The setup for my question is an embedded surface $\Sigma\to M$ in a smooth, compact 4-manifold $M$. Assuming one knows the induced metric $g_\Sigma$ on $\Sigma$ , I would like to know if there is a way to find the metric on a tubular neighborhood of $\Sigma$ without knowing the metric on $M$.
I was thinking that the tubular neighborhood is "just like" the normal bundle of the surface. If the normal bundle is $\pi:N\Sigma \to \Sigma$, then locally we have a trivialization like $\Sigma \times \pi^{-1}(\Sigma) \subset M$ and we should be able to put a product metric on the tubular neighborhood. If the normal bundle is trivial, does this splitting become global (meaning I can now write $\Sigma \times \pi^{-1}(\Sigma) = M$)? Can someone give some references on determining when the normal bundle is trivial?