紀要論文

紀要論文

text

Osaka Journal of Mathematics. 2013, 50(3), p. 749-763

THE MAGNETIC FLOW ON THE MANIFOLD OF ORIENTED GEODESICS OF A THREE DIMENSIONAL SPACE FORM

Osaka University and Osaka City University, Departments of Mathematics

英語

text

2013-09

Let M be the three dimensional complete simply connected manifold of constant sectional curvature 0, 1 or -1. Let L be the manifold of all (unparametrized) complete oriented geodesics of M, endowed with its canonical pseudo-Riemannian metric of signature (2, 2) and Kähler structure J. A smooth curve in L determines a ruled surface in M. We characterize the ruled surfaces of M associated with the magnetic geodesics of L, that is, those curves σ in L satisfying ∇_σ σ = Jσ. More precisely: a timelike (space-like) magnetic geodesic determines the ruled surface in M given by the binormal vector field along a helix with positive (negative) torsion. Null magnetic geodesics describe cones, cylinders or, in the hyperbolic case, also cones with vertices at infinity. This provides a relationship between the geometries of L and M.

10.18910/26012

26012

THE MAGNETIC FLOW ON THE MANIFOLD OF ORIENTED GEODESICS OF A THREE DIMENSIONAL SPACE FORM

Osaka University and Osaka City University, Departments of Mathematics

英語

2013-09

Osaka Journal of Mathematics

50

3

749

763

https://hdl.handle.net/11094/26012

10.18910/26012

https://doi.org/10.18910/26012