# Bundle morphisms and automorphism

1. Oct 7, 2010

### 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 .

2. Oct 7, 2010

### lavinia

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

The second is obvious.

3. Oct 8, 2010

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

4. Oct 8, 2010

### 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 ?

5. Oct 8, 2010

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

6. Oct 8, 2010

### math6

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 ?

7. Oct 8, 2010

### math6

8. Oct 8, 2010

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.

9. Oct 8, 2010

### math6

why we must have linear isomorphisme onto a fibre ?

10. Oct 8, 2010

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.

11. Oct 8, 2010

### 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 .

"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 ?

12. Oct 8, 2010