Conjugate points

In differential geometry, conjugate points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. Geodesics are locally length-minimizing, but, for example, on a sphere, any geodesic from the north-pole fail to be length-minimizing if it passes through the south-pole. In differential geometry, conjugate points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. Geodesics are locally length-minimizing, but, for example, on a sphere, any geodesic from the north-pole fail to be length-minimizing if it passes through the south-pole. Suppose p and q are points on a Riemannian manifold, and γ {displaystyle gamma } is a geodesic that connects p and q. Then p and q are conjugate points along γ {displaystyle gamma } if there exists a non-zero Jacobi field along γ {displaystyle gamma } 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 fields). Therefore, if p and q are conjugate along γ {displaystyle gamma } , one can construct a family of geodesics that start at p and almost end at q. In particular,if γ s ( t ) {displaystyle gamma _{s}(t)} is the family of geodesics whose derivative in s at s = 0 {displaystyle s=0} generates the Jacobi field J, then the end pointof the variation, namely γ s ( 1 ) {displaystyle gamma _{s}(1)} , 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.