nLab tubular neighborhood theorem




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

In that case, we have that for each xXx \in X, the tangent space T xXT_x X is included in the subspace T i(x)MT_{i(x)} M. We define the normal fiber N xN_x to be the quotient T i(x)M/T xXT_{i(x)} M/T_x X and the normal bundle (with respect to the embedding ii) to be the space NXN X consisting of pairs {(x,v):xX,vN x}\{(x, v): x \in X, v \in N_x\}, forming a vector bundle over XX in an evident way. We let i 0:XNXi_0 \colon X \to N X denote the zero section. An open neighborhood VV of the zero section is convex if its intersection with N xN_x is a convex subset of the vector space N xN_x.


(Tubular Neighborhood theorem)

For any submanifold i:XMi \colon X \hookrightarrow M, there is an open neighborhood UU of i(X)i(X) in MM and a convex open neighborhood VV of i 0(X)i_0(X) in NXN X and a diffeomorphism ϕ:UV\phi: U \to V such that the diagram

X i i 0 U ϕ V \array{ X & & \\ \mathllap{i} \downarrow & \searrow{}^{\mathrlap{i_0}} & \\ U & \underset{\phi}{\longrightarrow} & V }

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

See for instance (Silva 06, theorem 6.5)


Last revised on November 16, 2023 at 09:44:27. See the history of this page for a list of all contributions to it.