Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $z\in Z$.

share|cite|improve this question

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, $$ \psi^{-1}(z,\lambda_k,\dots,\lambda_k)=(z,\sum_i\lambda_i\nabla f_i(z)). $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.