Riemannian isometry and local isometry

An isometry $F:(M,g)\to (N,h)$ is a diffeomorphism with $F^{\ast }h=g$. A local isometry is a smooth local diffeomorphism with $F^{\ast }h=g$. An isometric immersion is a smooth immersion satisfying that same pullback identity.

Use Pullback of a riemannian metric as a tensor and Diffeomorphisms and local diffeomorphisms of manifolds. Positivity forces injective differential by Pullback of a riemannian metric is riemannian exactly for immersions. In equal dimensions on boundaryless manifolds, the inverse function theorem as applied in the next proposition makes a metric-preserving smooth map a local isometry. At a boundary the definition retains the local-diffeomorphism requirement. An isometric immersion need not have equal source and target dimensions.

Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 13, pp.328–332 and 341–342.