Vector field as smooth embedding

[huyichen]

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?

 

[arkajad]

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.

 

[zhentil]

Yep, it's the straight line homotopy.