Sections of the vector bundle

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In summary: Yes, I can prove this. Suppose that E is a vector bundle over a manifold M and (s_ {1}, ..., s_ {n}) is a family of sections of E. Then every point x in M has a unique sectional representation (s_ {1} (x), ...,s_ {n} (x)) in E. Moreover, every sectional representation is a basis for the vector space E_{x} consisting of all linear combinations of the sectional representations. Therefore, every point x in M has a unique vector space representation (s_ {1} (x), ...,s_ {n} (x)) in E.
  • #1
math6
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Hi Friends :))
my little problem is :
Let E be a vector bundle over a manifold M, and (s_ {1}, ..., s_ {n}) a family of sections of E. This family is generating bundle E, ​​ that is for every point x in M, (s_ {1} (x), ..., s_ {n} (x)) is generator of the vector space E_{x} ? is that we have only (s_ {1} (x1), ..., s_ {n} (x1)) is a generator of E_ {x1} and (s_ {1} (x 2), .. ..., s_ {n} (x2)) is not generating E_ {x2}??
Thank you for making me understand this confusion on sections of a vector bundle generator ...
 
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  • #2
A set of sections may be lineally independent at one point but not another. It cannot span the fiber above a point where they are not linearly independent.

But over any point where there are n linearly independent sections they span the fiber. Over another point where they are independent they also space the fiber. That means that on that fiber some linear combination of the sections equals any v.

But more is true: they simultaneously span all of the fibers where they from a basis. Can you prove this?
 
  • #3
My problem is: if a family of sections generate E_ {x1}? is that this family engandrent E_ {x2} or any other fiber, or a family of sections if (free generator, form a basis ...) is that above all point x variety, (S_{1}, ...,s_{n}) are kept the same properties?
 
  • #4
math6 said:
My problem is: if a family of sections generate E_ {x1}? is that this family engandrent E_ {x2} or any other fiber, or a family of sections if (free generator, form a basis ...) is that above all point x variety, (S_{1}, ...,s_{n}) are kept the same properties?

Read my post carefully.
 
  • #5
If I understand what you mean, a family of sections can be generating a vector bundle at a point that if they are linearly independent
"" It cannot span the fiber above a point where they are not linearly independent. ""
 
  • #6
math6 said:
If I understand what you mean, a family of sections can be generating a vector bundle at a point that if they are linearly independent
"" It cannot span the fiber above a point where they are not linearly independent. ""

What about the last sentence in the post?
 
  • #7
you said " But over any point where there are n linearly independent sections they span the fiber. Over another point where they are independent they also space the fiber " . you want to say dependent, they also generate the vector bundle ?
 
  • #8
math6 said:
you said " But over any point where there are n linearly independent sections they span the fiber. Over another point where they are independent they also space the fiber " . you want to say dependent, they also generate the vector bundle ?

I said

But more is true: they simultaneously span all of the fibers where they from a basis. Can you prove this?
 

1. What is a section of a vector bundle?

A section of a vector bundle is a function that assigns to each point in the base space a vector in the associated fiber. Essentially, it is a continuous assignment of a vector to each point of the base space.

2. What is the difference between a section and a vector field?

A section and a vector field are essentially the same thing, but the terminology depends on the context. In general, a section refers to a function defined on a vector bundle, whereas a vector field refers to a function defined on a smooth manifold.

3. How are sections of a vector bundle used in physics?

Sections of a vector bundle are used in many areas of physics, including classical mechanics, electromagnetism, and quantum field theory. They are used to describe physical quantities, such as the position and momentum of a particle, and to study the behavior of physical systems.

4. What is the importance of understanding sections of a vector bundle?

Understanding sections of a vector bundle is crucial in many mathematical and scientific fields, including differential geometry, topology, and physics. It allows for the study of vector fields and their properties, which are essential in understanding many physical phenomena.

5. How are sections of a vector bundle visualized?

Sections of a vector bundle can be visualized as arrows or vectors attached to each point on a base space. In a 2-dimensional space, for example, the sections can be represented as arrows at each point on a plane. In higher dimensions, they can be visualized using vector fields or through mathematical representations.

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