# Bundle morphisms and automorphism

• math6
In summary, the conversation discusses the definition of an isomorphism of vector bundles covering a map between two manifolds. It is established that for the map to be a diffeomorphism, we need the base manifold and total space to be diffeomorphic. The conversation then moves on to discussing the isomorphism between any pair of fibers of the vector bundle and its pullback, and how an automorphism of a trivial bundle can be defined by a smooth map. The conversation ends with a question about the definition of bundle automorphisms and the importance of preserving the base.
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 .

math6 said:
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.

math6 said:
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

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 ?

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

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 ?

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.

why we must have linear isomorphisme onto a fibre ?

math6 said:
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.

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 ?

math6 said:
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.

## 1. What are bundle morphisms and automorphisms?

Bundle morphisms and automorphisms are mathematical concepts that describe the relationship between two bundles, or collections of mathematical objects. A bundle morphism is a function that maps the elements of one bundle to the elements of another bundle in a way that preserves the structure of the bundles. An automorphism is a type of bundle morphism that maps a bundle to itself, essentially describing a symmetry or transformation of the bundle.

## 2. What is the significance of bundle morphisms and automorphisms?

Bundle morphisms and automorphisms are important in mathematics because they allow us to study the properties and symmetries of bundles in a more abstract way. By understanding the relationships between bundles, we can gain a deeper understanding of the mathematical structures they represent.

## 3. How do you determine if two bundles are isomorphic?

To determine if two bundles are isomorphic, we can look for a bundle morphism between them that is both one-to-one and onto. This means that the function maps every element of one bundle to a unique element in the other bundle, and that every element in the second bundle has a corresponding element in the first bundle. If such a function exists, then the two bundles are isomorphic.

## 4. Can a bundle have multiple automorphisms?

Yes, a bundle can have multiple automorphisms. In fact, the set of all automorphisms of a bundle forms a group, meaning that they can be composed and inverted to form new automorphisms.

## 5. How are bundle morphisms and automorphisms used in practical applications?

Bundle morphisms and automorphisms have applications in many areas of mathematics, including topology, algebra, and differential geometry. They are also used in physics to describe symmetries and transformations of physical systems. Additionally, these concepts have practical applications in fields like computer science and engineering, where they can be used to study and analyze complex systems.

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