Riemannian Metrics and Geodesics

A Riemannian metric is a smoothly varying inner product on the tangent spaces of a manifold, and geodesics are its locally length-minimizing curves — the manifold's intrinsic notion of "straight lines." Key result: geodesics satisfy a second-order ODE (the geodesic equation) determined by the Christoffel symbols of the metric, and they are exactly the critical points of the energy/length functional. Applies wherever curved spaces appear: general relativity, geometric optics, optimization on manifolds, shape analysis, and pure differential geometry. Introduced by Bernhard Riemann (1854) and developed by Christoffel, Ricci, Levi-Civita, and Cartan.

A Riemannian metric $g$ on a smooth manifold $M$ assigns to each point $p \in M$ a positive-definite symmetric bilinear form $g_{p}$ on the tangent space $T_{p} M$ , varying smoothly with $p$ . This single structure lets one measure lengths of curves, angles between vectors, volumes of regions, and ultimately curvature — turning bare topology into geometry. Geodesics are the curves that locally minimize length (equivalently, that have zero covariant acceleration), generalizing straight lines to curved spaces. The surprising fact: although the metric is purely local data, it rigidly determines global phenomena — through the Hopf–Rinow theorem, geodesic completeness, conjugate points, and the way curvature forces geodesics to focus or spread. Riemannian geometry is the mathematical backbone of Einstein's general relativity and modern geometric analysis.

The Core Insight

Give each tangent space an inner product, and all of metric geometry follows: lengths via integration of speed, distance via infimum over paths, "straight lines" via the variational principle, and curvature via how nearby geodesics fail to stay parallel. The metric is the only input; the Levi-Civita connection, geodesics, and curvature are uniquely forced by compatibility and symmetry.

Formal Statement

Riemannian Metric

Let $M$ be a smooth $n$ -dimensional manifold. A Riemannian metric $g$ is a smooth $(0, 2)$ -tensor field such that each $g_{p} : T_{p} M \times T_{p} M \to R$ is symmetric and positive-definite. In local coordinates $(x^{1}, \dots, x^{n})$ with coordinate basis $\partial_{i} = \partial / \partial x^{i}$ , the metric is encoded by the component matrix $g_{ij}$ :

g = g_{ij} d x^{i} \otimes d x^{j}, g_{ij} (p) = g_{p} (\partial_{i}, \partial_{j}) = g_{j i} (p) .

⟳Evaluate the bilinear form on coordinate basis vectors; symmetry of

g

gives

g_{ij} = g_{j i}

, positive-definiteness gives

g_{ij} v^{i} v^{j} > 0

for

v \neq = 0

The pair $(M, g)$ is a Riemannian manifold. The squared norm and inner product of tangent vectors $u, v$ at $p$ are

⟨ u, v ⟩_{g} = g_{ij} u^{i} v^{j}, ∣ v ∣_{g} = g_{ij} v^{i} v^{j} .

⟳Expand

u = u^{i} \partial_{i}

v = v^{j} \partial_{j}

, apply bilinearity of

g

, and use

g (\partial_{i}, \partial_{j}) = g_{ij}

Length, Energy, and Riemannian Distance

For a piecewise-smooth curve $γ : [a, b] \to M$ , define its length and energy:

L (γ) = \int_{a}^{b} ∣ \overset{γ}{˙} (t) ∣_{g} d t = \int_{a}^{b} g_{ij} (γ) \overset{γ}{˙}^{i} \overset{γ}{˙}^{j} d t, E (γ) = \frac{1}{2} \int_{a}^{b} g_{ij} (γ) \overset{γ}{˙}^{i} \overset{γ}{˙}^{j} d t .

⟳

L

integrates instantaneous speed

∣ \overset{γ}{˙} ∣

;

E

uses the squared integrand, which is smoother and reparametrization-sensitive.

The Riemannian distance between $p, q \in M$ is the infimum of lengths over connecting curves:

d_{g} (p, q) = in f {L (γ) : γ piecewise-smooth, γ (a) = p, γ (b) = q} .

⟳Length is reparametrization-invariant, so the infimum is well-defined;

(M, d_{g})

is a metric space whose topology equals the manifold topology.

