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Bundle morphisms and automorphism

  1. Oct 7, 2010 #1
    In an article of differential Geometry righted by ALEXI KOVALEV , he said that to define an isomorphism of vector bundle covering a map f: B-> M ( B and M are two manifolds ) we need that f must be a diffeomorphism.

    then an other question he consider an exemple of morphism vector bundle F between a vector bundle E and his pull back . why we are certain that we have a linear isomorphism between any pairs of fibres (E) and a fiber of the pull back of E.

    Finally , when we take trivial bundle E=BXV ( B manifold and V typical fibre of E) any automorphism of E is defined by a smooth map B->G ( when G= group of invertible matrix)

    thnx a lot to explain me this point .
     
  2. jcsd
  3. Oct 7, 2010 #2

    lavinia

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    The first question can be answered directly from the definition of the induced bundle.

    The second is obvious.
     
  4. Oct 8, 2010 #3
    This is not very precise. First, there are two kinds of bundle automorphisms: those preserving the base and those that induce a diffeomorphism on the base. Evidently in your question we are dealing with the first case. Then the only change is in the fibers, and this, for each point x of the base, should be an endomorphism (invertible linear transformation) of the fiber. It is only when we endow V with a linear basis that such an endomorphism is described by an invertible matrix. Change the basis and the matrix representation will change
     
  5. Oct 8, 2010 #4
    for the first question how we see clearly the answer from the definition ? if we look to the definition of induced bundle we must so have a linear map g between a fibre E\f(p) ( a fibre of pullback bundle) and a fibre also E\f(p) to have a morphisme between the vector bundle E nad his pull back ?

    why it is clearly g is a linear map ?
     
  6. Oct 8, 2010 #5
    If you would give an exact place in Kovalev's notes where you are having a problem - it would be easier to help you.
     
  7. Oct 8, 2010 #6
    thnx arkajad for your first answer you help me realy to understand the meaning . just i would ask you about something .
    you say " First, there are two kinds of bundle automorphisms: those preserving the base and those that induce a diffeomorphism on the base "
    can you explain more these ?
     
  8. Oct 8, 2010 #7
  9. Oct 8, 2010 #8
    Kovalev in his "Pulling back" section does not say exactly how the fibers of the pullback are defined. He refers to a "commutative diagram". But when, after (2.5) he says "isomorphism onto a fibre" he means a linear isomorphism.
     
  10. Oct 8, 2010 #9
    why we must have linear isomorphisme onto a fibre ?
     
  11. Oct 8, 2010 #10
    Well, we want to stay in vector bundles category. So, we want a pullback of a vector bundle to be again a vector bundle. Or, alternatively: we can take any pullback and then define a vector space structure in the fibers over M by defining

    [tex]p+q=F^{-1}(F(p)+F(q))[/tex]

    [tex]ap= F^{-1}(aF(p))[/tex]

    where a is a number and p,q are two points in the same fibre over M. Then we get the isomorphism by the very construction.
     
  12. Oct 8, 2010 #11
    ohhhh i get it it is very simple i just didn't concentrate :shy:
    it is just because we have the same fibre so we have the map id( fibre ) that'is clearly linear isomorphism .
    thnx arkajad. but you didn't answer me for

    "First, there are two kinds of bundle automorphisms: those preserving the base and those that induce a diffeomorphism on the base."

    if you look to definition in the article you always preserve the same base ?
     
  13. Oct 8, 2010 #12
    Different authors may have different terminology. For instance Husemoller in his "Fibre Bundles" will use the name B-automorphism for an automorphism that keeps the points at the base fixed. And it nice to be able to say, for instance, that every diffeomorphism of B lifts to an automorphism of its tangent bundle TM.

    But many authors define a bundle automorphism demanding that it induces the identity map on the base. So, as long as you know the definition in a given book - you are ok. But when you move to a different book or a paper, it is better to be prepared for a possible change.
     
  14. Oct 8, 2010 #13
    thnx arkajad .
     
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