= Local isometry
{wiki}
A local isometry between <Riemannian manifolds> is a <local diffeomorphism> $f$ with $f^*g_N=g_M$. Equivalently, its differentials are isometric <linear isomorphisms> between <tangent spaces>. It preserves the <Levi-Civita connection>, intrinsic <sectional curvature>, and affinely parametrized <geodesics>. A map whose differentials are merely isometric injections is an <isometric immersion>; the <local diffeomorphism> requirement distinguishes these notions.
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