Complete manifold

In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, there are straight paths extending infinitely in all directions.

Formally, a manifold ${\displaystyle M}$ is (geodesically) complete if for any maximal geodesic ${\displaystyle \ell :I\to M}$, it holds that ${\displaystyle I=(-\infty ,\infty )}$.^[1] A geodesic is maximal if its domain cannot be extended.

Equivalently, ${\displaystyle M}$ is (geodesically) complete if for all points ${\displaystyle p\in M}$, the exponential map at ${\displaystyle p}$ is defined on ${\displaystyle T_{p}M}$, the entire tangent space at ${\displaystyle p}$.^[1]

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

The Hopf–Rinow theorem states that ${\displaystyle (M,g)}$ is (geodesically) complete if and only if it satisfies one of the following equivalent conditions:^[2]

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

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

A simple example of a non-complete manifold is given by the punctured plane ${\displaystyle \mathbb {R} ^{2}\smallsetminus \lbrace 0\rbrace }$ (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.

If ${\displaystyle M}$ is geodesically complete, then it is not isometric to an open proper submanifold of any other Riemannian manifold. The converse does not hold.^[3]