Geodesic mapping: Difference between revisions

Latest revision as of 19:41, 18 May 2008

Definition

Let $M$ and $N$ be Riemannian manifolds. A geodesic mapping from $M$ to $N$ is a diffeomorphism from $M$ to $N$ that sends geodesics to geodesics, such that its inverse also sends geodesics to geodesics.

Another way of saying this is that if $M$ is a differential manifold, then two Riemannian metrics $g_{1}$ and $g_{2}$ of $M$ are related by a geodesic mapping if the geodesics for $g_{1}$ are precisely the same as the geodesics for $g_{2}$ .

Revision as of 08:02, 2 September 2007 (view source) Vipul (talk \| contribs) No edit summary		Latest revision as of 19:41, 18 May 2008 (view source) Vipul (talk \| contribs) m (2 revisions)
(One intermediate revision by the same user not shown)
Line 2:		Line 2:

	Let <math>M</math> and <math>N</math> be [[Riemannian manifold]]s. A '''geodesic mapping''' from <math>M</math> to <math>N</math> is a [[diffeomorphism]] from <math>M</math> to <math>N</math> that sends geodesics to geodesics, such that its inverse also sends geodesics to geodesics.		Let <math>M</math> and <math>N</math> be [[Riemannian manifold]]s. A '''geodesic mapping''' from <math>M</math> to <math>N</math> is a [[diffeomorphism]] from <math>M</math> to <math>N</math> that sends geodesics to geodesics, such that its inverse also sends geodesics to geodesics.

			Another way of saying this is that if <math>M</math> is a [[differential manifold]], then two [[Riemannian metric]]s <math>g_1</math> and <math>g_2</math> of <math>M</math> are related by a geodesic mapping if the geodesics for <math>g_1</math> are precisely the same as the geodesics for <math>g_2</math>.