Levi-Civita Connection and Christoffel Symbols

There is a unique affine connection $\nabla$ that is torsion-free ( $\nabla_{X} Y - \nabla_{Y} X = [X, Y]$ ) and metric-compatible ( $\nabla g = 0$ ): the Levi-Civita connection. Its Christoffel symbols in coordinates are

Γ_{ij}^{k} = \frac{1}{2} g^{k ℓ} (\partial_{i} g_{j ℓ} + \partial_{j} g_{i ℓ} - \partial_{ℓ} g_{ij}),

⟳Impose

\nabla g = 0

on cyclic permutations of indices

(i, j, ℓ)

, subtract one equation from the sum of the other two, and use

Γ_{ij}^{k} = Γ_{j i}^{k}

(torsion-free) to solve;

g^{k ℓ}

is the inverse metric.

where $g^{k ℓ}$ denotes the inverse matrix, $g^{k ℓ} g_{ℓ m} = δ_{m}^{k}$ .

Geodesic Equation

A curve $γ$ is a geodesic (zero covariant acceleration, $\nabla_{\overset{γ}{˙}} \overset{γ}{˙} = 0$ ) iff its components satisfy

\overset{γ}{¨}^{k} + Γ_{ij}^{k} (γ) \overset{γ}{˙}^{i} \overset{γ}{˙}^{j} = 0, k = 1, \dots, n .

⟳Write

\nabla_{\overset{γ}{˙}} \overset{γ}{˙} = (\overset{γ}{¨}^{k} + Γ_{ij}^{k} \overset{γ}{˙}^{i} \overset{γ}{˙}^{j}) \partial_{k}

using

\nabla_{\partial_{i}} \partial_{j} = Γ_{ij}^{k} \partial_{k}

and the chain rule for

\overset{γ}{˙} = \overset{γ}{˙}^{i} \partial_{i}

Geodesics are precisely the critical points of the energy functional $E$ , and (after arclength reparametrization) the locally length-minimizing curves. They have constant speed: $∣ \overset{γ}{˙} ∣_{g}$ is conserved along $γ$ .

Why It Holds / Derivation

Geodesics from the Energy Functional

We show that critical points of $E$ satisfy the geodesic equation, justifying the second-order ODE intrinsically.

Step 1 — Variation setup. Let

γ_{s} (t) = γ (t, s)

be a smooth one-parameter family with fixed endpoints,

γ_{0} = γ

, variation field

V = \partial_{s} γ ∣_{s = 0}

vanishing at

a, b

. The energy is

E (γ_{s}) = \frac{1}{2} \int_{a}^{b} g_{ij} (γ_{s}) \partial_{t} γ_{s}^{i} \partial_{t} γ_{s}^{j} d t .

⟳Substitute the family into the energy integral [3]; differentiation in

s

will produce the first variation.

Step 2 — Differentiate in $s$ . Using

\partial_{s} g_{ij} = \partial_{ℓ} g_{ij} V^{ℓ}

and

\partial_{s} \partial_{t} γ^{i} = \partial_{t} V^{i}

\frac{d}{d s}_{0} E = \int_{a}^{b} (\frac{1}{2} \partial_{ℓ} g_{ij} V^{ℓ} \overset{γ}{˙}^{i} \overset{γ}{˙}^{j} + g_{ij} \overset{γ}{˙}^{i} \dot{V}^{j}) d t .

⟳Apply the product rule under the integral; the first term comes from varying the metric components, the second from varying velocity.

Step 3 — Integrate by parts. On the second term,

\int g_{ij} \overset{γ}{˙}^{i} \dot{V}^{j} = - \int \frac{d}{d t} (g_{ij} \overset{γ}{˙}^{i}) V^{j}

(boundary terms vanish since

V (a) = V (b) = 0

). Relabel indices and collect coefficients of

V^{k}

\frac{d}{d s}_{0} E = - \int_{a}^{b} (g_{k j} \overset{γ}{¨}^{j} + (\partial_{i} g_{k j} - \frac{1}{2} \partial_{k} g_{ij}) \overset{γ}{˙}^{i} \overset{γ}{˙}^{j}) V^{k} d t .

⟳Expand

\frac{d}{d t} (g_{ij} \overset{γ}{˙}^{i}) = \partial_{ℓ} g_{ij} \overset{γ}{˙}^{ℓ} \overset{γ}{˙}^{i} + g_{ij} \overset{γ}{¨}^{i}

, then symmetrize

\partial_{ℓ} g_{ij}

(ℓ, i)

Step 4 — Fundamental lemma. Since

V^{k}

is arbitrary, the bracket must vanish. Multiply by

g^{mk}

and symmetrize to recover the Christoffel symbols:

\overset{γ}{¨}^{m} + g^{mk} \frac{1}{2} (\partial_{i} g_{k j} + \partial_{j} g_{k i} - \partial_{k} g_{ij}) \overset{γ}{˙}^{i} \overset{γ}{˙}^{j} = \overset{γ}{¨}^{m} + Γ_{ij}^{m} \overset{γ}{˙}^{i} \overset{γ}{˙}^{j} = 0.

