Let $(M^3,g)$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the immersion $\phi: \Sigma \times [0,\varepsilon)\to M$ given by
when $N$ is a unit normal vector field along to $\Sigma$.
I would like to show that if we take the pullback metric $\phi^*g$ in $\Sigma\times [0,\varepsilon)$, then $\phi^*g=d\sigma_t^2+dt^2$, where $d\sigma_t^2$ is a smooth family of metrics in $\Sigma$.
I saw already argument like that in many papers, but I failed in prove it.
Anyone has a little help? Thanks so much.