 Pullback of a Riemannian metric

For a smooth map $f:M\to N$, the pullback of a Riemannian metric is $(f^{\ast }g{)}_p(u,v)=g_{f(p)}(d{f}_{p}u,d{f}_{p}v)$. It is positive definite precisely when $d{f}_p$ is injective at every point, that is, when $f$ is an immersion. In coordinates, $(f^{\ast }g{)}_{ij}=g_{ab}(f(y)){\partial }_i{f}^a{\partial }_j{f}^b$.