⟳Use

\overset{γ}{˙}^{i} \overset{γ}{˙}^{j}

symmetry to replace

\partial_{i} g_{k j}

\frac{1}{2} (\partial_{i} g_{k j} + \partial_{j} g_{k i})

, then apply definition [5].

□

Constant speed. Differentiate

∣ \overset{γ}{˙} ∣^{2} = g_{ij} \overset{γ}{˙}^{i} \overset{γ}{˙}^{j}

and substitute [6]; metric compatibility forces

\frac{d}{d t} ∣ \overset{γ}{˙} ∣^{2} = 0

Bernhard Riemann | 1826–1866

In his 1854 Göttingen habilitation lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen ("On the Hypotheses which lie at the Foundations of Geometry"), delivered before an audience including Gauss, Riemann introduced the concept of an $n$ -dimensional manifold equipped with a positive-definite quadratic form measuring infinitesimal length — what we now call a Riemannian metric. He proposed that the geometry of physical space is an empirical question determined by its curvature, anticipating general relativity by six decades. Riemann gave the curvature in terms of the metric and treated geodesics as the analog of straight lines, though the explicit machinery (Christoffel symbols, covariant differentiation, the tensor calculus) was supplied later by Christoffel (1869), Ricci-Curbastro, and Levi-Civita (1900).

Building Intuition

Imagine an ant living on a curved surface that cannot see the ambient space — it has access only to a ruler (the metric) measuring distances within the surface. To walk "as straight as possible," the ant minimizes turning: at each instant it never veers left or right intrinsically. That is exactly $\nabla_{\overset{γ}{˙}} \overset{γ}{˙} = 0$ — the acceleration has no component tangent to the surface; any acceleration is purely the unavoidable bending needed to stay on the surface. The Christoffel symbols are the "correction terms" that tell the ant how the coordinate grid itself curves, so that subtracting them isolates true geometric acceleration. On a sphere, great circles are geodesics; on a flat plane, ordinary straight lines. Curvature reveals itself when you release two ants walking parallel from a baseline: on a sphere they converge (positive curvature), on a saddle they diverge (negative curvature). Geodesics minimize length only locally — over long distances a great circle path is beaten by going the other way around.

Geodesics on a sphere: two great-circle arcs from p; the shorter one is the minimizer. Parallel geodesics converge under positive curvature.

Worked Examples

Geodesics on the 2-Sphere of Radius R

Given. The round sphere

S_{R}^{2}

in spherical coordinates

(θ, ϕ)

, colatitude

θ \in (0, π)

, longitude

ϕ

. The induced metric is

g = R^{2} d θ^{2} + R^{2} sin^{2} θ d ϕ^{2}, so g_{θ θ} = R^{2}, g_{ϕϕ} = R^{2} sin^{2} θ .

⟳Embed

S_{R}^{2} \subset R^{3}

via

(R sin θ cos ϕ, R sin θ sin ϕ, R cos θ)

and pull back the Euclidean metric.

Step 1 — Christoffel symbols. With

g_{ϕϕ} = R^{2} sin^{2} θ

, the nonzero symbols are

Γ_{ϕϕ}^{θ} = - sin θ cos θ, Γ_{θ ϕ}^{ϕ} = Γ_{ϕθ}^{ϕ} = cot θ .

⟳Apply [5]; only

\partial_{θ} g_{ϕϕ} = 2 R^{2} sin θ cos θ

is nonzero, giving these two via

g^{θ θ} = 1/ R^{2}

g^{ϕϕ} = 1/ (R^{2} sin^{2} θ)

Step 2 — Geodesic equations.

\ddot{θ} - sin θ cos θ \dot{ϕ}^{2} = 0, \ddot{ϕ} + 2 cot θ \dot{θ} \dot{ϕ} = 0.

