If ${M}$ is a manifold and ${N}$ a compact submanifold, then a tubular neighborhood of ${N}$ consists of an open set ${U \supset N}$ diffeomorphic to a neighborhood of the zero section in some vector bundle ${E}$ over ${N}$, by which $N$ corresponds to the zero section.

Theorem 1 Hypotheses as above, ${N}$ has a tubular neighborhood. (more…)