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Could someone tell me how to prove the following result?

Let $Z$ be a submanifold of codimension $k$ in $Y$. Prove that the normal bundle $N(Z;Y)=\left\{(z,v):z\in Z, v\in T_{z}(Y)\text{ and } v\bot T_{z}(Z)\right\}$ is locally trivial, that is, each point $z\in Z$ has a neighborhood $V$ in $Z$ such that $N(V;Y)$ is trivial.

Note: We say that the normal bundle $N(Z;Y)$ is trivial if there exists a diffeomorphism $\psi:N(Z;Y)\longrightarrow Z\times\mathbb{R}^k$ that restricts to a linear isomorphism $N_{z}(Z;Y)\longrightarrow\left\{z\right\}\times\mathbb{R}^k$ for each point $z\in Z$.

Mhenni Benghorbal
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R.Shaug
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1 Answers1

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Being a local matter, we can think $Y$ is an affine space $R^n$. Each point in $Z$ has a neighborhood $V$ in $R^n$ on which $Z$ has regular equations, that is smooth functions $f_1,\dots,f_k:V\to R$ whose jacobian has rank $k$ everywhere in $V$ and $Z\cap V=\{f_1=\cdots=f_k=0\}$. Then the normal bundle is generated on $V$ by the gradients $\nabla f_1,\dots,\nabla f_k$. In other words, $$ \psi^{-1}(z,\lambda_k,\dots,\lambda_k)=(z,\sum_i\lambda_i\nabla f_i(z)). $$

Jesus RS
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