A Jacobi field is a vector field along a geodesic in a Riemannian manifold that describes the behavior of nearby geodesics and captures the notion of how the geometry of the manifold influences the divergence or convergence of these paths. It is crucial for understanding stability and the properties of geodesics, especially when analyzing the Jacobi equation, which governs the evolution of these fields. Jacobi fields play an essential role in studying variations of geodesics and have implications in the fields of differential geometry and mathematical physics.
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