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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. ==Definition== Suppose ''p'' and ''q'' are points on a Riemannian manifold, and is a geodesic that connects ''p'' and ''q''. Then ''p'' and ''q'' are conjugate points along if there exists a non-zero Jacobi field 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 fields). 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「conjugate points」の詳細全文を読む スポンサード リンク
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