Riemannian Geometry and Geodesics | Metrics, Length, Distance and Shortest Paths

Riemannian geometry begins when a smooth manifold is given a rule for measuring tangent vectors. From that one rule come length, angle, volume, geodesics, distance and a precise language for curvature.

A smooth manifold by itself knows how to differentiate, but it does not know whether one tangent vector is longer than another. It does not know whether two directions are perpendicular, how far two points are apart, or which path is shortest. A Riemannian metric supplies those measurements point by point while remaining compatible with the smooth structure.

This guide continues the differential-geometry route built by Curves and Surfaces, Smooth Manifolds and Maps, Tangent, Cotangent and Tensor Fields and Differential Forms and Integration. Here the metric becomes the organising object.

Level: undergraduate differential geometry and advanced enrichment. Prerequisites are multivariable calculus, linear algebra, smooth manifolds, tangent vectors and basic differential forms. This is not a school syllabus checklist. Use the BTT Mathematics Hub for stage-specific learning.

Reading route: Riemannian metrics → length and distance → Levi-Civita geometry → geodesics → worked models → completeness → Jacobi fields → practice → solutions.

1. A Riemannian metric is a smoothly varying inner product

On a smooth n-manifold M, a Riemannian metric g assigns to every point p a positive-definite inner product gₚ on TₚM. Smoothness means that if X and Y are smooth vector fields, then the function p↦gₚ(Xₚ,Yₚ) is smooth.

Positive definiteness gives gₚ(v,v)>0 for every nonzero tangent vector v. The norm is ‖v‖₍g₎=√g(v,v), and the angle between nonzero vectors is defined by cosθ=g(v,w)/(‖v‖‖w‖).

In local coordinates x¹,…,xⁿ, write gᵢⱼ=g(∂ᵢ,∂ⱼ). The matrix (gᵢⱼ) is symmetric and positive definite. For v=vⁱ∂ᵢ, its squared norm is gᵢⱼvⁱvʲ, with summation over repeated indices.

The matrix is a coordinate representation, not the metric itself. Under a change of coordinates its entries transform as a type (0,2) tensor, preserving the scalar g(v,w). This is exactly the distinction developed in the preceding tensor guide.

A concise university route through metrics, distance, connections and geodesics appears in the Georgia Tech differential-geometry lecture notes index. A current graduate-level syllabus with metrics, geodesics, connections and curvature is also listed by CUHK Riemannian Geometry I.

2. Euclidean space is one Riemannian model, not the definition

On Rⁿ with standard coordinates, the Euclidean metric is g=Σ(dxⁱ)². Its matrix is the identity. Tangent vectors have their familiar lengths and dot products.

But a manifold can carry many different metrics. On R², g=dx²+4dy² is a valid Riemannian metric. A tangent vector (a,b) has squared length a²+4b². The underlying smooth plane is unchanged; the geometry used to measure it has changed.

More generally, g=e²ᶠ(x,y)(dx²+dy²) is a conformal metric. It preserves Euclidean angles but rescales all lengths at a point by eᶠ. If f varies, two equal coordinate displacements can have different physical lengths at different locations.

This is why the phrase “the manifold is curved” needs context. Curvature belongs to a metric geometry. The same differentiable manifold may support metrics with very different curvature behaviour.

3. An embedded surface inherits a metric from Euclidean space

Let X(u,v) be a regular surface patch in R³. The ambient Euclidean dot product restricts to the tangent plane. In coordinates, the induced metric is the first fundamental form I=E du²+2F du dv+G dv², where E=Xᵤ·Xᵤ, F=Xᵤ·Xᵥ and G=Xᵥ·Xᵥ.

Thus the classical surface calculation is already Riemannian geometry. The metric contains exactly the tangent measurements available to an observer constrained to the surface.

