Complete manifold

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title: "Complete manifold" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["differential-geometry", "geodesic-(mathematics)", "manifolds", "riemannian-geometry"] description: "Riemannian manifold in which geodesics extend infinitely in all directions" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Complete_manifold" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Riemannian manifold in which geodesics extend infinitely in all directions ::

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_pM, the entire tangent space at p.

Hopf–Rinow theorem

Main article: Hopf–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.

Examples 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

::figure[src="https://upload.wikimedia.org/wikipedia/commons/7/72/Punctured_plane_is_not_geodesically_complete.svg" caption="The punctured plane \mathbb R^2 \backslash {(0,0)} is not geodesically complete because the maximal geodesic with initial conditions p = (1,1), v = (1,1) does not have domain \mathbb R."] ::

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.

Extendibility

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.

References

Notes

Sources

• {{citation | last = do Carmo |first=Manfredo Perdigão |authorlink=Manfredo do Carmo | title = Riemannian geometry | series = Mathematics: theory and applications | publisher = Birkhäuser | location = Boston | year = 1992 | pages = xvi+300 | isbn = 0-8176-3490-8

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