In mathematics, a complete manifold (or geodesically complete manifold) is a (pseudo-) Riemannian manifold for which, starting at any point, there are straight paths extending infinitely in all directions.
Formally, a manifold
M
\ell:I\toM
I=(-infty,infty)
Equivalently,
M
p\inM
p
TpM
p
See main article: Hopf-Rinow theorem.
The Hopf–Rinow theorem gives alternative characterizations of completeness. Let
(M,g)
dg:M x M\to[0,infty)
The Hopf–Rinow theorem states that
(M,g)
(M,dg)
dg
M
Rn
Sn
Tn
All compact Riemannian manifolds and all homogeneous manifolds are geodesically complete. All symmetric spaces are geodesically complete.
A simple example of a non-complete manifold is given by the punctured plane
R2\smallsetminus\lbrace0\rbrace
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.
If
M