For the cylinder X(u,v)=(R cosu,R sinu,v), the metric is R²du²+dv². Setting x=Ru and y=v makes it locally dx²+dy². This explains the familiar fact that a cylinder can be unrolled locally without changing lengths.

For the sphere X(θ,φ)=(R sinθ cosφ,R sinθ sinφ,R cosθ), the metric is R²dθ²+R²sin²θ dφ². The shrinking coefficient of dφ² records the convergence of longitude directions near a pole.

4. The metric turns tangent speed into curve length

For a piecewise smooth curve γ:[a,b]→M, its Riemannian length is L(γ)=∫ₐᵇ√g(γ′,γ′)dt. This definition is invariant under orientation-preserving reparametrisation and unchanged by reversing direction.

The formula depends on the metric. For g=dx²+4dy², the straight coordinate segment γ(t)=(t,t), 0≤t≤1, has speed √5 and length √5. In the Euclidean metric it would have length √2.

A curve can be reparametrised by arc length when its speed is nonzero. Then g(γ′,γ′)=1. Unit-speed parametrisations remove the artificial effect of moving through the same geometry at different clock rates.

Length is nonnegative and additive under concatenation. It is not usually equal to the norm of a coordinate displacement, because a manifold need not have one global coordinate vector connecting two points and the metric may vary along the route.

5. Distance is defined by an optimisation over curves

For points p and q in the same connected component, define d(p,q) as the infimum of the lengths of piecewise smooth curves joining p to q. This is the Riemannian distance.

It satisfies symmetry, nonnegativity and the triangle inequality. The nontrivial point is positive separation: distinct points have positive Riemannian distance. Locally, a smooth positive-definite metric is comparable to the Euclidean metric, preventing arbitrarily short curves between distinct nearby points.

The metric topology induced by d agrees with the original manifold topology. Thus the Riemannian metric enriches the manifold without changing which sets are open.

The infimum need not automatically be attained. A length-minimising curve is a stronger object than a sequence of curves whose lengths merely approach the infimum. Completeness will later supply an important existence theorem.

6. Geodesics are not defined as globally shortest paths

A geodesic is a curve whose tangent vector is parallel along itself: ∇_{γ′}γ′=0 for the Levi-Civita connection. Equivalently, in local coordinates it satisfies a second-order differential equation.

Geodesics are locally straight with respect to the connection. Sufficiently short geodesic segments minimise length locally, but a geodesic can fail to be globally shortest after travelling far enough.

On a round sphere, every great circle is a geodesic. Between non-antipodal points, the shorter great-circle arc is distance-minimising. The longer arc is also geodesic but not minimising. Between antipodal points, infinitely many great semicircles are minimisers.

This distinction is essential: “geodesic” is a differential equation; “global minimiser” is an optimisation property over all competing curves.

7. The Levi-Civita connection is the metric’s preferred derivative

A connection ∇ differentiates vector fields while respecting the vector-bundle structure. A Riemannian metric determines a unique connection satisfying two conditions: it is torsion-free and metric-compatible. This is the Levi-Civita connection.

Metric compatibility means X[g(Y,Z)]=g(∇ₓY,Z)+g(Y,∇ₓZ). Parallel transport therefore preserves inner products. Torsion-free means ∇ₓY−∇ᵧX=[X,Y].

These two requirements determine the connection through the Koszul formula. In coordinates the connection is represented by Christoffel symbols Γᵏᵢⱼ. For the Levi-Civita connection,

Γᵏᵢⱼ=(1/2)gᵏˡ(∂ᵢgⱼₗ+∂ⱼgᵢₗ−∂ₗgᵢⱼ).

Christoffel symbols are not tensor components. They can vanish at a point in suitable coordinates even when curvature there is nonzero. Their job is to correct ordinary coordinate derivatives so that the resulting covariant derivative has geometric meaning.

For a broader connection-focused treatment, see Chris Wendl’s lecture notes on bundles and connections, which separate connections, curvature on bundles and curvature in Riemannian geometry.

