Local isometries are determined by first-order data
= Local isometries are determined by first-order data
Two <local isometries> from a connected <Riemannian manifold> that have the same value and <differential> at one point agree everywhere. They preserve <geodesics>, so their equality follows on a <normal neighborhood> from $f(\exp_pv)=\exp_{f(p)}(df_pv)$. The set of points where both value and differential agree is consequently open, and continuity makes it closed. <Connectedness> completes the argument.