Hi everyone, a couple of technical questions :(adsbygoogle = window.adsbygoogle || []).push({});

1) Definition: Anyone know the definition of the induced orientation of a submanifold S of an orientable manifold M?

2)Dividing sets in contact manifolds: We have a contact 3-manifold (M^{3},ζ ). We

define a surface S embedded in M^{3}to be a convex surface if there exists a contact

vector field X that is transverse to S, i.e., X does not live on the surface; X is not in the span of any basis for TpS. Now, we define the dividing set of the surface S to be the set of points of X that live in the contact planes , i.e., p is in the dividing set if X(p) is in ζ(p) ; ζ(p) is the contact plane at the point p, and X(p) is the contact vector field at p ( a contact vector field for (M^{3},ζ ) is a vector field whose flow preserves ζ , i.e., L_{X}ζ=gζ , where L is the

Lie derivative of the form ζ along the vector field X, and g is a positive smooth function.

So, say I have the standard contact structure in R^{3}given by ker(cos(πr)dx+sin(πr)dθ) . I know ∂/∂z is a contact field , so that it is transverse to any disk in the xy-plane. How do I find the dividing set in this case? I need to find the points in R^{3}so that ∂/∂z (p)

( basically, the z-axis "based at p " ) lies in the contact plane at p.

I'm kind of stuck in a loop here; any suggestions, please ?

Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Dividing Sets in Contact Structures, and Induced Orientations

**Physics Forums | Science Articles, Homework Help, Discussion**