8. The geodesic equation comes from parallel velocity

Write γ(t) in local coordinates xᵏ(t). Then ∇_{γ′}γ′=0 becomes

ẍᵏ+Γᵏᵢⱼẋⁱẋʲ=0.

This is a second-order nonlinear ODE. A point p and initial velocity v determine a unique local geodesic. The speed of an affinely parametrised geodesic is constant because metric compatibility gives d[g(γ′,γ′)]/dt=2g(∇_{γ′}γ′,γ′)=0.

If the parameter is changed nonlinearly, the same trace may no longer satisfy this equation. An affine change t↦at+b preserves geodesic parametrisation. Thus the trace and the distinguished affine parameter should be kept separate.

In Euclidean Cartesian coordinates all Christoffel symbols vanish, so ẍᵏ=0 and geodesics are straight lines with constant velocity. The curved-coordinate corrections appear when the basis changes from point to point or when the metric itself varies.

9. Worked model: Euclidean polar coordinates

In polar coordinates on the Euclidean plane, g=dr²+r²dθ². The inverse metric is diag(1,1/r²). Computing from the metric gives Γʳ_{θθ}=−r and Γ^θ_{rθ}=Γ^θ_{θr}=1/r, with the other basic symbols zero.

The geodesic equations are r̈−rθ̇²=0 and θ̈+2(ṙ/r)θ̇=0. These equations look nonlinear even though the underlying plane is flat.

A radial line θ=θ₀ has θ̇=0, so the first equation reduces to r̈=0. Thus constant-speed radial lines are geodesics.

A coordinate circle r=R with θ̇≠0 is not a geodesic because the radial equation would require −Rθ̇²=0. This matches Euclidean intuition: moving around a circle requires continuous inward acceleration.

The nonzero Christoffel symbols therefore do not imply intrinsic curvature. They can arise solely from curvilinear coordinates on a flat space.

10. Worked model: great circles on a sphere

For the round sphere of radius R, use coordinates θ,φ with metric R²dθ²+R²sin²θ dφ². Relevant Christoffel symbols include Γ^θ_{φφ}=−sinθ cosθ and Γ^φ_{θφ}=Γ^φ_{φθ}=cotθ.

The geodesic equations are θ̈−sinθcosθ φ̇²=0 and φ̈+2cotθ θ̇φ̇=0.

The equator θ=π/2 with constant φ̇ satisfies both equations because cosθ=0 and θ̇=0. Hence the equator is a geodesic.

A latitude θ=θ₀ different from π/2 with nonzero φ̇ fails the first equation because sinθ₀cosθ₀ is nonzero. So ordinary latitude circles are not geodesics, even though they look geometrically simple in spherical coordinates.

Every great circle is obtained by rotating the equator through an ambient Euclidean rotation, and rotations are isometries of the round metric. Isometries carry geodesics to geodesics, so every great circle is geodesic.

11. The energy functional produces the same geodesic equation

For a curve γ:[a,b]→M, define its energy E(γ)=(1/2)∫ₐᵇg(γ′,γ′)dt. The length is invariant under increasing reparametrisation; energy is not. Energy prefers constant-speed parametrisations.

For fixed endpoints, critical points of the energy functional are geodesics. In coordinates, the Euler–Lagrange equations for L=(1/2)gᵢⱼẋⁱẋʲ reproduce the geodesic equation.

Cauchy–Schwarz gives L(γ)²≤2(b−a)E(γ), with equality precisely when speed is constant almost everywhere. Thus a length minimiser can be reparametrised to constant speed and then becomes an energy minimiser among suitable competitors.

This variational viewpoint is important because later geometric analysis studies critical points of functionals on spaces of curves, maps and metrics rather than solving one coordinate ODE at a time.

12. The exponential map packages all geodesics from one point

Fix p∈M. For a tangent vector v for which the geodesic γᵥ with γᵥ(0)=p and γᵥ′(0)=v is defined up to time 1, define expₚ(v)=γᵥ(1).

