Complete manifold explained

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 is (geodesically) complete if for any maximal geodesic \ell : I \to M , it holds that I = ( -inf ty, inf ty) . 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

The Hopf–Rinow theorem gives alternative characterizations of completeness. Let

(M,g) be a connected Riemannian manifold and let d _{g} : M x M \to [ 0 , inf ty) 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

R ^{n} , the sphere S ^{n} , and the tori 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

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

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

Sources

• • Book: Lee, John. Introduction to Riemannian Manifolds. Graduate Texts in Mathematics. Springer International Publishing AG. 2018.
• Book: O'Neill, Barrett. Semi-Riemannian Geometry. Academic Press. 1983. 0-12-526740-1. Chapter 3.