Let \(F\colon M\to N\) and \(g\) a Riemannian metric on \(N\). Then we call the pullback \(F^*g\) pullback metric if it is positive definite.
A pullback metric is a Riemannian metric on \(M\).
Remark
- \(F^*g\) is symmetric due to definition
- \(F^*g\) is a pullback metric if and only if \(F\) is a smooth immersion [@lee2013smooth_manifolds, Proposition 13.9].