Geodesic flows on compact surfaces

[lavinia]

Does a geodesic flow on a compact surface - compact 2 dimensional Riemannian manifold without boundary - always have a geodesic that is orthogonal to the flow?

 

[mathwonk]

some words from an ignorant rookie: harmonic? isothermal?

 

[lavinia]

some words from an ignorant rookie: harmonic? isothermal?

Not sure if it is an interesting question but here is an example.

On the sphere take the great circles connecting the north and south poles. The orthogonal circles to them are not geodesics except at the equator which is itself a great circle. So in this case the answer is yes and the geodesic is unique.

On a flat torus the orthogonal curves are always geodesics.

The underlying question I am really asking is when can a vector field with isolated singularities be everywhere tangent to geodesics. For the sphere with the usual metric the vector field must be tangent to great circles. But what is the metric is different? Can a vector field with a singularity of index -1 be tangent to geodesics?