Conjugate points: Difference between revisions

Revision as of 02:00, 13 September 2007

Revision as of 16:52, 14 September 2007

Line 1:

In [[differential geometry]], conjugate points are points that can be connected with geodesics in more than one way. For example, on a [[Spherical geometry|sphere]], the north-pole and south-pole are connected by any [[Meridian (geography)|meridian]].

geodesics in more than one way. For example, on a [[Spherical geometry|sphere]], the north-pole and south-pole are connected by any [[Meridian (geography)|meridian]].

==Formal definition==

Suppose ''p'' and ''q'' are points on a [[Riemannian manifold]], and ''c'' is a [[geodesic]] that connects ''p'' and ''q''. Then ''p'' and ''q'' are '''conjugate points''' if there is a non-zero [[Jacobi field]] on ''c'' that is orthogonal to ''c'' and that vanishes at ''p'' and ''q''.

[[geodesic]] that connects ''p'' and ''q''. Then ''p'' and ''q''

are '''conjugate points''' if there is a non-zero [[Jacobi field]] on ''c'' that vanishes

on ''p'' and ''q''.

Let us recall that any Jacobi field can be written as the derivative of a geodesic

derivative of a geodesic variation. Therefore one can construct a family of geodesics that connect conjugate points.

==Examples==

* On the sphere <math>S^n2</math>, any two[[antipodal]] points are conjugate.

* On <math>\mathbb{R}^n</math>, there are no conjugate points.

{{Uncategorized|date=September 2007}}

[[Category:Riemannian geometry]]