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K-dimensional manifold problem

  • Thread starter Frillth
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Homework Statement



Suppose X ⊂ R^n is a k-dimensional manifold and Y ⊂ R^p is an l-dimensional manifold. Prove that:

X × Y = {[x,y] ∈ R^n × R^p : x ∈ X and y ∈ Y}

is a (k+l)-dimensional manifold in R^(n+p). (Hint: Recall that X is locally a graph over a k-dimensional coordinate plane in R^n and Y is locally a graph over an l-dimensional coordinate plane in R^p.)

Homework Equations



None known.

The Attempt at a Solution



Our book doesn't even give a really solid definition of a manifold, so I don't really know where to start.
 

Answers and Replies

  • #2
HallsofIvy
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It doesn't give a definition of a manifold? Then how can it ask you to recall "that X is locally a graph over a k-dimensional coordinate plane in R^n and Y is locally a graph over an l-dimensional coordinate plane in R^p"? That's all you need.
 

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