Nonvanishing section for direct sum of Mobius band

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SUMMARY

The discussion centers on the existence of nonvanishing sections for the direct sum of a Möbius band. The user presents two sections, s1 and s2, defined as s1=(E^(i*theta), (Cos(theta/2), Sin(theta/2))) and s2=(E^(i*theta), (-Sin(theta/2), Cos(theta/2)), respectively. These sections are confirmed to be linearly independent and nonvanishing, establishing that the direct sum of the Möbius band indeed possesses the required properties.

PREREQUISITES
  • Understanding of differential geometry concepts, particularly related to vector bundles.
  • Familiarity with the properties of the Möbius band.
  • Knowledge of linear independence in the context of vector spaces.
  • Basic proficiency in complex numbers and trigonometric functions.
NEXT STEPS
  • Study the properties of vector bundles over non-orientable surfaces.
  • Explore the implications of linear independence in differential geometry.
  • Research the applications of the Möbius band in topology and physics.
  • Learn about the construction and properties of nonvanishing sections in vector bundles.
USEFUL FOR

Mathematicians, particularly those specializing in topology and differential geometry, as well as physics researchers interested in the applications of non-orientable surfaces.

huyichen
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For a direct sum of Mobius band, it is trivial if it has two linear independent nonvanishing sections. I have the following as my sections：
s1=(E^(i*theta), (Cos(theta/2), Sin(theta/2))
s2=(E^(i*theta), (-Sin(theta/2), Cos(theta/2))
Clearly, the above sections are linearly independent and nonvanishing, but I am not sure if they are indeed the correct ones, need help to confirm!
 
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