Isometry and Immersion between two riemannian manifolds I am confuse with these concepts: Isometry and Immersion. Let  $M$ and $N$ be riemannian manifolds.
If $f:M\to N$ is a smooth isometry and will it be a immersion...
If $g$ is a immersion then i know that it need not be isometry... but whether converse is true?

An immersion is a map such that its differential at every point is injective. An isometry is a map such that its differential at every point is a linear isometry between the corresponding tangent spaces. Since every linear isometry is injective, the result follows.
