In mathematics, a complete manifold (or geodesically complete manifold) is a (

pseudo The prefix pseudo- (from Greek ψευδής, ''pseudes'', "false") is used to mark something that superficially appears to be (or behaves like) one thing, but is something else. Subject to context, ''pseudo'' may connote coincidence, imitation, ...

-) Riemannian manifold for which, starting at any point , you can follow a "straight" line indefinitely along any direction. More formally, the exponential map at point , is defined on , the entire tangent space at . Equivalently, consider a maximal geodesic

\ell\colon I\to M

. Here

I

is an open interval of

\mathbb

, and, because geodesics are parameterized with "constant speed", it is uniquely defined up to transversality. Because

I

is maximal,

\ell

maps the

ends End, END, Ending, or variation, may refer to: End *In mathematics: **End (category theory) ** End (topology) **End (graph theory) ** End (group theory) (a subcase of the previous) **End (endomorphism) *In sports and games ** End (gridiron footbal ...

I

to points of , and the length of

I

measures the distance between those points. A manifold is geodesically complete if for any such geodesic

\ell

, we have that

I=(-\infty,\infty)

Examples and non-examples

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

\mathbb^n

, the

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...

\mathbb^n

, and the tori

\mathbb^n

(with their natural

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...

s) are all complete manifolds. All

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

Riemannian manifolds and all homogeneous manifolds are geodesically complete. All

symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...

s are geodesically complete. Every finite-dimensional path-connected Riemannian manifold which is also a

complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...

(with respect to the Riemannian distance) is geodesically complete. In fact, geodesic completeness and metric completeness are equivalent for these spaces. This is the content of the

Hopf–Rinow theorem Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931. Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem ...

Non-examples

A simple example of a non-complete manifold is given by the punctured plane

\mathbb^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 geometry, the Clifton–Pohl torus is an example of a compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete (by the Hopf–Rinow theorem), this space shows that the ...

. In the theory of

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

, 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 The Penrose–Hawking singularity theorems (after Roger Penrose and Stephen Hawking) are a set of results in general relativity that attempt to answer the question of when gravitation produces singularities. The Penrose singularity theorem is ...

References

* {{DEFAULTSORT:Complete Manifold Differential geometry Geodesic (mathematics) Manifolds Riemannian geometry