this seems to be an easy question but I'm stuck anyway. Let $\Gamma$ be a submanifold of a Riemannian manifold $(M,g)$. Let further $U$ be a coodinate neighborhood of $\Gamma$ such that a point $p\in\Gamma$ is given by $\phi(x)$, $x\in U$.
Consider now a vector field on $X$ on $M$ which is normal to $\Gamma$ (locally on $U$) and consider the map $f: U\times(-\varepsilon,\varepsilon)\to M$ given by $(x,t)\mapsto \exp_{\phi(x)}(t X(\phi(x)))$ How can I calculate the pullback of the metric $g$ under $f$?
I fail already when trying to compute $\partial_if(x,t)$, the main issue being that the "footpoint" of the exponential map varies also, as $x$ varies. Thanks for any explications, comments and help.