In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, conjugate points or focal points are, roughly, points that can almost be joined by a 1-parameter family of
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s. For example, on a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, the north-pole and south-pole are connected by any
meridian
Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to
Science
* Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon
* ...
. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. All geodesics are ''locally'' length-minimizing, but not globally. For example on a sphere, any geodesic passing through the north-pole can be extended to reach the south-pole, and hence any geodesic segment connecting the poles is not (uniquely) ''globally'' length minimizing. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.
[Cheeger, Ebin. ''Comparison Theorems in Riemannian Geometry''. North-Holland Publishing Company, 1975, pp. 17-18.]
Definition
Suppose ''p'' and ''q'' are points on a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
, and
is a
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
that connects ''p'' and ''q''. Then ''p'' and ''q'' are conjugate points along
if there exists a non-zero
Jacobi field In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic for ...
along
that vanishes at ''p'' and ''q''.
Recall that any Jacobi field can be written as the derivative of a geodesic variation (see the article on
Jacobi field In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic for ...
s). Therefore, if ''p'' and ''q'' are conjugate along
, one can construct a family of geodesics that start at ''p'' and ''almost'' end at ''q''. In particular,
if
is the family of geodesics whose derivative in ''s'' at
generates the Jacobi field ''J'', then the end point
of the variation, namely
, is the point ''q'' only up to first order in ''s''. Therefore, if two points are conjugate, it is not necessary that there exist two distinct geodesics joining them.
Examples
* On the sphere
,
antipodal point
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true d ...
s are conjugate.
* On
, there are no conjugate points.
* On Riemannian manifolds with non-positive
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...
, there are no conjugate points.
See also
*
Cut locus
The cut locus is a mathematical structure defined for a closed set S in a space X as the closure of the set of all points p\in X that have two or more distinct shortest paths in X from S to p.
Definition in a special case
Let X be a metric s ...
*
Jacobi field In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic for ...
References
{{DEFAULTSORT:Conjugate Points
Riemannian geometry