sections of the vector bundle


by math6
Tags: bundle, sections, vector
math6
math6 is offline
#1
Feb15-13, 02:01 PM
P: 67
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 ...
Phys.Org News Partner Science news on Phys.org
Lemurs match scent of a friend to sound of her voice
Repeated self-healing now possible in composite materials
'Heartbleed' fix may slow Web performance
lavinia
lavinia is offline
#2
Feb15-13, 03:26 PM
Sci Advisor
P: 1,716
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?
math6
math6 is offline
#3
Feb16-13, 11:17 AM
P: 67
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?

lavinia
lavinia is offline
#4
Feb18-13, 03:11 PM
Sci Advisor
P: 1,716

sections of the vector bundle


Quote Quote by math6 View Post
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.
math6
math6 is offline
#5
Feb18-13, 05:01 PM
P: 67
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. ""
lavinia
lavinia is offline
#6
Feb18-13, 05:12 PM
Sci Advisor
P: 1,716
Quote Quote by math6 View Post
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?
math6
math6 is offline
#7
Feb18-13, 05:32 PM
P: 67
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 ?
lavinia
lavinia is offline
#8
Feb18-13, 05:35 PM
Sci Advisor
P: 1,716
Quote Quote by math6 View 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 ?
I said

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


Register to reply

Related Discussions
holomorphic vector bundle Differential Geometry 2
What does it mean by a Riemannian metric on a vector bundle? Differential Geometry 9
dual vector bundle E* is isomorphic to Hom(E, MXR) Differential Geometry 4
tangent bundle of vector space Differential Geometry 2
What is a Vector Bundle? Differential Geometry 3