gamma(t)\bigr)}\,dt.\]

Riemannian distance. \(d_g(p,q) = \inf_\gamma L(\gamma)\), the infimum being taken over all piecewise smooth curves from \(p\) to \(q\). \((M, d_g)\) becomes a metric space whose topology agrees with the underlying manifold topology.

Volume form. In oriented charts:

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

This gives volumes of subsets, integrals of smooth functions and – through the Hessian of \(g\) – curvature quantities (see Article 02).

4. Examples

Euclidean space. On \(\mathbb{R}^n\) the flat standard metric is \(g_{\mathrm{eucl}} = \delta_{ij}\,dx^i\,dx^j\) — lengths, angles and volume are the familiar ones.

Sphere \(S^n\). Embedded in \(\mathbb{R}^{n+1}\), \(S^n\) inherits the round metric \(g_{\mathrm{round}}\) as the restriction of the Euclidean inner product to the tangent space. Great circles are geodesics; the sectional curvature is constant equal to \(1\) on the unit sphere.

Hyperbolic space \(\mathbb{H}^n\). On the upper half space \(\{(x^1,\dots,x^n) : x^n > 0\}\) the metric \(g_{\mathrm{hyp}} = (x^n)^{-2}\,\delta_{ij}\,dx^i\,dx^j\) has constant sectional curvature \(-1\).

Product metrics. On \(M \times N\), \(g_M \oplus g_N\) combines the factors — for example the standard metric on \(T^2 = S^1 \times S^1\) or on \(S^2 \times \mathbb{R}\).

Squashed sphere. Scaling the round \(S^2\) in one direction yields an ellipsoid. Topologically still a 2-sphere; geometrically curvature, geodesics and volume change.

5. Existence and Variety

A basic observation: every paracompact smooth manifold admits at least one Riemannian metric. The proof glues local coordinate metrics with a partition of unity, transferring positive definiteness to the global object (Lee 2018, Prop. 2.4).

More importantly: on a fixed smooth manifold the infinite-dimensional space \(\mathcal{M}(M)\) of Riemannian metrics is the actual playground of the Ricci flow: a starting metric \(g_0\) becomes a one-parameter family \(g(t)\) — a trajectory in \(\mathcal{M}(M)\) (see Article 03).

6. The Levi-Civita Connection

A Riemannian metric canonically induces a connection \(\nabla\) on the tangent bundle — the Levi-Civita connection. It is characterised by two properties:

  • Torsion-free: \(\nabla_X Y - \nabla_Y X = [X,Y]\).
  • Metric-compatible: \(X(g(Y,Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)\).

The fundamental theorem of Riemannian geometry says these two conditions determine \(\nabla\) uniquely. The associated Christoffel symbols

\[\Gamma^k_{ij} = \tfrac{1}{2}\,g^{kl}\bigl(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}\bigr)\]

are the computational backbone of curvature and geodesics.

7. Geodesics

A geodesic is a curve \(\gamma\) with \(\nabla_{\dot\gamma}\dot\gamma = 0\) — it travels "as straight as possible". Geodesics generalise straight lines in \(\mathbb{R}^n\) and great circles on \(S^n\). In coordinates:

\[\ddot\gamma^k + \Gamma^k_{ij}\,\dot\gamma^i\,\dot\gamma^j = 0.\]

Locally geodesics are the shortest connections; globally that holds only with extra assumptions (such as completeness). The Hopf-Rinow theorem ties metric completeness to geodesic completeness.

8. Why the Metric Is Central in Act 2

Hamilton's Ricci flow is the evolution equation

\[\frac{\partial g(t)}{\partial t} = -2\,\mathrm{Ric}(g(t)).\]

It is a partial differential equation for the metric itself. All geometric quantities the next articles work with — curvature tensors, volume, diameter, entropy functionals — are functions of \(g\). Anyone who wants to understand the flow must first understand what a metric is and how it changes when one "wiggles" it.

Sources

  • John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer (2018), Ch. 2.
  • Manfredo do Carmo, Riemannian Geometry, Birkhäuser (1992), Ch. 1–3.
  • Peter Petersen, Riemannian Geometry, 3rd ed., Springer (2016), Ch. 2.

Cross References