Geodesic variation refers to the process of analyzing how geodesics, which are the shortest paths between points on a manifold, can change when the manifold itself is varied. This concept is essential in understanding how small perturbations in a Riemannian metric affect the behavior of geodesics and provides insight into the stability and geometry of the manifold. In this context, geodesic variations are closely related to Jacobi fields, which describe how geodesics deviate from one another in a smooth manifold.
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