⟳Substitute [12] into [6] for

k = θ

and

k = ϕ

Step 3 — Solve along the equator. Try

θ (t) \equiv π /2

: then

\dot{θ} = 0

and the first equation reads

- sin (π /2) cos (π /2) \dot{ϕ}^{2} = 0

, automatically satisfied; the second gives

\ddot{ϕ} = 0

, so

ϕ = ω t

. This is a great circle traversed at constant angular speed. By rotational symmetry, every great circle is a geodesic. Step 4 — Distance from p to q. For

p, q

subtending central angle

Δ σ

d_{g} (p, q) = R Δ σ

Numbers. For Earth (

R = 6371 km

), the great-circle distance between two points

4 5^{\circ} = π /4 rad

apart is

d = 6371 \times π /4 \approx 5004 km

Geodesics in the Hyperbolic Upper Half-Plane

Given. The Poincaré upper half-plane

H = {(x, y) : y > 0}

with metric

g = \frac{d x ^{2} + d y ^{2}}{y ^{2}}, g_{xx} = g_{y y} = \frac{1}{y ^{2}} .

⟳This is the unique (up to scale) complete metric of constant curvature

- 1

on the half-plane; conformal to the Euclidean metric with factor

1/ y^{2}

Step 1 — Christoffel symbols. With

g_{xx} = g_{y y} = y^{- 2}

\partial_{y} g_{xx} = \partial_{y} g_{y y} = - 2 y^{- 3}

Γ_{x y}^{x} = Γ_{y x}^{x} = - \frac{1}{y}, Γ_{xx}^{y} = \frac{1}{y}, Γ_{y y}^{y} = - \frac{1}{y} .

⟳Apply [5] with

g^{xx} = g^{y y} = y^{2}

; only

y

-derivatives are nonzero.

Step 2 — Geodesic equations.

\overset{x}{¨} - \frac{2}{y} \overset{x}{˙} \overset{y}{˙} = 0, \overset{y}{¨} + \frac{1}{y} (\overset{x}{˙}^{2} - \overset{y}{˙}^{2}) = 0.

⟳Substitute [15] into [6].

Step 3 — Identify the curves. Vertical lines

x = const

solve the system (

\overset{x}{˙} = 0 \Rightarrow \overset{x}{¨} = 0

, and

\overset{y}{¨} = \overset{y}{˙}^{2} / y

). The other geodesics are semicircles centered on the $x$ -axis:

(x - c)^{2} + y^{2} = r^{2}

. One checks the first integral

\overset{x}{˙} / y^{2} =

const (a conserved momentum), which combined with constant speed yields these circles. Step 4 — Hyperbolic distance. For two points on a common vertical line

x = c

, heights

y_{1} < y_{2}

d_{g} = \int_{y_{1}}^{y_{2}} \frac{d y}{y} = ln \frac{y _{2}}{y _{1}}

Numbers. From

(0, 1)

(0, e^{2})

d_{g} = ln (e^{2} /1) = 2 (hyperbolic units)

. Note the Euclidean separation is

e^{2} - 1 \approx 6.39

, but hyperbolic distance compresses heights logarithmically.

Common Misconceptions

"Geodesics are always the shortest paths." False globally. Geodesics are only locally length-minimizing. Past a conjugate point or cut point, a geodesic ceases to minimize: on the sphere, a great-circle arc longer than

π R

is no longer the shortest path between its endpoints. Geodesics are critical points of energy, not necessarily minima. "The metric determines distances, so it must be the Euclidean dot product in some coordinates." False in general. A metric can be brought to the identity at a single point (orthonormal frame) and even to "flat to second order" (normal coordinates), but it can be flattened in a whole neighborhood iff the Riemann curvature tensor vanishes there. "Energy and length minimizers are the same." They give the same images but length is reparametrization-invariant while energy is not — energy minimizers are automatically constant-speed (affinely parametrized) geodesics, whereas length minimizers can be parametrized arbitrarily. This is why one usually varies

E

, not

L

. Incompleteness wrecks existence of minimizers. Take the punctured plane

M = R^{2} ∖ {0}

with the flat Euclidean metric. The points

p = (- 1, 0)

and

q = (1, 0)

have

d_{g} (p, q) = 2

(the infimum), but the straight segment realizing it passes through the missing origin. No geodesic of length 2 exists in $M$ ; minimizing sequences leave every compact set. This shows the Hopf–Rinow hypothesis (geodesic or metric completeness) is essential: on a complete Riemannian manifold any two points are joined by a minimizing geodesic, but completeness cannot be dropped.

