Sections of a line bundles

In summary, the conversation discusses the concept of sections of vector bundles, using the examples of the Mobius band and a twisted plane. It is noted that while a nontrivial line bundle like the Mobius band does not have any nonzero sections from the circle to the Mobius strip, a twisted plane, which is a trivial line bundle over a line, does have a nonzero section. The conversation then suggests trying to construct a nonzero section for the twisted plane to understand how this is possible.
  • #1
duranduran
1
0
Hey guys,
I am confused about the concept of sections of vector bundles. Mobius band is nontrivial line bundle over circle so we can not find any nonzero section from circle to mobius strip. However a plane which is twisted once is a trivial line bundle over line. That means there is a nonzero section for that bundle. How is it possible?
 
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  • #2
because the missing point of that twisted plane is the point where the section would have been zero.
 
  • #3
duranduran said:
Hey guys,
I am confused about the concept of sections of vector bundles. Mobius band is nontrivial line bundle over circle so we can not find any nonzero section from circle to mobius strip. However a plane which is twisted once is a trivial line bundle over line. That means there is a nonzero section for that bundle. How is it possible?

Try constructing a non-zero section
 

1. What is a line bundle?

A line bundle is a mathematical construct used in algebraic geometry and topology to describe a space that varies smoothly in one direction. It is a generalization of the concept of a tangent bundle, which describes the variation of a space in all directions.

2. What are the sections of a line bundle?

The sections of a line bundle are the functions that map points in the base space to points in the total space. These functions are continuous and smooth, and they preserve the structure of the line bundle.

3. How are line bundles used in physics?

Line bundles are used in physics to describe gauge fields, which are fundamental forces such as electromagnetism and the strong and weak nuclear forces. These forces are described mathematically by connections on line bundles.

4. What is the importance of holomorphic line bundles?

Holomorphic line bundles are important in complex geometry and algebraic geometry. They are used to study complex manifolds and algebraic varieties, and they have applications in string theory and mirror symmetry.

5. How do line bundles relate to divisors?

In algebraic geometry, divisors are used to describe the zeros and poles of a function on a variety. Line bundles can be associated with divisors, and they provide a way to study the geometry of the variety through the algebraic properties of the line bundle.

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