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 {\displaystyle M} is (geodesically) complete if for any maximal geodesic ℓ : I → M {\displaystyle \ell :I\to M}, it holds that I = ( − ∞ , ∞ ) {\displaystyle I=(-\infty ,\infty )}. A geodesic is maximal if its domain cannot be extended.

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

Hopf–Rinow theorem

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

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

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

Examples and non-examples

Euclidean space R n {\displaystyle \mathbb {R} ^{n}}, the sphere S n {\displaystyle \mathbb {S} ^{n}}, and the tori T n {\displaystyle \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

The punctured plane R 2 ∖ { ( 0 , 0 ) } {\displaystyle \mathbb {R} ^{2}\backslash \{(0,0)\}} is not geodesically complete because the maximal geodesic with initial conditions p = ( 1 , 1 ) {\displaystyle p=(1,1)}, v = ( 1 , 1 ) {\displaystyle v=(1,1)} does not have domain R {\displaystyle \mathbb {R} }.

A simple example of a non-complete manifold is given by the punctured plane R 2 ∖ { 0 } {\displaystyle \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 {\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.[further explanation needed]

Notes

Sources

  • 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.