= Cut-point dichotomy for complete Riemannian manifolds
A <Riemannian cut point> along a <geodesic> in a complete <Riemannian manifold> is either its first <conjugate point> or the endpoint of another minimizing <geodesic> from the initial point. To prove this, approach the cut time from above and choose minimizing geodesics to the later endpoints using the <Hopf-Rinow theorem>. A subsequence of their initial unit vectors converges. If its limit is the original initial vector and the cut endpoint is not conjugate, the <inverse function theorem> for the <Riemannian exponential map> identifies the two geodesics locally and contradicts the shorter length of the chosen minimizers. A conjugate point earlier than the cut time is excluded by the negative <Riemannian index form> direction obtained by extending an endpoint-vanishing <Jacobi field> by zero and perturbing its endpoint corner.
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