Connections

Riemannian metrics sit at the crossroads of several theories. The metric uniquely determines the Levi-Civita Connection, from which the Riemann Curvature Tensor is built; curvature in turn governs geodesic behavior via the Jacobi Equation and conjugate points. The variational viewpoint links geodesics to Calculus of Variations and the Euler–Lagrange formalism, and the energy functional connects to Morse Theory (geodesics as critical points of $E$ on the path space, with index = number of conjugate points). The distance function $d_{g}$ makes $(M, d_{g})$ a Metric Space, bridging to metric geometry and Gromov–Hausdorff Convergence. In physics, the Lorentzian analog underlies General Relativity, where geodesics are the worldlines of free-falling particles. Optimization on the manifold of positive-definite matrices, on Lie groups, and on Stiefel/Grassmann manifolds all use Riemannian Optimization built on this same machinery.

Real-World Applications

General Relativity (Einstein, 1915). Spacetime is a Lorentzian 4-manifold; freely falling bodies and light rays follow geodesics of the metric $g_{μν}$ . GPS satellites must correct for the resulting time dilation (~38 µs/day) or positions would drift kilometers daily.
Geometric Optics / Fermat's Principle. Light in a medium of refractive index $n (x)$ follows geodesics of the conformal metric $n (x)^{2} δ_{ij} d x^{i} d x^{j}$ ; this explains mirages and gradient-index lenses.
Robotics & Path Planning. Configuration spaces of robots are manifolds; shortest collision-free motions are geodesics in suitably penalized metrics, used in motion planning and minimum-energy trajectories.
Medical Imaging (DTI). Diffusion tensor MRI encodes water diffusion as a metric in each voxel; tractography traces neural fibers as geodesics through the brain's white matter.
Machine Learning / Information Geometry. The Fisher information metric makes a statistical model a Riemannian manifold; natural gradient descent follows geodesic-aware directions, and geodesic distances quantify model separation.
Computer Graphics & Shape Analysis. Geodesic distances on triangulated surfaces (heat method, fast marching) drive texture mapping, mesh parameterization, and shape correspondence.
Riemannian Optimization. Algorithms on manifolds of orthogonal matrices, SPD matrices, and low-rank factorizations use geodesics (or retractions approximating them) for problems in PCA, dictionary learning, and quantum control.

Historical Development

1827 — Gauss, Disquisitiones generales circa superficies curvas. Introduces the first fundamental form (metric of a surface) and proves the Theorema Egregium: Gaussian curvature is intrinsic, computable from the metric alone.
1854 — Riemann, habilitation lecture at Göttingen. Defines $n$ -dimensional manifolds with an infinitesimal length form and the notion of curvature for higher dimensions.
1869 — Christoffel introduces the symbols $Γ_{ij}^{k}$ and the covariant transformation rules central to tensor calculus.
1888–1901 — Ricci-Curbastro & Levi-Civita develop the absolute differential calculus (tensor analysis), systematizing covariant differentiation and the curvature tensor.
1900 — Bianchi publishes the differential identities (Bianchi identities) for the curvature tensor.
1915 — Einstein uses Riemannian (Lorentzian) geometry as the language of general relativity, with geodesics modeling free fall and the field equations relating curvature to matter.
1917 — Levi-Civita defines parallel transport, giving geometric meaning to the connection and clarifying the role of curvature as holonomy.
1926 — Élie Cartan reformulates the subject with moving frames and differential forms; introduces symmetric spaces and the modern global perspective.
1931 — Hopf & Rinow prove the equivalence of geodesic completeness, metric completeness, and the geodesic-connectedness of complete manifolds.

Subtleties & Limits

Even with a complete classical theory, deep questions persist. The closed-geodesic problem: does every closed Riemannian manifold admit infinitely many distinct prime closed geodesics? This is known for surfaces (Bangert–Franks, 1992, via dynamics) and many cases, but remains open in general for higher dimensions, e.g. it is not fully settled for all metrics on $S^{n}$ , $n \geq 3$ . Spectral rigidity / "Can one hear the shape of a drum?" asks how much of the metric is determined by the Laplace spectrum; isospectral non-isometric manifolds exist (Milnor 1964, Gordon–Webb–Wolpert 1992), so the metric is not fully recoverable from eigenvalues. The boundary rigidity and geodesic X-ray transform problems — whether boundary distance data determine the interior metric — are central in inverse problems and only partially resolved (Pestov–Uhlmann for simple surfaces). In optimal transport and metric measure spaces, extending Ricci curvature bounds to non-smooth spaces (the Lott–Sturm–Villani synthetic theory) actively generalizes geodesic geometry beyond manifolds.