Near v=0, the exponential map is a diffeomorphism from a neighbourhood of the origin in TₚM to a neighbourhood of p. The derivative d(expₚ)₀ is the identity after the natural identification T₀(TₚM)≈TₚM.

Coordinates obtained from expₚ are called normal coordinates. At p, the metric matrix becomes the identity after choosing an orthonormal basis, and the Christoffel symbols vanish. First-order metric variation can be removed at one point.

Curvature generally remains in the second-order behaviour. A coordinate system can remove connection coefficients at a chosen point, but it cannot remove genuine curvature over a whole neighbourhood when curvature is nonzero.

13. Minimising geodesics can stop minimising

For a point p, a cut point along a geodesic is roughly where that geodesic ceases to be the unique distance-minimising continuation from p. The cut locus collects such points.

On a round sphere, the cut locus of p is its antipode. Geodesics from p minimise until they reach the antipode, where infinitely many minimising great semicircles arrive.

A conjugate point is detected infinitesimally: nearby geodesics issued from p focus so that d(expₚ) loses invertibility. Conjugate points and cut points are related but not identical notions.

This is the deeper reason a geodesic need not remain shortest forever. Local straightness does not provide unlimited global optimality.

14. Completeness connects geodesics, metric distance and compactness

A Riemannian manifold is geodesically complete if every geodesic can be extended for all real parameter values. It is metrically complete if every Cauchy sequence converges with respect to the Riemannian distance.

The Hopf–Rinow theorem states that for a connected finite-dimensional Riemannian manifold, metric completeness, geodesic completeness and several compactness/minimising-geodesic properties are equivalent.

In particular, on a complete connected Riemannian manifold, any two points can be joined by a distance-minimising geodesic. Also, closed and bounded subsets are compact in the Riemannian distance.

The open unit disc with the Euclidean metric is not complete. The radial sequence pₙ=(1−1/n,0) is Cauchy but converges to a boundary point absent from the manifold. A radial geodesic reaches the missing boundary in finite length and cannot be continued within the disc.

The same underlying open disc can be given a complete hyperbolic metric whose boundary is infinitely far away. Completeness therefore belongs to the chosen metric, not merely to the point set or topology.

15. Isometries preserve the complete metric structure

A diffeomorphism F:(M,g)→(N,h) is an isometry when F*g=h. Equivalently, h(dFv,dFw)=g(v,w) for tangent vectors at every point.

An isometry preserves lengths of curves, Riemannian distance, angles, volume, the Levi-Civita connection, geodesics and curvature tensors. It is far stronger than a diffeomorphism.

A local isometry satisfies the metric-preservation condition locally but need not be globally one-to-one. The covering map R→S¹ with the standard metrics, t↦e^{it}, is the model: it preserves local length while wrapping the source around the circle repeatedly.

Thus local metric agreement does not automatically determine global topology. Geometry and topology interact, but neither simply replaces the other.

16. Jacobi fields measure how nearby geodesics separate

Consider a smooth one-parameter family of geodesics γₛ(t). The variation field J(t)=∂γₛ/∂s at s=0 satisfies the Jacobi equation

∇ₜ²J+R(J,γ′)γ′=0.

Here R is the Riemann curvature tensor. This equation describes the infinitesimal relative acceleration of neighbouring geodesics. Curvature enters as the coefficient controlling their focusing or spreading.

In Euclidean space R=0, so in parallel coordinates Jacobi fields are affine in t: nearby straight lines separate linearly. On a sphere of radius R, perpendicular Jacobi fields along a unit-speed geodesic satisfy j″+j/R²=0 and can refocus after a finite distance.

Zeros of a nontrivial Jacobi field with J(0)=0 signal conjugate points. At such points the exponential map loses local one-to-one differential behaviour in the corresponding direction.

