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K-dimensional manifold problem

  1. Dec 3, 2008 #1
    1. The problem statement, all variables and given/known data

    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.)

    2. Relevant equations

    None known.

    3. 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.
     
  2. jcsd
  3. Dec 3, 2008 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Re: Manifolds

    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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