Riemannian Metric

Why Riemannian metric matters

• General relativity. Spacetime is a 4D Lorentzian manifold; the metric tensor g_μν is the dynamical field. Einstein's equations R_μν − ½ g_μν R = 8πG T_μν / c⁴ determine the metric from matter; geodesics are world-lines of free-falling particles.
• Mesh processing and computer graphics. Discrete differential geometry on triangle meshes uses cotan-Laplacian metrics for parameterization, smoothing, and remeshing.
• Manifold learning. Many machine-learning data sets lie near low-dimensional submanifolds of ℝ^d. Diffusion maps, ISOMAP, and Riemannian gradient descent treat parameter space as Riemannian.
• Ricci flow and Perelman. Hamilton's Ricci flow ∂g/∂t = −2 Ric(g) deforms a metric toward uniform curvature. Perelman (2002–2003) used Ricci flow with surgery to prove the Poincaré conjecture; the only Fields Medal that has been declined.
• Optimization on manifolds. Many problems (Stiefel, Grassmann, SO(n)) live on manifolds; gradient descent generalizes via Riemannian gradient and exponential map.
• Information geometry. The Fisher information metric makes a parametrized family of probability distributions a Riemannian manifold; geodesics give natural-gradient descent.
• Robotics and motion planning. Configuration spaces are manifolds with metrics from kinetic energy; minimum-action paths are geodesics.

Formal definition

Let M be a smooth n-manifold. A Riemannian metric g is a smooth section of the symmetric tensor bundle T*M ⊗ T*M, such that for every p ∈ M, the bilinear form g_p: TₚM × TₚM → ℝ is symmetric and positive-definite. Equivalently, in any chart (U, x¹, ..., xⁿ), g is given by a smooth matrix-valued function:

g = g_ij(x) dx^i ⊗ dx^j     (Einstein summation)

with g_ij = g_ji and (g_ij(x)) positive-definite for every x ∈ U. The matrix transforms tensorially under coordinate change: g'_kl = (∂x^i / ∂x'^k)(∂x^j / ∂x'^l) g_ij.

Arc length and distance

For a piecewise C¹ curve γ: [a, b] → M:

ℓ(γ) = ∫_a^b √(g_γ(t)(γ̇(t), γ̇(t))) dt

The distance d(p, q) between two points is the infimum of ℓ(γ) over all curves from p to q. (M, d) is a metric space. The Hopf–Rinow theorem says: for a connected Riemannian manifold, the following are equivalent: (i) (M, d) is complete; (ii) every closed bounded set is compact; (iii) geodesics extend for all time. Under any of these, every pair of points is joined by a length-minimizing geodesic.

Canonical examples

• Euclidean. M = ℝⁿ, g_ij = δ_ij. Geodesics are straight lines; curvature is zero everywhere.
• Round sphere. M = S² ⊂ ℝ³, g induced from ambient. In coordinates (φ, θ): g = R²(dφ² + sin²φ dθ²). Geodesics are great circles. Constant positive sectional curvature 1/R².
• Hyperbolic plane (Poincaré disc). M = {z ∈ ℂ : |z| < 1}, g = 4/(1 − |z|²)² (dx² + dy²). Geodesics are arcs of circles meeting the boundary at right angles. Constant negative curvature −1.
• Flat torus. M = ℝ²/ℤ², g = dx² + dy². Compact, flat (zero curvature), but not isometric to a subset of ℝ²; can't be embedded in ℝ³ flatly.
• Schwarzschild metric. Outside a non-rotating massive body of mass M: ds² = −(1 − r_s/r) c² dt² + (1 − r_s/r)⁻¹ dr² + r²(dθ² + sin²θ dφ²) with r_s = 2GM/c². Lorentzian, not Riemannian, but the spatial slices are Riemannian.
• Fubini–Study metric. On ℂP^n: the unique (up to scale) U(n+1)-invariant metric. Foundation of complex projective geometry and Calabi–Yau studies.

Christoffel symbols and the Levi-Civita connection

The fundamental theorem of Riemannian geometry: there is a unique connection ∇ on M that is (i) torsion-free and (ii) compatible with g (i.e., ∇g = 0). It's called the Levi-Civita connection. In coordinates:

Γ^k_ij = (1/2) g^{kl} (∂_i g_jl + ∂_j g_il − ∂_l g_ij)

where g^{ij} is the inverse matrix of g_ij. Christoffel symbols are not tensors — they vanish in normal coordinates at any chosen point and transform with an inhomogeneous correction. The covariant derivative of a vector field Y in direction X is (∇_X Y)^k = X^i (∂_i Y^k + Γ^k_ij Y^j).

