Complete manifold: Difference between revisions
grammar |
Incorporated def from Hopf-Rinow theorem and clarified original definition Tags: nowiki added Visual edit |
||
Line 1: | Line 1: | ||
{{context|date=January 2013}} |
{{context|date=January 2013}} |
||
⚫ | In [[mathematics]], a '''complete manifold''' (or '''geodesically complete manifold''') {{Mvar|M}} is a ([[Pseudo-Riemannian manifold|pseudo]]-) [[Riemannian manifold]] for which {{Math|exp{{sub|''p''}}}}, the [[Exponential map (Riemannian geometry)|exponential map]] at a point {{Mvar|p}}, is defined on {{Math|''TM''{{sub|''p''}}}}, the entire tangent space at {{Mvar|p}}. |
||
{{formula missing descriptions|date=January 2013}} |
|||
Equivalently, consider a maximal [[geodesic]] {{Math|''l'':''I''→''M''}}. {{Mvar|I}} is an open interval of {{Mvar|ℝ}}, and, because geodesics travel at fixed speed, uniquely defined up to translation. Because {{Mvar|l}} is maximal, {{Mvar|l}} maps the [[End (topology)|ends]] of {{Mvar|I}} to points of {{Math|∂''M''}}, and the length of {{Mvar|I}} measures the distance between those points. A manifold is geodesically complete if for any such {{Mvar|l}}, {{Math|1=''I''=(-∞,∞)}}. |
|||
⚫ | |||
==Examples== |
==Examples and non-examples== |
||
All [[compact space|compact]] Riemannian manifolds and all [[homogeneous space|homogeneous]] manifolds are geodesically complete. |
All [[compact space|compact]] Riemannian manifolds and all [[homogeneous space|homogeneous]] manifolds are geodesically complete. |
||
[[Euclidean space]] |
[[Euclidean space]] {{Math|ℝ{{sup|n}}}}, the [[sphere]]s {{Math|𝕊{{sup|n}}}}, and the [[torus|tori]] {{Math|𝕋{{sup|n}}}} (with their natural [[Riemannian metric]]s) are all complete manifolds. |
||
=== Non-examples === |
|||
A simple example of a non-complete manifold is given by the punctured plane |
A simple example of a non-complete manifold is given by the punctured plane {{Math|ℝ{{sup|2}}\{0<nowiki>}</nowiki>}} (with its induced metric). Geodesics going to the origin cannot be defined on the entire real line. |
||
There exist non-geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds. It is the case for example of the [[Clifton–Pohl torus]]. |
There exist non-geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds. It is the case for example of the [[Clifton–Pohl torus]]. |
||
Line 17: | Line 18: | ||
==References== |
==References== |
||
* {{Citation | last1=O'Neill | first1=Barrett | title=Semi-Riemannian Geometry | publisher=[[Academic Press]] | isbn=0-12-526740-1 | year=1983}}. ''See chapter 3, pp. 68''. |
|||
* {{Cite book|title=Semi-Riemannian Geometry|last=O'Neill|first=Barrett|publisher=[[Academic Press]]|year=1983|isbn=0-12-526740-1|location=|pages=|at=Chapter 3}} |
|||
{{DEFAULTSORT:Complete Manifold}} |
{{DEFAULTSORT:Complete Manifold}} |
||
[[Category:Riemannian geometry]] |
[[Category:Riemannian geometry]] |
Revision as of 19:07, 18 June 2019
This article provides insufficient context for those unfamiliar with the subject.(January 2013) |
In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which expp, the exponential map at a point p, is defined on TMp, the entire tangent space at p.
Equivalently, consider a maximal geodesic l:I→M. I is an open interval of ℝ, and, because geodesics travel at fixed speed, uniquely defined up to translation. Because l is maximal, l maps the ends of I to points of ∂M, and the length of I measures the distance between those points. A manifold is geodesically complete if for any such l, I=(-∞,∞).
Examples and non-examples
All compact Riemannian manifolds and all homogeneous manifolds are geodesically complete.
Euclidean space ℝn, the spheres 𝕊n, and the tori 𝕋n (with their natural Riemannian metrics) are all complete manifolds.
Non-examples
A simple example of a non-complete manifold is given by the punctured plane ℝ2\{0} (with its induced metric). Geodesics going to the origin cannot be defined on the entire real line.
There exist non-geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds. It is the case for example of the Clifton–Pohl torus.
Path-connectedness, completeness and geodesic completeness
It can be shown that a finite-dimensional path-connected Riemannian manifold is a complete metric space (with respect to the Riemannian distance) if and only if it is geodesically complete. This is the Hopf–Rinow theorem. This theorem does not hold for infinite-dimensional manifolds. The example of a non-complete manifold (the punctured plane) given above fails to be geodesically complete because, although it is path-connected, 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.
References
- O'Neill, Barrett (1983). Semi-Riemannian Geometry. Academic Press. Chapter 3. ISBN 0-12-526740-1.