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Math Help - involutive subbundle

  1. #1
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    involutive subbundle

    Hello,

    I have two questions about involutive subbundles and foliations.
    If we have a foliation on a manifold M of codimension q, then this submits an involutive subbundle E of T(M).
    We can take E_{|U_i}=ker(d(pr_2\circ\phi_i)), whereas (U_i,\phi_i ) is a foliation chart.

    I see that (\frac{d}{dx^i})_{|p} \;, i=1,...,n-q is a basis for E_p. But now i don't know why it is involutive.
    My guess is, that if X,Y are smooth sections for E, then also [X,Y] is a smooth section since: [X,Y](p)=X_p Y - Y_p X. \in E_p \subset T_p M, since E_p is the kernel of the linear map above and therefore a linear subspace of T_p M.


    Is this correct? I ask because in my book it is shown very complicated, s.t. i don't understand it really.

    My second question is, why the induced subbundle E is unique? what happens if i choose a different chart?
    I think that the elm. of the basis changes of course. But why is the subspace the same?

    Regards
    Last edited by Sogan; February 14th 2011 at 11:47 AM.
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  2. #2
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    As a geometry object, the definition of a foliation doesn't depend on the choice of the charts. The foliation defines a collection of n-q dimensional sub-manifolds. In each chart (x,y), a plaque is defined as y=c where c is constant, though in another chart (x1,y1) the same plaque is represented by the same type of equation y1=c1, with a (possible) different constant c1. So the plaque( a part of a submanifold) is defined independent of charts. Then the tangent space of this plaque is well-defined. Piece together all those tangent spaces we get the sub-bundle.
    To show it's involutive you need the following equation:
    \langle X\wedge Y, d\omega\rangle=X\langle Y, \omega \rangle - Y \langle X, \omega \rangle - \langle [X,Y], \omega\rangle
    Let \omega=dy^j and notice that \langle Y, dy^j\rangle=\langle X, dy^j \rangle = 0, we get \langle [X,Y], dy^j\rangle=0, which shows [X,Y] belongs to the sub-space.
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  3. #3
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    Hello,

    thank you for your help.

    What do you mean by "<.,.>"?

    Regards
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  4. #4
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    Tensor contraction - Wikipedia, the free encyclopedia. < X, df > = df(X) = X(f) = the directional derivative of f along X.
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  5. #5
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    Hello. Thank you for your help. now i understand you better. One more question:
    X,Y are vector fields, not alternating tensors. Why X \wedge Y makes sense? what is its definition?

    Regards
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