Complete manifold

Short description: Riemannian manifold in which geodesics extend infinitely in all directions

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

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

Hopf–Rinow theorem

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

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

• The metric space [math]\displaystyle{ (M,d_g) }[/math] is complete (every [math]\displaystyle{ d_g }[/math]-Cauchy sequence converges),
• All closed and bounded subsets of [math]\displaystyle{ M }[/math] are compact.

Examples and non-examples

Euclidean space [math]\displaystyle{ \mathbb{R}^n }[/math], the sphere [math]\displaystyle{ \mathbb{S}^n }[/math], and the tori [math]\displaystyle{ \mathbb{T}^n }[/math] (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 [math]\displaystyle{ \mathbb R^2 \backslash \{(0,0)\} }[/math] is not geodesically complete because the maximal geodesic with initial conditions [math]\displaystyle{ p = (1,1) }[/math], [math]\displaystyle{ v = (1,1) }[/math] does not have domain [math]\displaystyle{ \mathbb R }[/math].

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

• Riemannian geometry, Mathematics: theory and applications, Boston: Birkhäuser, 1992, 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.