Complete manifold

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

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

Formally, a manifold <math>M</math> is (geodesically) complete if for any maximal geodesic <math>\ell : I \to M</math>, it holds that <math>I=(-\infty,\infty)</math>.{{sfn|ps=|Lee|2018|p=131}} A geodesic is '''maximal''' if its domain cannot be extended.

Equivalently, <math>M</math> is (geodesically) complete if for all points <math>p \in M</math>, the exponential map at <math>p</math> is defined on <math>T_pM</math>, the entire tangent space at <math>p</math>.{{sfn|ps=|Lee|2018|p=131}}

== Hopf–Rinow theorem == {{Main|Hopf–Rinow theorem}}

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

The Hopf–Rinow theorem states that <math>(M,g)</math> is (geodesically) complete if and only if it satisfies one of the following equivalent conditions:{{sfn|ps=|do Carmo|1992|pp=146-147}} * The metric space <math>(M,d_g)</math> is complete (every <math>d_g</math>-Cauchy sequence converges), * All closed and bounded subsets of <math>M</math> are compact.

== Examples and non-examples ==

Euclidean space <math>\mathbb{R}^n</math>, the sphere <math>\mathbb{S}^n</math>, and the tori <math>\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 ===

thumb|The punctured plane <math>\mathbb R^2 \backslash \{(0,0)\}</math> is not geodesically complete because the maximal geodesic with initial conditions <math>p = (1,1)</math>, <math>v = (1,1)</math> does not have domain <math>\mathbb R</math>.

A simple example of a non-complete manifold is given by the punctured plane <math>\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>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.{{sfn|ps=|do Carmo|1992|p=145}}{{explain|reason=What the converse being considered here is should be made explicit.|date=July 2025}}

== 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