Inmathematics, acomplete manifold (orgeodesically complete manifold)M is a (pseudo-)Riemannian manifold for which, starting at any pointp ofM, there are straight paths extending infinitely in all directions.

Formally, a manifold${\displaystyle M}$ is (geodesically) complete if for any maximalgeodesic${\displaystyle \ell :I\to M}$, it holds that${\displaystyle I=(-\mathrm{\infty },\mathrm{\infty })}$.^[1] A geodesic ismaximal if its domain cannot be extended.

Equivalently,${\displaystyle M}$ is (geodesically) complete if for all points${\displaystyle p\in M}$, theexponential map at${\displaystyle p}$ is defined on${\displaystyle T_{p}M}$, the entiretangent space at${\displaystyle p}$.^[1]

TheHopf–Rinow theorem gives alternative characterizations of completeness. Let${\displaystyle (M,g)}$ be aconnected Riemannian manifold and let${\displaystyle d_g:M\times M\to [0,\mathrm{\infty })}$ be itsRiemannian distance function.

The Hopf–Rinow theorem states that${\displaystyle (M,g)}$ is (geodesically) complete if and only if it satisfies one of the following equivalent conditions:^[2]

• The metric space${\displaystyle (M,d_g)}$ iscomplete (every${\displaystyle d_g}$-Cauchy sequence converges),
• All closed and bounded subsets of${\displaystyle M}$ arecompact.

Euclidean space${\displaystyle {\mathbb{R}}^n}$, thesphere${\displaystyle {\mathbb{S}}^n}$, and thetori${\displaystyle {\mathbb{T}}^n}$ (with their naturalRiemannian metrics) are all complete manifolds.

Allcompact Riemannian manifolds and allhomogeneous manifolds are geodesically complete. Allsymmetric spaces are geodesically complete.

A simple example of a non-complete manifold is given by the punctured plane${\displaystyle {\mathbb{R}}^2\setminus \{0\}}$ (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 theClifton–Pohl torus.

In the theory ofgeneral 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 aBig Bang. The fact that such incompleteness is fairly generic in general relativity is shown in thePenrose–Hawking singularity theorems.

If${\displaystyle M}$ is geodesically complete, then it is not isometric to an open proper submanifold of any other Riemannian manifold. The converse does not hold.^{[3][further explanation needed]}

• do Carmo, Manfredo Perdigão (1992),Riemannian geometry, Mathematics: theory and applications, Boston: Birkhäuser, pp. xvi+300,ISBN 0-8176-3490-8
• Lee, John (2018).Introduction to Riemannian Manifolds. Graduate Texts in Mathematics. Springer International Publishing AG.
• O'Neill, Barrett (1983).Semi-Riemannian Geometry.Academic Press. Chapter 3.ISBN 0-12-526740-1.

• Differential geometry
• Geodesic (mathematics)
• Manifolds
• Riemannian geometry

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