Conjugate points

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Definition

A pair of points in a Riemannian manifold are said to be conjugate points if there is a Jacobi field that vanishes at both points. In other words, we can construct a geodesic variation of a geodesic joining the two points, or there is a smoothly varying infinite family of geodesics joining the two points.

To any pair of conjugate points, we can associate a corresponding multiplicity, which is the nullity of a certain space associated with the energy functional.

If a Riemannian manifold has negative sectional curvature everywhere then there does not exist any pair of conjugate points.