The cut locus is a set of points on a Riemannian manifold where geodesics originating from a given point cease to be minimizing. When considering the exponential map, the cut locus can help us understand the behavior of geodesics and how they behave as they reach certain limits in normal coordinates. This concept is crucial for understanding conjugate points and focal points, as these features relate directly to the structure of geodesics in the manifold.
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