Let M a $k$-dimensional manifold without boundary of $\mathbb{R}^{n}$ and N a $l$-dimensional manifold of $\mathbb{R}^{m}$ with or withour boundary. Show that $\partial(M\times N)=M\times\partial(N)$

My approach: First I prove that, if M a $k$-dimensional manifold of $\mathbb{R}^{n}$ and N a $l$-dimensional manifold of $\mathbb{R}^{m}$ then $M\times N$ is a manifold of $\mathbb{R}^{k+l}$ of dimension $k+l$. But now, I cannot see the relation $\partial(M\times N)=M\times\partial(N)$, any hint. Thanks!

  • $\begingroup$ Well you've shown that $M\times N$ is a $k+l$ dim manifold. Now you need to show that if $N$ is a manifold with boundary, then $M\times N$ is a manifold with boundary. From there, try looking at coordinate charts for points in the boundary of $M\times N$, which might give you an idea of how to construct a diffeomorphism. $\endgroup$ – Moya Jan 18 '16 at 3:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.