# Sections of the vector bundle

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 ...

## Answers and Replies

lavinia
Gold Member
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?

lavinia
Gold Member
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. ""

lavinia
Gold Member
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 ?

lavinia