Bundle morphisms and automorphism

[math6]

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 .

 

[lavinia]

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 .

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

The second is obvious.

 

[arkajad]

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).

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

 

[math6]

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 ?

 

[arkajad]

If you would give an exact place in Kovalev's notes where you are having a problem - it would be easier to help you.

 

[math6]

thnx arkajad for your first answer you help me really 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 ?

 

[math6]

http://www.dpmms.cam.ac.uk/~agk22/vb.pdf

this is the article ? my probleme is in section 2.1 in Examples 2 and last paragraph .

thnx very much .

 

[arkajad]

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.

 

[math6]

why we must have linear isomorphisme onto a fibre ?

 

[arkajad]

why we must have linear isomorphisme onto a fibre ?

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

$$p+q=F^{-1}(F(p)+F(q))$$

$$ap= F^{-1}(aF(p))$$

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.

 

[math6]

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 ?

 

[arkajad]

if you look to definition in the article you always preserve the same base ?

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.

 

[math6]

thnx arkajad .