Isometry

Definition

Let two Riemannian manifolds $(M,g^M)$ and $(N,g^N)$, and a Diffeomorphism $f:M\to N$. Then $f$ is called an isometry if the following condition holds: $\forall p\in M,\mathbf{u},\mathbf{v}\in T_p(M),g^M(\mathbf{u},\mathbf{v})=g^N(f(\mathbf{u}),f(\mathbf{v}))$