Pullback of a Riemannian metric
= Pullback of a Riemannian metric
{title2=$f^*g$}
For a smooth map $f:M\to N$, the pullback of a <Riemannian metric> is $(f^*g)_p(u,v)=g_{f(p)}(df_pu,df_pv)$. It is positive definite precisely when $df_p$ is injective at every point, that is, when $f$ is an <immersion>. In coordinates, $(f^*g)_{ij}=g_{ab}(f(y))\partial_i f^a\partial_j f^b$.