Jacobi fields convert curvature from an abstract tensor into an observable statement about families of geodesics. They are also central to second-variation tests for whether a geodesic minimises length.

17. Sectional curvature is curvature assigned to a tangent 2-plane

For a 2-plane σ=span{u,v} in TₚM, its sectional curvature is K(σ)=g(R(u,v)v,u)/(g(u,u)g(v,v)−g(u,v)²), using one common sign convention for R.

On a two-dimensional surface there is only one tangent 2-plane at each point, so sectional curvature reduces to Gaussian curvature. On higher-dimensional manifolds, different tangent 2-planes through the same point can have different sectional curvatures.

The round sphere of radius R has constant sectional curvature 1/R². Euclidean space has zero. Hyperbolic space of curvature scale R has constant negative sectional curvature −1/R².

The sign influences geodesic behaviour but should not be reduced to a slogan. Positive curvature tends to promote focusing, negative curvature tends to promote spreading, and zero curvature is the flat benchmark. Precise conclusions require hypotheses and comparison theorems.

18. Ricci and scalar curvature are contractions, not replacements for the full tensor

The Riemann tensor contains detailed directional curvature information. Contracting one input-output pair gives the Ricci tensor. Taking the metric trace of Ricci gives scalar curvature.

These contractions preserve less information. In two dimensions the curvature tensor is determined by Gaussian curvature, but in higher dimensions equal scalar curvature does not imply equal sectional curvature or local isometry.

Ricci curvature appears in volume comparison, geodesic focusing and geometric evolution equations. Scalar curvature appears in global integral formulas and variational problems. They serve particular jobs; neither should be described as “the curvature” without context.

The next guide on connections and curvature will build the curvature tensor directly from covariant derivatives and parallel transport, including sign-convention checks.

19. A dependable Riemannian calculation workflow

First state the manifold, coordinate domain and metric. Check positive definiteness. Compute the inverse metric before raising indices. If geodesics are required, derive only the Christoffel symbols that actually enter the chosen coordinates.

Keep local and global claims separate. A geodesic equation is local differential information. Distance-minimising behaviour needs a comparison with competing curves. Global existence may require completeness.

Use invariant checks whenever possible: constant speed along an affine geodesic, conservation laws produced by symmetries, known flatness in Euclidean coordinates, or agreement between a direct length calculation and a coordinate change.

Finally, record the convention for R before comparing curvature signs across sources. The sign of the Riemann tensor is not universal across textbooks, even though geometric consequences are consistent once the convention is translated correctly.

20. Independent practice: fourteen questions

1. For g=dx²+4dy² on R², find the length of γ(t)=(t,2t), 0≤t≤1.
2. For g=e²x(dx²+dy²), find the length of the segment γ(t)=(t,0), 0≤t≤a with a>0.
3. In Euclidean polar coordinates, verify Γʳ_{θθ}=−r and Γ^θ_{rθ}=1/r.
4. Show that θ=constant, r=at+b is a geodesic where r>0.
5. Explain why r=R, θ=ct with c≠0 is not a Euclidean geodesic.
6. For the unit sphere, verify that the equator θ=π/2, φ=ct satisfies the geodesic equations.
7. For the sphere of radius R, what is the geodesic distance between two points separated by central angle α with 0≤α≤π?
8. Give an example of a geodesic segment that is not globally distance-minimising.
9. Show that affine reparametrisation preserves the geodesic equation but a general nonlinear reparametrisation need not.
10. Why is the open Euclidean unit disc incomplete? Give both a Cauchy-sequence and geodesic interpretation.
11. In Rⁿ with the Euclidean metric, compute expₚ(v).
12. Along a Euclidean unit-speed geodesic, solve J″=0 with J(0)=0 and J(1)=w.
13. What are the sectional, Ricci and scalar curvatures of the round 2-sphere of radius R under the convention K=1/R²?
14. Explain why vanishing Christoffel symbols at one point do not prove zero curvature there.

