Can a Set of Sections Globally Generate a Vector Bundle?

  • Context: Graduate 
  • Thread starter Thread starter math6
  • Start date Start date
  • Tags Tags
    Vector
Click For Summary

Discussion Overview

The discussion revolves around the conditions under which a set of sections of a vector bundle can be considered to generate the bundle at various points on a manifold. Participants explore the implications of linear independence and the spanning properties of sections across different fibers of the vector bundle.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether a family of sections generating the vector bundle at one point implies they generate it at another point, raising concerns about the consistency of properties across fibers.
  • Another participant states that a set of sections can be linearly independent at one point but not at another, which affects their ability to span the fiber at those points.
  • There is a suggestion that if sections are linearly independent at a point, they span the fiber at that point, but this does not guarantee the same for other points unless they maintain linear independence.
  • Some participants seek clarification on whether the dependence of sections at one point implies they can still generate the vector bundle at another point.
  • One participant asserts that if sections form a basis at any point, they span all fibers simultaneously, inviting proof of this claim.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between linear independence of sections and their ability to generate the vector bundle across different fibers. The discussion remains unresolved, with multiple competing perspectives on the implications of these properties.

Contextual Notes

There are limitations regarding the assumptions made about the nature of sections and their linear independence, as well as the definitions of generating sets and bases in the context of vector bundles.

math6
Messages
66
Reaction score
0
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 ...
 
Physics news on Phys.org
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?
 
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?
 
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.
 
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. ""
 
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?
 
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 ?
 
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?
 

Similar threads

Graduate Problem in Understanding notation of distributional section
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Why are sheafs defined using abelian groups?
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad About the maximal extension of local charts on a manifold
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Diffeomorphism invariance and contracted Bianchi identity
  • · Replies 7 ·
Replies
7
Views
4K
Graduate Penrose paragraph on Bundle Cross-section?
  • · Replies 15 ·
Replies
15
Views
4K
Graduate Two questions for vector bundles
  • · Replies 18 ·
Replies
18
Views
2K
Graduate On the relationship between Chern number and zeros of a section
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Pushforward of Smooth Vector Fields
  • · Replies 1 ·
Replies
1
Views
4K
Undergrad What is the construction of charts for the tangent bundle on the unit circle?
  • · Replies 8 ·
Replies
8
Views
4K
Graduate Connection between 1-Forms and Fourier Transform
  • · Replies 3 ·
Replies
3
Views
3K