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Let $M\subset{E^{n}}$ be an r manifold and $N\subset{E^{m}}$ be an s manifold. Regarding $E^{m+n}$ as the Cartesian product $E^{n}\times{E^{m}}$, show that $M\times{N}$ is an (r+s)manifold. Show that the tangent space at a point of $M\times{N}$ is the Cartesian product of the tangent spaces at the corresponding points of M and N. Can any one help me to do this problem.

Robert
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  • It is just a remark not a proof. I think a intuitive way is to admit Nash's embedding theorem. In this way, we just need to prove it in Euclidean space. – yaoxiao Apr 14 '14 at 15:52
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    What is your definition of the tangent space? –  Apr 14 '14 at 15:58

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