Hey all. The setup for my question is an embedded surface $\Sigma \hookrightarrow M$ 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 so locally we have something like $\Sigma \times N\Sigma \subset M$ and we should be able to put a product metric on the tubular neighborhood. Does that sound correct?