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Book on Curvature wrong or am I confused

  1. Jan 11, 2012 #1

    I was doing some exercises from the book on curvature by Lee to buff up my differential geometry. I came a cross the following question and it seems to me the question isn't completely correct, but I'm not so good at differential geometry that I am confident. Maybe someone else is!

    the question is:
    Suppose N ⊂ M is an embedded submanifold.

    If X is a vector field on M , show that X is tangent to N at points
    of N if and only if Xf = 0 whenever f is a smooth function on M that
    vanishes on N.

    What looks to be wrong is Xf only needs to vanish at points of N not all of M.

    I came up with the example:

    the vector field X=[itex]\partial_x[/itex] + y[itex]\partial_y[/itex] on M=ℝ² where the submanifold N is the real line (so set y to 0).

    It seems that although at points of N X=[itex]\partial_x[/itex] (so at p \in N)
    which is tangent to N.
    the smooth function f(x,y)=y which vanishes on the real line has Xf=y so this only vanishes on N not on all of M.

    So the question is is their something wrong with this reasoning or is the question wrong?

  2. jcsd
  3. Jan 11, 2012 #2


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    You're right, of course. The book should say "[...] if and only if Xf=0 on N [...]".
  4. Jan 12, 2012 #3
    Ok thank you, I should probably doubt myself less!
  5. Jan 12, 2012 #4


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    Lee has errata for all his books in his web site.
  6. Jan 12, 2012 #5
    Lee's also wonderful about answering emails with questions about his books. I spotted an error in that book and sent him an email, and he had emailed me back and posted the erratum within 24 hours.
  7. Jan 12, 2012 #6
    Oh that's awesome! so next time I should check the website first and then e-mail!
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