I'm trying to prove that if $Y_1, Y_2 : P → V$ is an orthonormal frame with respect to $g$, then so is $X_1, X_2 : P → V$ defined by $X_i(F(p)) = F'(p)(Y_ i(p))$.
It is given that $F : P → P$ is an isometry, from the Riemannian chart $(P,g)$ back to itself. So $F∗g = g$.
How can I prove $X_1, X_2 : P → V$ is also an orthonormal frame?
For $p \in P$ we have $g(F(p))(X_i(F(p)),X_j(F(p))) = g(F(p))(F'(p)(Y_i(p)),F'(p)(Y_j(p)))$. Now, you need to apply the definition of the pull-back together with the fact that $F_*g = g.$