nLab tubular neighborhood theorem

Statement

Let $M$ be a smooth manifold. By an (embedded) submanifold we mean a smooth immersion of smooth manifolds $i \colon X \to M$ that is a topological embedding of $X$ as a closed subspace of $M$.

In that case, we have that for each $x \in X$, the tangent space $T_x X$ is included in the subspace $T_{i(x)} M$. We define the normal fiber $N_x$ to be the quotient $T_{i(x)} M/T_x X$ and the normal bundle (with respect to the embedding $i$) to be the space $N X$ consisting of pairs $\{(x, v): x \in X, v \in N_x\}$, forming a vector bundle over $X$ in an evident way. We let $i_0 \colon X \to N X$ denote the zero section. An open neighborhood $V$ of the zero section is convex if its intersection with $N_x$ is a convex subset of the vector space $N_x$.

Theorem

(Tubular Neighborhood theorem)

For any submanifold $i \colon X \hookrightarrow M$, there is an open neighborhood $U$ of $i(X)$ in $M$ and a convex open neighborhood $V$ of $i_0(X)$ in $N X$ and a diffeomorphism $\phi: U \to V$ such that the diagram

$$\array{ X & & \\ \mathllap{i} \downarrow & \searrow{}^{\mathrlap{i_0}} & \\ U & \underset{\phi}{\longrightarrow} & V }$$

commutes. Such $U$ is called a tubular neighbourhood of $i(X)$.

See for instance (Silva 06, theorem 6.5)

References

• Ana Cannas da Silva, §6.2 in: Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer (2008) [doi:10.1007/978-3-540-45330-7, pdf]