Geodesics

A curve γ is a geodesic if its tangent is parallel-transported along itself: ∇_{γ̇} γ̇ = 0. In coordinates this becomes the geodesic equation:

d²x^k/dt² + Γ^k_ij(γ(t)) (dx^i/dt)(dx^j/dt) = 0

Equivalently, geodesics are critical points of the length functional or of the energy functional E(γ) = ½ ∫ g(γ̇, γ̇) dt. For initial conditions (p, v) ∈ TM, there's a unique geodesic γ with γ(0) = p, γ̇(0) = v on a maximal interval. The exponential map exp_p: TₚM → M sends v to γ_v(1) — the time-1 endpoint.

Curvature tensors

The Riemann curvature tensor:

R(X, Y) Z = ∇_X ∇_Y Z − ∇_Y ∇_X Z − ∇_{[X,Y]} Z

measures how covariant differentiation fails to commute. In coordinates:

R^l_{ijk} = ∂_i Γ^l_jk − ∂_j Γ^l_ik + Γ^l_im Γ^m_jk − Γ^l_jm Γ^m_ik

Contractions:

• Ricci tensor R_ij = R^k_{ikj}, symmetric (n × n) matrix.
• Scalar curvature R = g^{ij} R_ij, a single number at each point.
• Sectional curvature K(σ) for a 2-plane σ ⊂ TₚM: K(σ) = R(X, Y, X, Y) / (‖X‖²‖Y‖² − ⟨X, Y⟩²) for any basis X, Y of σ.

Constant sectional curvature: model spaces — sphere (K > 0), Euclidean (K = 0), hyperbolic (K < 0).

Volume form

The metric induces a volume form on an orientable manifold:

dV = √(det g_ij) dx¹ ∧ ... ∧ dxⁿ

Total volume Vol(M) = ∫_M dV. For a sphere S² of radius R: dV = R² sin φ dφ dθ; total area 4πR². The volume form is invariant under isometries — the natural measure for integration on a Riemannian manifold.

General relativity

In GR, spacetime is a 4D Lorentzian manifold — signature (−, +, +, +) so that g(v, v) can be negative. Tangent vectors are classified:

• Timelike: g(v, v) < 0. Worldlines of massive particles.
• Null: g(v, v) = 0. Light rays.
• Spacelike: g(v, v) > 0. Spatial separations.

The Einstein field equations R_μν − ½ g_μν R + Λ g_μν = (8πG/c⁴) T_μν are second-order PDEs for g_μν. Free-falling particles follow timelike geodesics; light follows null geodesics. Gravitational time dilation, perihelion precession of Mercury, gravitational lensing, and gravitational waves all emerge from the geometry.

Ricci flow

Hamilton (1982) introduced Ricci flow:

∂g_ij/∂t = −2 R_ij(g)

An evolution equation for the metric, analogous to the heat equation for functions. Tends to homogenize curvature. Perelman (2002–2003) used Ricci flow with surgery — pinching off neck-like singularities — to prove Thurston's geometrization conjecture, which implies the Poincaré conjecture: every simply connected closed 3-manifold is homeomorphic to S³. Earned a Fields Medal Perelman declined.

Common misconceptions

• "A metric is a function." A Riemannian metric is a (0, 2)-tensor field — a section of T*M ⊗ T*M. At each point it's a bilinear form; varying smoothly across M.
• "Always Euclidean locally." Locally, you can find normal coordinates where g_ij = δ_ij at one point, but the second derivatives encode curvature and can't be eliminated. Manifolds are flat only if the Riemann tensor vanishes everywhere.
• "Only for surfaces." Riemannian geometry works in any dimension; GR uses dimension 4, string theory uses 10 or 11.
• "Metric = distance." The metric tensor g defines distances by integrating along curves — distance is a derived notion. Two metrics can give the same distance function (conformally equivalent metrics agree on angles, not lengths).
• "Geodesics minimize globally." Only locally. Going the long way around the equator solves the same geodesic equation but isn't a global minimizer.
• "Christoffel symbols are tensors." No — they transform inhomogeneously. The covariant derivative ∇ is the geometrically meaningful object; Christoffels are coordinate components of ∇ minus the partial derivatives.
• "Lorentzian = Riemannian with extra signs." Mathematically yes (signature change), but the analytic theory differs sharply: Lorentzian metrics have causal structure, light cones, and singularities not present in Riemannian geometry.