21. Worked solutions and checks

1. The velocity is (1,2). Its squared norm is 1²+4·2²=17, so the constant speed is √17 and the length is √17.

2. Along γ, the metric factor is e²t and the coordinate velocity is (1,0), so speed is eᵗ. The length is ∫₀ᵃeᵗdt=eᵃ−1.

3. For g_rr=1 and g_θθ=r², the inverse entries are 1 and 1/r². The only relevant derivative is ∂ᵣg_θθ=2r. Substitution gives Γʳ_{θθ}=−r and Γ^θ_{rθ}=Γ^θ_{θr}=1/r.

4. With θ̇=0, both angular terms vanish. The radial equation becomes r̈=0, which r=at+b satisfies. The angular equation is also zero. The restriction r>0 keeps the chosen polar chart valid.

5. If r=R and θ̇=c, then r̈=0 but the radial geodesic equation gives −Rc²=0, impossible for R>0 and c≠0.

6. At θ=π/2, cosθ=0 and θ̇=0. Hence θ̈−sinθcosθ φ̇²=0. Also cot(π/2)=0, so the second equation reduces to φ̈=0, satisfied by φ=ct.

7. The shorter great-circle arc has length Rα. For α≤π it is minimising, so d=Rα.

8. On a round sphere, take a great-circle arc longer than a semicircle. It remains a geodesic but the complementary shorter arc joins the same endpoints with less length.

9. If s=at+b, then d²x/ds² and the quadratic velocity term both gain the same factor 1/a², preserving the zero equation. A nonlinear parameter introduces an additional term proportional to the velocity, so the affine geodesic equation is generally lost.

10. The sequence (1−1/n,0) is Euclidean Cauchy but its limit is missing. The radial unit-speed geodesic γ(t)=(t,0), 0≤t<1, reaches the absent boundary after finite parameter time and cannot be extended within the disc.

11. Euclidean geodesics are γ(t)=p+tv. Hence expₚ(v)=p+v.

12. J(t)=tw satisfies J″=0, J(0)=0 and J(1)=w. Nearby Euclidean geodesics therefore separate linearly at first order.

13. The unique sectional curvature is 1/R². In two dimensions Ric=(1/R²)g, and scalar curvature is 2/R².

14. Normal coordinates can make Γᵏᵢⱼ(p)=0 at a chosen point. Curvature depends on derivatives and quadratic combinations of the connection coefficients, so those values can vanish while curvature remains nonzero. The sphere provides the standard example.

22. Questions that separate the concepts

Does every geodesic minimise distance?

No. Geodesics are locally straight in the connection sense. Short enough segments minimise locally, but global minimisation can fail past cut or conjugate behaviour.

Does every distance-minimising curve satisfy the geodesic equation?

A smooth interior length minimiser, parametrised at constant speed, is a geodesic. At nonsmooth boundaries, corners, obstacles or constrained spaces, the relevant theorem and class of allowed curves must be stated separately.

Can the same manifold be complete with one metric and incomplete with another?

Yes. Completeness is metric-dependent. The open disc with the Euclidean metric is incomplete, while standard hyperbolic metrics make its ideal boundary infinitely distant.

What is the next conceptual step?

The next guide will stop treating the connection as a supplied tool and build its curvature directly. Parallel transport around loops will reveal curvature operationally, and the Bianchi identities will show how the curvature tensor is constrained.

Sources and further study

The Georgia Tech differential-geometry lecture notes provide a structured route through metrics, distance, connections, geodesics and the exponential map. Chris Wendl’s Bundles and Connections notes separate bundle connections from Riemannian curvature. The CUHK Riemannian Geometry I course description gives a current graduate-level scope including affine connections, geodesics, curvature and global theory.

Continue to connections and curvature for covariant derivatives, parallel transport, Riemann curvature and holonomy. Return to the BTT Mathematics Hub.