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Math Help - involutive subbundle and Lie bracket

  1. #1
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    involutive subbundle and Lie bracket

    Hello,

    i have a question about involutive subbundles.
    If M is a manifold and E is a subbundle. We say E is involutive, if for each smooth vector fields X,Y of E, (that is X(p),Y(p) \in E_p) also the lie bracket [X,Y] is a smooth vector field to E.

    I think that not any subbundle is involutive, but i don't know why?
    Whats wrong with this proof?

    Claim:
    If X,Y are smooth vector fields to E, =>[X,Y] is also a vect. field:

    Pf:
    [X,Y](p)= X_p (Y) - Y_p (X). since X,Y are vector fields and E_p is a linear subspace, therefore the sum above is also in E_p??


    Where is the mistake? I don't see it.

    Regards
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  2. #2
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    Take M=R^3. Let X=\frac{\partial}{\partial x}, Y=\frac{\partial}{\partial y}+x \frac{\partial}{\partial z}.
    Obviously X, Y are nowhere dependent, So they span a sub-space span{X,Y} in the tangent space of each point of M. Let E=\cup_{p\in R^3} span\{X, Y\}. E is a smooth sub-bundle.
    Now you can get easily compute the result [X,Y]=\frac{\partial}{\partial z}, which cannot be expressed by linear combination of X and Y, thus doesn't belong to E.
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  3. #3
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    Actually, according to Frobenius Theorem

    In the vector field formulation, the theorem states that a subbundle of the tangent bundle of a manifold is integrable (or involutive) if and only if it arises from a regular foliation. In this context, the Frobenius theorem relates integrability to foliation;


    So involutivity means the subbundle is locally the tangent bundle of a sub-manifold.
    Last edited by xxp9; February 19th 2011 at 01:00 AM.
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