# Dividing Sets in Contact Structures, and Induced Orientations

• WWGD
In summary, the conversation discusses the definition of induced orientation of a submanifold in an orientable manifold, and the concept of dividing sets in contact manifolds. The main question is how to find the dividing set in a specific case using the standard contact structure. The conversation also mentions the need to derive a basis for the contact structure explicitly.
WWGD
Gold Member
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 (M3,ζ ). We
define a surface S embedded in M3 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 (M3,ζ ) is a vector field whose flow preserves ζ , i.e., LXζ=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 R3 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 R3 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.

Last edited:
Well, it seems I may have to go thru the pain of deriving a ( basis for ) the contact structure explicitly:

We set :

(cosrdz+ rsinrdθ)(a∂/∂z+ b∂/∂θ)=0 ; the r variable is free. Then:

acosr+ brsinr =0 , so that acosr = -brsinr

So that { ( 0, b, -brtanr ), (c,0,0) }; c any Real number is a basis for the contact planes, and the vector field clearly has singularities at r=k∏. But now I have to figure out how to crank out the contact planes using the basis. Not hard, but pretty tedious, since this is a curved system ( it is cylindrical-coordinates-based ). Would be great if someone had a simpler approach.

## 1. What is a contact structure?

A contact structure is a mathematical concept used in the study of geometry and topology. It involves a smooth manifold, which is a type of mathematical space, and a specific type of vector field known as a contact vector field. This structure is used to study the behavior of surfaces and their interactions with other objects.

## 2. How does dividing sets in contact structures work?

Dividing sets in contact structures involves separating a contact structure into smaller, simpler pieces, known as "dividing sets". These sets can then be studied and analyzed individually, which can provide insight into the behavior of the overall structure.

## 3. What are induced orientations?

Induced orientations refer to a way of assigning a direction to a surface or object within a contact structure. This direction is determined by the contact vector field and is used to study the behavior and interactions of the surface or object within the structure.

## 4. How are dividing sets and induced orientations related?

Dividing sets and induced orientations are closely related in the study of contact structures. The dividing sets can help identify and analyze the behavior of surfaces within the structure, and the induced orientations provide a way to assign directions to these surfaces for further analysis and understanding.

## 5. What are some real-world applications of dividing sets in contact structures and induced orientations?

This mathematical concept has various applications in fields such as physics, engineering, and robotics. For example, it can be used to study the behavior of fluids and surfaces in fluid dynamics, analyze the motion of particles in a magnetic field, and design efficient robotic movements and interactions with surfaces.

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