A Family of curves tangent to a smooth distribution of lines

[Lodeg]

Hi,
Given a smooth distribution of lines in R2, could we assert that there is a unique distribution of curves such that:
- the family of curves "fill in" R2 completely
- every curve is tangent at every point to one of the smooth distribution of lines

 

[lavinia]

Can you be more precise about what you mean by a smooth distribution of lines?

 

[Lodeg]

I mean: to each point p in R2 is associsted a line Lp. The family Lp depends smoothly on p. It is somewhat a smooth subbundle of TR2

 

[lavinia]

I mean: to each point p in R2 is associsted a line Lp. The family Lp depends smoothly on p. It is somewhat a smooth subbundle of TR2

So can these lines intersect? Or do you mean directions at each point not straight lines in space?

 

[Lodeg]

Indeed, directions at each point.

 

[lavinia]

Indeed, directions at each point.

A smooth set of directions is called a vector field and I think a fundamental theorem on solutions of ordinary differential equations guarantees an integral curve through any point.