# Vector fields transverse to the boundary of a manifold

• jford1906
In summary, a flow being transverse to the boundary means that the flow is never tangent to the boundary and must either point inward or outward on each component of the boundary. Since the solid torus only has one boundary component, a flow transverse to its boundary must be everywhere outward or inward. Additionally, any flow transverse to the boundary of a manifold cannot be periodic, as it would carry all points back to where they started in a certain time.
jford1906
I'm trying to work up some examples to help me understand this concept. Would the periodic flow on a solid torus be transverse to it's boundary?

Transverse I think means here not lying in the tangent space to the boundary points.

So since it's a periodic flow, say given by $$dx = -y, dy=x \mbox{ and }dz = 0,$$ all trajectories are parallel to the boundary, so they would not lie in the tangent space to any boundary points, and the flow would be transverse?

That sounds right.

Rock. Thanks!

jford1906 said:
So since it's a periodic flow, say given by $$dx = -y, dy=x \mbox{ and }dz = 0,$$ all trajectories are parallel to the boundary, so they would not lie in the tangent space to any boundary points, and the flow would be transverse?

Transverse to the boundary means that the part of the flow that is on boundary does not lie in the tangent space to the boundary. So the flow must point either outwards or inwards.

jford1906 said:
I'm trying to work up some examples to help me understand this concept. Would the periodic flow on a solid torus be transverse to it's boundary?

First of all, there are a lot of different periodic flows on a solid torus, so no one flow is "the" periodic flow.*

Secondly (as Lavinia has said), a flow being transverse to a manifold's boundary means that the flow is never tangent to the boundary. It follows that on each component of the boundary, the flow is either pointing only inwards, or else pointing only outwards on that component. Since the solid torus has only one boundary component, of course, a flow transverse to its boundary must be everywhere outward or everywhere inward on the boundary.

Clearly, any flow transverse to the boundary of any manifold cannot be periodic, since a periodic flow means that at a certain time T > 0, the flow carries all points back to where they started.
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* For example: If θ is the angular coordinate in the solid torus D2 × S1 (the disk cross the circle = {exp(iθ) : 0 <= θ < 2π}), then the simplest periodic flow is given by the vector field that is d/dθ everywhere. But now we can imagine cutting D2 × S1 along say the disk D2 × {exp(i0)}, giving one exposed disk a rotation by angle (2πp/q) where p/q is any rational number, and the reattaching the exposed disks to make a solid torus again. In this case, the flow will carry all points back to where they started at time 2πq. The core circle of the solid torus will flow back to where it started in less time, namely 2π, than any other trajectory.

## 1. What is a vector field?

A vector field is a mathematical concept that assigns a vector to each point in a given space, such as a plane or a three-dimensional space.

## 2. What is a manifold?

A manifold is a mathematical object that is locally similar to a Euclidean space. It can be thought of as a surface that can be smoothly bent or curved without tearing or stretching.

## 3. What does it mean for a vector field to be transverse to the boundary of a manifold?

A vector field is transverse to the boundary of a manifold if at each point on the boundary, the vector is perpendicular to the surface of the boundary. This means that the vector field is not tangent to the boundary, but instead crosses it at a right angle.

## 4. Why is it important for a vector field to be transverse to the boundary of a manifold?

Having a vector field that is transverse to the boundary of a manifold allows for a better understanding of the behavior of the vector field near the boundary. It also helps in solving certain differential equations and in studying the dynamics of a system.

## 5. What are some applications of vector fields transverse to the boundary of a manifold?

Some applications include fluid dynamics, electromagnetism, and general relativity. In fluid dynamics, the velocity of a fluid can be represented as a vector field that is transverse to the boundary of a manifold. In electromagnetism, the electric and magnetic fields can be represented as vector fields that are transverse to the boundary of a manifold. In general relativity, the curvature of spacetime can be represented as a vector field that is transverse to the boundary of a manifold.

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