Why Can Some Line Bundles Have Nonzero Sections While Others Cannot?

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SUMMARY

The discussion centers on the concept of sections of vector bundles, specifically contrasting the Mobius band and a twisted plane. The Mobius band is identified as a nontrivial line bundle over the circle, which prevents the existence of a nonzero section from the circle to the Mobius strip. In contrast, a twisted plane is classified as a trivial line bundle over a line, allowing for the existence of a nonzero section due to the missing point where the section would be zero. This highlights the fundamental differences in the topology of these bundles.

PREREQUISITES
  • Understanding of vector bundles
  • Familiarity with the Mobius band and its properties
  • Knowledge of trivial vs. nontrivial line bundles
  • Basic concepts of topology
NEXT STEPS
  • Study the properties of vector bundles in algebraic topology
  • Explore the implications of nontrivial line bundles in differential geometry
  • Learn about the construction of sections in vector bundles
  • Investigate the relationship between topology and geometry in twisted planes
USEFUL FOR

Mathematicians, particularly those specializing in topology and differential geometry, as well as students seeking to deepen their understanding of vector bundles and their sections.

duranduran
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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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because the missing point of that twisted plane is the point where the section would have been zero.
 
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
 

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