tubular neighborhood theorem

**manifolds** and **cobordisms**

cobordism theory, *Introduction*

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 $N_x$.

**(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)

- Ana Cannas da Silva,
*Lectures on Symplectic Geometry*, Springer Lecture Notes in Math. 1764. Revised online version, 2006. (pdf)

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