Geodesic flows on compact surfaces

Click For Summary
SUMMARY

A geodesic flow on a compact 2-dimensional Riemannian manifold without boundary can exhibit unique behaviors depending on the surface. For example, on a sphere, great circles connecting the poles have orthogonal circles that are only geodesics at the equator. Conversely, on a flat torus, orthogonal curves are consistently geodesics. The discussion raises the question of when a vector field with isolated singularities can be tangent to geodesics, particularly under varying metrics.

PREREQUISITES
  • Understanding of Riemannian geometry
  • Familiarity with geodesics and their properties
  • Knowledge of vector fields and singularities
  • Basic concepts of differential geometry
NEXT STEPS
  • Explore the properties of geodesics on Riemannian manifolds
  • Study the implications of different metrics on geodesic flows
  • Investigate vector fields with isolated singularities in differential geometry
  • Learn about harmonic and isothermal coordinates in Riemannian geometry
USEFUL FOR

Mathematicians, physicists, and students interested in Riemannian geometry, differential geometry, and the study of geodesic flows on compact surfaces.

lavinia
Science Advisor
Messages
3,385
Reaction score
760
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?
 
Last edited:
Physics news on Phys.org
some words from an ignorant rookie: harmonic? isothermal?
 
mathwonk said:
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?
 

Similar threads

Undergrad Compact manifold cannot be represented by (single) parametric equation
  • · Replies 5 ·
Replies
5
Views
1K
Graduate Understanding: double of conformally flat manifold is conformally flat
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Parallel Transport Along Geodesic
  • · Replies 6 ·
Replies
6
Views
5K
Undergrad Get geodesics & parallel transport on 2D surfaces (discrete timestep)
  • · Replies 12 ·
Replies
12
Views
3K
Graduate Geodesics on three dimensional graph
  • · Replies 9 ·
Replies
9
Views
3K
High School Can Curvature and Geodesics be Generalized in Higher Dimensions?
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Pseudo-Riemaniann isometries
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Natural parametrization of a curve
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Notion of parallel worldlines in curved geometry
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad About global inertial frame in GR - revisited
  • · Replies 56 ·
2
Replies
56
Views
4K