HelpA problem about embedded surface.

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SUMMARY

The discussion centers on demonstrating that the torus T^2, parametrized by t(u,v)=((2+cosu)cosv,(2+cosu)sinv,sinu), is an embedded surface in R^3. The key method to establish this is by proving that the parametrization is a one-to-one immersion. Additionally, it is noted that the compactness of the torus plays a crucial role in this proof. Furthermore, the discussion touches on the triviality of the tangent bundle T(T^2).

PREREQUISITES
  • Understanding of differential geometry concepts, particularly embedded surfaces.
  • Familiarity with parametrizations of surfaces in R^3.
  • Knowledge of the properties of compact spaces in topology.
  • Basic comprehension of tangent bundles and their triviality.
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  • Study the concept of one-to-one immersions in differential geometry.
  • Research the properties of compact surfaces and their implications.
  • Learn about tangent bundles and conditions for their triviality.
  • Explore examples of parametrizations of other surfaces in R^3.
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Mathematicians, students of differential geometry, and anyone interested in the properties of surfaces and their parametrizations in three-dimensional space.

aquarius
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I have a parametrization of a torus T^2:t(u,v)=((2+cosu)cosv,(2+cosu)sinv,sinu). How to show that T^2 is an embedded surface in R^3? How to show that the tangent bundleT(T^2) is trivial? Thanks.
 
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There's only one way to show that's it's an embedded surface. Show it's a 1-1 immersion, and note that the torus is compact.
 

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