Vector field as smooth embedding

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SUMMARY

The discussion focuses on demonstrating that any vector field V: M -> TM is a smooth embedding of the manifold M. Participants explore the concept of smoothly homotopic embeddings and propose constructing a homotopy by connecting points in the tangent space using straight line paths. The straight line homotopy is identified as a viable method for achieving this connection, emphasizing the importance of understanding the tangent bundle and homotopy theory in this context.

PREREQUISITES
  • Understanding of smooth manifolds and vector fields
  • Familiarity with tangent bundles, specifically TM
  • Basic knowledge of homotopy theory
  • Concept of smooth embeddings in differential geometry
NEXT STEPS
  • Study the properties of tangent bundles in differential geometry
  • Learn about smooth homotopies and their applications
  • Explore the concept of straight line homotopy in more detail
  • Investigate examples of smooth embeddings in various manifolds
USEFUL FOR

Mathematicians, particularly those specializing in differential geometry, topology, and anyone interested in the applications of vector fields and homotopy theory.

huyichen
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We can show that any vector field V:M->TM(tangent bundle of M) is smooth embedding of M, but how do we show that these smooth embeddings are all smoothly homotopic? How to construct such a homotopy?
 
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I do not know much about homotopy, but my guess is that for every p in M you smoothly connect X(p) to Y(p) using the straight line connecting V(p) and W(p) in the tangent vector space at p.
 
Yep, it's the straight line homotopy.
 

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