fix a point formula_1 in a complete riemannian manifold formula_4, and consider the tangent space formula_5. it is a standard result that for sufficiently small formula_6 in formula_7, the curve defined by the riemannian exponential map, formula_8 for formula_9 belonging to the interval formula_10 is a minimizing geodesic, and is the unique minimizing geodesic connecting the two endpoints. here formula_11 denotes the exponential map from formula_1. the cut locus of formula_1 in the tangent space is defined to be the set of all vectors formula_6 in formula_5 such that formula_16 is a minimizing geodesic for formula_17 but fails to be minimizing for formula_18 for any formula_19. the cut locus of formula_1 in formula_21 is defined to be image of the