What flows on a surface can be geodesic flows?

[lavinia]

The question is what flows on a surface can be geodesic flows. Specifically, starting with a smooth vector field on a surface - perhaps with isolated singularities - when is there a Riemannian metric so that the vector field has constant length and is tangent to geodesics on the surface?

Here is an attempt but much more needs to be done.

If I have not made a mistake in computation then here goes.

If the vector field is length 1 it may be viewed as a map into the tangent unit circle bundle. Under this map, the connection 1 form pulls back to a 1 form on the surface. If e1 is the vector field and e2 is the orthogonal vector field with e1,e2 a positively oriented basis for the tangent space at each point, then -[e1,e2], negative the Lie bracket of e1 and e2, is dual to this 1 form - I think. If e1 is tangent to geodesics, then [e1,e2] is a multiple of e2 as can be seen from direct computation - unless I have made an error.

Write [e1,e2] = se2

Let w be the pull back of the connection 1 form. Then dw = -Kvol where K is the Gauss curvature and vol is the volume element. So

dw(e1,e2) = -K = e1.w(e2) - w([e1,e2]) = e1.<e2,-[e1,e2]> - <-[e1,e2],[e2,e2]>

Note that I omitted 1 term on the right because it is zero by assumption.

= -e1.s - s$^{2}$ so one has the differential equation

ds/dx = K + s$^{2}$ where x is the arc length parameter along the geodesic.From this one gets information right away. For instance suppose the Gauss curvature is positive. Then s is increasing along each curve. Therefore the geodesic can not be closed - although it may actually close off at the singularities. But if K = 0 and s = 0 the geodesic can be closed as is illustrated in the flat cylinder.

Further if K > 0 so that s is increasing, the geodesics can not be dense in any region (Cauchy sequence argument I think) and any spiral must have finite length.

Interestingly, in the case of constant positive curvature this integrates to the tangent - forgetting the constant K$^{1/2}$ - which diverges in finite time. So one also gets a bound on the length of the geodesics. On the sphere it diverges right at the north and south poles.So I am asking for ideas on how to push this further.

BTW: s can always be solved for in an open region of any point. So this problem is really asking whether it can be solved for globally on the surface. For instance suppose I have a vector field with a singularity of index -1. Cant it be globally tangent to geodesics?

If this is all wrong - please cut it to pieces.

 

[jvicens]

Thank you for your interesting post and your attempt at solving this problem. I have some thoughts and ideas that may help push this further.

Firstly, I believe your computation is correct and it is a good starting point for solving this problem. However, I think there are some additional considerations that need to be taken into account.

One important aspect is the existence of a Riemannian metric that satisfies the conditions. In general, not every smooth vector field on a surface can be realized as a geodesic flow with a Riemannian metric. This is due to the fact that the vector field may have "twists" or "turns" that cannot be captured by a Riemannian metric. So, in order for a vector field to be tangent to geodesics, it must satisfy certain geometric conditions.

One such condition is that the vector field must be divergence-free. This means that the flow of the vector field must preserve the volume of the surface. In other words, the vector field cannot "squeeze" or "stretch" the surface. This condition is important because it guarantees the existence of a Riemannian metric that satisfies the conditions.

Another important consideration is the behavior of the vector field at the singularities. As you mentioned, the vector field may have isolated singularities. In order for the vector field to be tangent to geodesics, these singularities must be "nice" in some sense. For example, they cannot be too "wild" or "chaotic" as this would prevent the existence of a Riemannian metric.

In terms of pushing this further, I suggest looking at specific examples of vector fields on surfaces and trying to see if they can be realized as geodesic flows with a Riemannian metric. This may give some insight into the necessary conditions for a vector field to be tangent to geodesics.

Additionally, it may be helpful to look at the global behavior of the vector field. For example, can the vector field be realized as a complete geodesic flow on the surface? Are there any obstructions to this? These are some questions that may help in further understanding this problem.

Overall, I think your approach is a good starting point and with some additional considerations and specific examples, we can make progress in solving this problem. I hope this helps and I look forward to hearing more about your thoughts and ideas on this topic.