Complete manifold

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 is (geodesically) complete if for any maximal geodesic \ell : I \to M, it holds that I=(-\infty,\infty).^[1] 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.^[1]

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 \times M \to [0,\infty) 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:^[2]

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

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

A simple example of a non-complete manifold is given by the punctured plane \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 is geodesically complete, then it is not isometric to an open proper submanifold of any other Riemannian manifold. The converse does not hold.^[3]

References

Notes

^ Lee 2018, p. 131.
^ do Carmo 1992, pp. 146–147.
^ do Carmo 1992, p. 145.

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.