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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.


Local isometry

{title2=$f^*g_N=g_M$}

An isometric immersion is an <immersion> $f:M\to N$ between <Riemannian manifolds> whose derivative preserves the <inner products> of <tangent vectors>; equivalently, $f^*g_N=g_M$. Its derivative is injective and the target may have larger dimension. A <local isometry> additionally requires a <local diffeomorphism>, so its tangent maps are <linear isomorphisms>. For example, a <sphere> with its induced <Riemannian metric> is isometrically immersed in <Euclidean space>, although its positive intrinsic <sectional curvature> differs from that of the ambient space.


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 Incoming links: Isometric immersion