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Complete manifold

Riemannian manifold in which geodesics extend infinitely in all directions

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Source updated July 8, 2025

In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p of M, there are straight paths extending infinitely in all directions.

Formally, a manifold $M$ is (geodesically) complete if for any maximal geodesic $\ell :I\to M$ , it holds that $I=(-\infty ,\infty )$ . A geodesic is maximal if its domain cannot be extended.

Equivalently, $M$ is (geodesically) complete if for all points $p\in M$ , the exponential map at $p$ is defined on $T_{p}M$ , the entire tangent space at $p$ .

01Hopf-Rinow theorem

The Hopf-Rinow theorem gives alternative characterizations of completeness. Let $(M,g)$ be a connected Riemannian manifold and let $d_{g}:M\times M\to [0,\infty )$ be its Riemannian distance function.

The Hopf-Rinow theorem states that $(M,g)$ is (geodesically) complete if and only if it satisfies one of the following equivalent conditions:

The metric space $(M,d_{g})$ is complete (every $d_{g}$ -Cauchy sequence converges),
All closed and bounded subsets of $M$ are compact.

02Examples and non-examples

Euclidean space $\mathbb {R} ^{n}$ , the sphere $\mathbb {S} ^{n}$ , and the tori $\mathbb {T} ^{n}$ (with their natural Riemannian metrics) are all complete manifolds.

All compact Riemannian manifolds and all homogeneous manifolds are geodesically complete. All symmetric spaces are geodesically complete.

Non-examples

A simple example of a non-complete manifold is given by the punctured plane $\mathbb {R} ^{2}\smallsetminus \lbrace 0\rbrace$ (with its induced metric). Geodesics going to the origin cannot be defined on the entire real line. By the Hopf-Rinow theorem, we can alternatively observe that it is not a complete metric space: any sequence in the plane converging to the origin is a non-converging Cauchy sequence in the punctured plane.

There exist non-geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds. An example of this is the Clifton-Pohl torus.

In the theory of general relativity, which describes gravity in terms of a pseudo-Riemannian geometry, many important examples of geodesically incomplete spaces arise, e.g. non-rotating uncharged black-holes or cosmologies with a Big Bang. The fact that such incompleteness is fairly generic in general relativity is shown in the Penrose-Hawking singularity theorems.

03Extendibility

If $M$ is geodesically complete, then it is not isometric to an open proper submanifold of any other Riemannian manifold. The converse does not hold.

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Sources and credits

This article is adapted from the Wikipedia article “Complete manifold”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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