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Hi everyone, a couple of technical questions :

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

define a surface S embedded in M

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

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

( 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.

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 ). Wedefine a surface S embedded in M

^{3}to be a convex surface if there exists a contactvector 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 theLie 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.

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