Kähler Magnetic Flows for a Manifold of Constant Holomorphic Sectional Curvature

Adachi, Toshiaki

doi:10.3836/tjm/1270043477

Cited by 88 publications

(63 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the study of (uniform) magnetic fields, since we have no nontrivial uniform magnetic fields on non-flat real space form of dimension greater than 2, the author thinks that complex space forms, complex projective spaces, complex Euclidean spaces, and complex hyperbolic spaces, play as model spaces. In the preceeding papers [1] and [2] we studied trajectories for Kahler magnetic fields on complex space forms, and pointed out that the feature of trajectories depends on the curvature condition of the base manifold and on the strength of a uniform magnetic field. In this context it is quite natural to study comparison theorems associated to uniform magnetic fields.…”

Section: -V 2 B X )mentioning

confidence: 99%

“…In order to get rid of the choice of the orientation of R 3 it is natural to identify IB with the 2-form B = B t dx 2 A dxj + B 2 <fx 3 A dx x + B 3 dx t A dx 2 . Under this identification the Gauss formula is equivalent to dB = 0, and the Newton With this observation we call a closed 2-form IB on a Riemannian manifold (M, (,)) a magnetic field. Let fi = fi B : TM -> TM denote the skew symmetric operator satisfying B(u, v) = (u, Q(v)) for every u, v e TM.…”

Section: -V 2 B X )mentioning

confidence: 99%

“…Solving these equations we get the following. +x -> CH" is the standard S'-fibration with f/f +l ={ze C + l | «z, z» = -1 } , and y is a horizontal lift of y into / / j " + l (see [2]). Therefore normal fe • By-Jacobi fields along y with 7(0) = 0 are given by…”

Section: Kahler Magnetic Jacobi Fields On Complex Space Formsmentioning

confidence: 99%

“…On a charged particle with charge q and velocity vector v = (u,, v 2 , u 3 ), this yields the Lorentz force ¥ = q-\ xE-<j(u 2 B 3 -u 3 B 2 , u 3 B, -u,B 3 , u,B 2 …”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A comparison theorem on magnetic jacobi fields

Adachi

1997

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

A scalar multiple of the Kahler form of a Kahler manifold is called a Kahler magnetic field. We are focused on trajectories of charged particles under this action. As a variation of trajectories we define a magnetic Jacobi field. In this paper we discuss a comparison theorem on magnetic Jacobi fields, which corresponds to the Rauch's comparison theorem.1991 Mathematics subject classification: 53C22.53C55.

show abstract

Section: -V 2 B X )mentioning

confidence: 99%

Section: -V 2 B X )mentioning

confidence: 99%

Section: Kahler Magnetic Jacobi Fields On Complex Space Formsmentioning

confidence: 99%

“…On a charged particle with charge q and velocity vector v = (u,, v 2 , u 3 ), this yields the Lorentz force ¥ = q-\ xE-<j(u 2 B 3 -u 3 B 2 , u 3 B, -u,B 3 , u,B 2 …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A comparison theorem on magnetic jacobi fields

Adachi

1997

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…In our recent works we gave a light on geometric properties of circles on a Kähler manifold. For example, we interpreted holomorphic circles in terms of magnetic fields ( [1], [3]). We defined Kähler magnetic flows associated to these circles, and showed that Kähler magnetic flows of small geodesic curvature on a complex hyperbolic space are conjugate to the geodesic flow, hence are of Anosov type.…”

Section: §1 Introductionmentioning

confidence: 99%

Distribution of length spectrum of circles on a complex hyperbolic space

Adachi¹

1999

Nagoya Mathematical Journal

Self Cite

View full text Add to dashboard Cite

Abstract. It is well-known that all geodesics on a Riemannian symmetric space of rank one are congruent each other under the action of isometry group. Being concerned with circles, we also know that two closed circles in a real space form are congruent if and only if they have the same length. In this paper we study how prime periods of circles on a complex hyperbolic space are distributed on a real line and show that even if two circles have the same length and the same geodesic curvature they are not necessarily congruent each other. §1. IntroductionThe aim of this paper is to study the distribution of the length of closed circles on a complex hyperbolic space. In our recent works we gave a light on geometric properties of circles on a Kähler manifold. For example, we interpreted holomorphic circles in terms of magnetic fields ([1], [3]). We defined Kähler magnetic flows associated to these circles, and showed that Kähler magnetic flows of small geodesic curvature on a complex hyperbolic space are conjugate to the geodesic flow, hence are of Anosov type. This result gives us information on length spectrum of holomorphic circles of small geodesic curvature on a compact manifold of constant holomorphic sectional curvature: The number of closed holomorphic circles grows exponentially with respect to their length (cf. [18], [8], [14] for more detail). In this paper we add another result on the feature of the length spectrum of circles of large geodesic curvature on a complex hyperbolic space in connection with the action of the isometry group.A smooth curve γ: R −→ M on a complete Riemannian manifold M is called a circle of geodesic curvature κ (≥ 0) if it is parametrized by its arc-length and satisfies the following equation: [17]). Needless to say, circles are classified by their geodesic curvature. When κ = 0, this equation is nothing but the equation of geodesics. One may think that the notion of circles is just a natural extension of the notion of geodesics. But when M is a Kähler manifold we have another tool for classification of circles which is associated with the complex structure J: For a circle γ we define its complex torsion τ by γ, J∇ tγ / ∇ tγ . This does not depend on t and satisfies |τ | ≤ 1. It might be natural to think that some properties of circles are related to the Kähler geometry of the base manifold.We call a circle γ closed if there exists a constant T withThis condition is equivalent to the condition that γ(t + T ) = γ(t) for every t. The minimum positive constant with these properties is called the prime period of γ and is denoted by length(γ). We put length(γ) = ∞ for an open circle γ, a circle which is not closed. We are interested in how prime periods of closed circles are distributed on the real line. It is well known that on a compact rank one symmetric space every geodesic is closed and has the same length. Moreover all these geodesics are congruent each other, and the same thing holds for geodesics on a rank one symmetric space of noncompact type. Here we call two circles γ 1 an...

show abstract