Definition of a pullback metric:
Given immersion f:M→N
from source M to target N manifold,
and metric tensor g, pullback metric is defined as
(f*g)_p(u,v) = g_{f(p)}(f*u,f*v)
where f*=df = pushfoward and df the differential of f.
g=Euclidean ⇒ f*g= transpose(Jac) × Jac