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 $$M$$ is (geodesically) complete if for any maximal geodesic $$\ell : I \to M$$, it holds that $$I=(-\infty,\infty)$$. 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 $$d_g : M \times M \to [0,\infty)$$ 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:
 * 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.

Extendiblity
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.