= Convex normal neighbourhood
= Convex normal neighborhood
{synonym}
A convex normal neighbourhood is an open set whose points have unique smoothly varying joining <geodesics>, with the joining geodesic contained in the set. The corresponding star-shaped restriction of each <exponential map> is a <diffeomorphism> onto the neighbourhood. Hence its differential is invertible and no such joining geodesic admits a nonzero <Jacobi field> vanishing at both endpoints. Every point of a <Riemannian manifold> has arbitrarily small such neighbourhoods. This is stronger than merely requiring the existence of a minimizing geodesic in the set.
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