Riemannian manifold is metric space - Revision history

Vipul: 2 revisions

2008-05-18T20:07:30Z

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Vipul: /* Geodesic completeness */

2007-08-31T11:09:52Z

Geodesic completeness

← Older revision		Revision as of 11:09, 31 August 2007
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	===Geodesic completeness===		===Geodesic completeness===

	A [[complete Riemannian manifold]], or [[geodesically complete Riemannian manifold]], is one where at any point, the exponential map from the tangent space at the point to the manifold, is well-defined~~, and surjective~~. Complete Riemannian manifolds ~~thus~~ give rise to [[geodesic metric space]]s. Not every geodesic metric space arises from a complete Riemannian manifold (in particular, for instance, the open disc is far from complete but it is a geodesic metric space).		A [[complete Riemannian manifold]], or [[geodesically complete Riemannian manifold]], is one where at any point, the exponential map from the tangent space at the point to the manifold, is well-defined. Complete Riemannian manifolds give rise to [[complete metric space]]s by the [[Hopf-Rinow theorem]]. The metric spaces they give rise to are also [[geodesic metric space]]s. Not every geodesic metric space arises from a complete Riemannian manifold (in particular, for instance, the open disc is far from complete but it is a geodesic metric space).



	===Submanifolds===		===Submanifolds===
	Given a [[Riemannian manifold]] and a [[Riemannian submanifold]] the metric space structure arising from the Riemannian metric on the submanifold, is different from the metric space structure arising from the Riemannian metric on the manifold, inherited to the submanifold.		Given a [[Riemannian manifold]] and a [[Riemannian submanifold]] the metric space structure arising from the Riemannian metric on the submanifold, is different from the metric space structure arising from the Riemannian metric on the manifold, inherited to the submanifold.

Vipul at 11:07, 31 August 2007

2007-08-31T11:07:52Z

New page

==Statement==

Any path-connected [[Riemannian manifold]] naturally acquires the structure of a [[metric space]].

==Explanation==

Suppose <math>M</math> is a path-connected [[differential manifold]] and <math>g</math> is a [[Riemannian metric]] on <math>M</math>. For two points <math>x,y \in M</math>, let <math>\gamma:[0,1] \to M</math> be a path from <math>x</math> to <math>y</math> and define the length of <math>\gamma</math> as:

<math>l(\gamma) = \int_0^1 |\gamma'(t)| dt</math>

where the modulus sign is the norm in the tangent space with respect to the Riemannian metric. The distance between <math>x<math> and <math>y</math> is defined as the infimum of the lengths of all paths from <math>x</math> to <math>y</math>.

==Facts==

===Geodesic completeness===

A [[complete Riemannian manifold]], or [[geodesically complete Riemannian manifold]], is one where at any point, the exponential map from the tangent space at the point to the manifold, is well-defined, and surjective. Complete Riemannian manifolds thus give rise to [[geodesic metric space]]s. Not every geodesic metric space arises from a complete Riemannian manifold (in particular, for instance, the open disc is far from complete but it is a geodesic metric space).

===Submanifolds===
Given a [[Riemannian manifold]] and a [[Riemannian submanifold]] the metric space structure arising from the Riemannian metric on the submanifold, is different from the metric space structure arising from the Riemannian metric on the manifold, inherited to the submanifold.