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Extending Vector Fields.

  1. Nov 20, 2007 #1

    WWGD

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    Hi: I am going over Lee's Riemm. mflds, and there is an exercise that asks:

    Let M<M' (< is subset) be an embedded submanifold.


    Show that any vector field X on M can be extended to a vector field on M'.

    Now, I don't know if he means that X can be extended to the _whole_ of

    M', because otherwise, there is a counterexample:


    dt/t on (0,1) as a subset of IR cannot be extended to the whole of IR.


    Anyone know?.


    What Would Gauss Do?
     
  2. jcsd
  3. Nov 20, 2007 #2

    Hurkyl

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    Are you sure M is not supposed to be a closed submanifold?
     
  4. Nov 21, 2007 #3

    WWGD

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    Hi. I did not see this stated, and I cannot see if it is being assumed somehow.

    Only conditions I saw where that M<M' , and M embedded submanifold of M',

    and a V.Field is defined in M.



    Re the second issue, of extensions, I guess these extensions are local, tho

    not necessarily global, right?. The answer would seem to be yes pretty

    clearly, but my intuition has failed me before.
     
  5. Nov 21, 2007 #4

    Chris Hillman

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    Consider the function with constant value one on (0,1). How many ways can you extend it to R?
     
  6. Nov 23, 2007 #5

    WWGD

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    Thanks, Chris, I am not sure I get the hint; there are uncountably many ways,

    for a continuous extension, most obvious extension being f==1 in IR (attach

    a line segment of slope m, m in (-oo,oo) , maybe

    smaller cardinality for smooth extensions . For smoothness I would imagine

    some combination of e^-x 's attached to both ends, or maybe some other

    bump functions ( Lee does not specify if X is C^1 , or C^k, C^oo).

    All I can think of when I think of immersed submanifolds is slice coordinates, tho

    this does not seem to make sense for 1-manifolds like (0,1) in IR.


    Am I on the right track?.
     
  7. Nov 23, 2007 #6

    Chris Hillman

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    Mrmph... never mind the hint, I was looking at the wrong textbook in Lee's excellent trilogy, my mistake! Unfortunately you are using the one I don't have, but can you state the what exercise you are attempting? I might be able to obtain a copy next week. I expect we will be able to figure it out!
     
    Last edited: Nov 23, 2007
  8. Nov 23, 2007 #7

    WWGD

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    Yes, this is Lee's Riemannian mflds, problem 2.3, part b, p.15 in my edition.

    Thanks.
     
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