Complete manifold

{{Short description|Riemannian manifold in which geodesics extend infinitely in all directions}}

In mathematics, a complete manifold (or geodesically complete manifold) {{Mvar|M}} is a (pseudo-) Riemannian manifold for which, starting at any point {{Math|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 $I=(-\infty,\infty)$ .{{sfn|Lee|2018|p=131}} 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 $p$ .{{sfn|Lee|2018|p=131}}

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:{{sfn|do Carmo|1992|p=146-147}}

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 =

File:Punctured plane is not geodesically complete.svg

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.{{sfn|do Carmo|1992|p=145}}

References

= Notes =

= Sources =

{{citation

| last = do Carmo |first=Manfredo Perdigão|authorlink=Manfredo do Carmo

| title = Riemannian geometry

| series = Mathematics: theory and applications

| publisher = Birkhäuser

| location = Boston

| year = 1992

| pages = xvi+300

| isbn = 0-8176-3490-8

}}

{{Cite book|last=Lee|first=John|title=Introduction to Riemannian Manifolds|series=Graduate Texts in Mathematics|publisher=Springer International Publishing AG|year=2018}}

{{Cite book|title=Semi-Riemannian Geometry|last=O'Neill|first=Barrett|publisher=Academic Press|year=1983|isbn=0-12-526740-1|at=Chapter 3}}

Category:Differential geometry

Category:Geodesic (mathematics)

Category:Manifolds

Category:Riemannian geometry