Geodesic Completeness and Cut Loci

Geodesic completeness, metric completeness, extendibility, conjugacy, and cut points answer different questions. Hopf–Rinow links several of them for connected Riemannian manifolds because a positive-definite metric produces a genuine distance. A Lorentzian metric does not, so the equivalence does not transfer unchanged: even compact Lorentzian manifolds can be geodesically incomplete.

Required background. Affine Connections and Covariant Derivatives supplies the general connection and covariant-derivative framework; Affine and Variational Geodesics supplies the affine geodesic equation and maximal-solution language. The Riemann Curvature Tensor supplies the curvature operator entering the Jacobi analysis used to detect conjugate points.

Helpful background. Calculus and Linear Algebra for Relativity supplies coordinate differentiation and linear maps; this page develops the ODE-extension and metric-space distinctions it needs. Presymplectic Potentials and Currents is an advanced field-space construction and is not assumed in the completeness or cut-locus discussion.

Affine and metric completeness

A connection is geodesically complete if every affinely parametrized maximal geodesic is defined for all $\lambda\in\mathbb R$. For a pseudo-Riemannian metric one may specify timelike, null, or spacelike completeness separately. The definition concerns affine parameter, not coordinate range. A curve that reaches a chart boundary at finite parameter may continue in another chart.

A Riemannian metric also defines a distance

$$d(p,q)=\inf_\gamma L(\gamma),$$

and metric completeness means every $d$-Cauchy sequence converges. A Lorentzian metric does not define a distance this way: null curves have zero length, timelike length is not positive definite, and the infimum construction degenerates. Any auxiliary Riemannian distance describes the manifold topology, not Lorentzian geodesic completeness.

Extendibility is different again. A geodesically incomplete manifold may embed isometrically as a proper open subset of a larger manifold, as the punctured Euclidean plane does after its missing point is restored. Other incomplete geometries may be inextendible in a specified regularity class. Incompleteness alone does not identify which case holds.

Hopf–Rinow in Riemannian geometry

For a connected finite-dimensional Riemannian manifold, the following are equivalent:

• the metric space $(M,d)$ is complete;
• closed bounded subsets are compact;
• for one, equivalently every, $p$, $\exp_p$ is defined on all of $T_pM$;
• every maximal geodesic is defined for all real affine parameters.

These conditions imply that every pair of points is joined by a length-minimizing geodesic. The proof uses positivity of length, compactness of bounded sets, and Gauss’s lemma; Calegari 2022, Theorem 6.4.1, pp. 51–52 states and proves this equivalence.

The conclusion “each pair is joined by some geodesic” is weaker than completeness. An open Euclidean ball is incomplete, yet any two points are joined by a straight segment contained in the ball. Thus even within Riemannian geometry, reversing only the final implication is invalid.

Cut loci and conjugate points

In a complete Riemannian manifold, follow a unit-speed geodesic $\gamma_v$ from $p$. Its cut time is the supremum of lengths for which the segment remains minimizing. The endpoint at finite cut time belongs to the cut locus of $p$. Loss of minimality can occur because another minimizing geodesic reaches the same point, because a conjugate point is reached, or through a limiting combination of these mechanisms.

On the round sphere $S^n_a$ with $n\ge2$, the cut locus of $p$ is the antipode. The cut time is $\pi a$, which is also the first conjugate time, while the manifold is geodesically complete: every great circle continues forever. The semicircle ending at the antipode still minimizes; extending it beyond the cut point loses that property. This proves that a cut point or conjugate point is not geodesic incompleteness.

For a fully explicit comparison, take the flat cylinder with metric $dz^2+a^2d\phi^2$, where $z\in\mathbb R$ and $\phi$ is periodic with period $2\pi$. Lift it to the Euclidean plane with coordinates $(z,a\phi)$. From $p=(0,0)$ to $(z,\phi)$, different winding lifts have lengths

$$L_k=\sqrt{z^2+a^2(\phi+2\pi k)^2},\qquad k\in\mathbb Z.$$

At $\phi=\pi$ the lifts $k=0,-1$ give two equally short segments. Thus the cut locus is the opposite generator $\{(z,\pi):z\in\mathbb R\}$. Yet $R=0$ and a Jacobi field vanishing initially has parallel components $J(\lambda)=\lambda B$, so it cannot vanish again unless $B=0$. There are no conjugate points. Global competition between paths has produced the cut locus without curvature focusing.

In a complete Riemannian manifold the first cut time is always no later than the first conjugate time, if the latter exists. Otherwise a still-minimizing segment would contain an interior conjugate point, contradicting the negative second variation constructed on the index-form page. The general cut-point alternatives are stated in Grant 2009, Proposition 7.20, p. 23, PDF.

Why Lorentzian geometry differs

There is no Lorentzian Hopf–Rinow theorem with the same equivalences. Compactness does not ensure affine completeness: the Clifton–Pohl torus is a compact Lorentzian surface with incomplete geodesics, a standard warning discussed in Burtscher 2023, §1. Indefinite signature permits geodesic velocities to escape in the tangent bundle even while their base points remain in a compact manifold.

Lorentzian time separation does have maximizing theorems under additional causal and compactness hypotheses, but those belong to global causality rather than a signature-blind replacement for Hopf–Rinow. Timelike, null, and spacelike completeness need not agree. Likewise, singularity theorems infer causal geodesic incompleteness only after global causal, focusing, and energy assumptions; they do not equate incompleteness with divergent curvature.

The safe procedure is to state the causal class, affine normalization, maximal domain, extension regularity, and global hypotheses separately. Cut behavior is about optimality; conjugacy is about the exponential-map differential; completeness is about indefinite continuation; curvature singularity and inextendibility require further diagnostics.

Exercises

Classify the antipode of $S^n_a$ for $n\ge2$ using the distinctions on this page. What changes for the circle $S^1_a$?

Solution

For $S^n_a$, every geodesic extends for all affine parameter, so the manifold is geodesically and metrically complete. At distance $\pi a$ from $p$, the antipode is the first cut point: semicircles minimize there, but extensions do not. It is also conjugate to $p$ because nonzero transverse Jacobi fields vanish at $0$ and $\pi a$. The point is smooth and has finite constant curvature, so it is not a curvature singularity. On $S^1_a$, the two semicircles still meet at a cut point and all geodesics remain complete, but there is no transverse Jacobi direction and therefore no conjugate point.

References

• Annegret Burtscher, “A Conformal Hopf–Rinow Theorem for Semi-Riemannian Spacetimes,” arXiv:2309.08682, 2023. arXiv §1.
• Danny Calegari, Introduction to Differential Geometry, preliminary version, May 26, 2022, University of Chicago. PDF.
• James D. E. Grant, Global Lorentzian Geometry, University of Vienna lecture notes, 2008, corrected February 12, 2009. PDF.

Further reading

• John M. Lee, Introduction to Riemannian Manifolds, second edition, Graduate Texts in Mathematics 176, Springer, 2018. DOI.