View source for Complete manifold

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

In [[HandWiki:Mathematics|mathematics]], a '''complete manifold''' (or '''geodesically complete manifold''') {{Mvar|M}} is a ([[Pseudo-Riemannian manifold|pseudo]]-) [[Riemannian manifold]] for which, starting at any point {{Math|''p''}}, there are straight paths extending infinitely in all directions.

Formally, a manifold <math>M</math> is (geodesically) complete if for any maximal [[Geodesic|geodesic]] <math>\ell : I \to M</math>, it holds that <math>I=(-\infty,\infty)</math>.{{sfn|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 (Riemannian geometry)|exponential map]] at <math>p</math> is defined on <math>T_pM</math>, the entire [[Tangent space|tangent space]] at <math>p</math>.{{sfn|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 manifold#Metric space structure|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|do Carmo|1992|p=146-147}}
* The metric space <math>(M,d_g)</math> is [[Complete metric space|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 [[N-sphere|sphere]] <math>\mathbb{S}^n</math>, and the [[Torus|tori]] <math>\mathbb{T}^n</math> (with their natural Riemannian metrics) are all complete manifolds.

All [[Compact space|compact]] Riemannian manifolds and all [[Homogeneous space|homogeneous]] manifolds are geodesically complete.  All [[Symmetric space|symmetric space]]s are geodesically complete.

=== Non-examples ===

[[File:Punctured plane is not geodesically complete.svg|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 [[Physics:General relativity|general relativity]], which describes gravity in terms of a pseudo-Riemannian geometry, many important examples of geodesically incomplete spaces arise, e.g. [[Astronomy:Schwarzschild metric|non-rotating uncharged black-holes]] or cosmologies with a [[Astronomy:Big Bang|Big Bang]]. The fact that such incompleteness is fairly generic in general relativity is shown in the [[Physics:Penrose–Hawking singularity theorems|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|do Carmo|1992|p=145}}

==References==

=== Notes ===

{{reflist}}

=== Sources ===
* {{citation
| 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=[[Company:Academic Press|Academic Press]]|year=1983|isbn=0-12-526740-1|at=Chapter 3}}

{{Manifolds}}
{{Riemannian geometry}}

{{DEFAULTSORT:Complete Manifold}}

[[Category:Differential geometry]]
[[Category:Geodesic (mathematics)]]
[[Category:Manifolds]]
[[Category:Riemannian geometry]]

{{Sourceattribution|Complete manifold}}