In this paper we classify the magnetic trajectories corresponding to contact magnetic fields in Sasakian manifolds of arbitrary dimension. Moreover, when the ambient is a Sasakian space form, we prove that the codimension of the curve may be reduced to 2. This means that the magnetic curve lies on a 3-dimensional Sasakian space form, embedded as a totally geodesic submanifold of the Sasakian space form of dimension (2n+1).

https://www.tandfonline.com/doi/pdf/10.1080/14029251.2015.1079426

Magnetic curves in Sasakian manifolds

In this paper we classify magnetic trajectories γ in ${{\mathbb{R}}^{2N+1}}$ endowed with a canonical quasi-Sasakian structure, corresponding to a magnetic field proportional to the fundamental 2-form. We prove that they are helices of order 5 and we show that there exists a totally geodesic ${{\mathbb{R}}^5}$ in ${\mathbb{R}^{2N+1}}$ such that γ lies in ${{\mathbb{R}}^5}$. Moreover, the quasi-Sasakian structure of ${{\mathbb{R}}^5}$ is that induced from the ambient manifold.

Magnetic Trajectories in an Almost Contact Metric Manifold {\mathbb{R}^{2N+1}}

On a Kahler manifold we have natural uniform magnetic fields which are constant multiples of the Kahler form. Trajectories, which are motions of electric charged particles, under these magnetic fields can be considered as generalizations of geodesics. We give an overview on a study of Kahler magnetic fields and show some similarities between trajectories and geodesics on Kahler manifolds of negative curvature.

Kähler magnetic fields on Kähler manifolds of negative curvature

A ruled real hypersurface in a complex space form is a real hypersurface having a codimension one foliation by totally geodesic complex hyperplanes of the ambient space. Our main purpose of this paper is to introduce a new viewpoint to investigate such hypersurfaces in complex hyperbolic space 
$$\mathbb {CH}^n$$

. As an application, we study minimal ruled real hypersurfaces in 
$$\mathbb {CH}^n$$

 and also those having constant scalar curvature.

New construction of ruled real hypersurfaces in a complex hyperbolic space and its applications

First, we classify proper biharmonic Hopf real hypersurfaces in $${\mathbb {C}}P^2$$. Next, we classify proper biharmonic real hypersurfaces with two distinct principal curvatures in $${\mathbb {C}}P^n$$, where $$n\ge 2$$. Finally, we prove that biharmonic ruled real hypersurfaces in $${\mathbb {C}}P^n$$ are minimal, where $$n\ge 2$$.

Classification Theorems for Biharmonic Real Hypersurfaces in a Complex Projective Space

In this paper we study congruency of minimal ruled real hypersurfaces in a nonflat complex space form with respect to the action of its isometry group. We show that those in a complex hyperbolic space are classified into 3 classes and show that those in a complex projective space are congruent to each other hence form just one class.

/pdf/congruence-classes-of-minimal-ruled-real-hypersurfaces-in-a-3xfvueoraa.pdf

Congruence classes of minimal ruled real hypersurfaces in a nonflat complex space form

In this paper we study which trajectories for Sasakian magnetic fields are circles on certain standard real hypersurfaces which are called hypersurfaces of type A in a nonflat complex space form. We also give a characterization of these real hypersurfaces by such a circular property of trajectories for Sasakian magnetic fields.

Circular trajectories on real hypersurfaces in a nonflat complex space form

On a real hypersurface in a Kahler manifold we can consider a natural closed 2-form associated with the almost contact metric structure induced by Kahler structure. We treat trajectories under magnetic fields which are constant multiples of this 2-form. We consider a condition for them to be also curves of order 2 on tubes around totally geodesic real hyperbolic spaces in a complex hyperbolic space.

Trajectories for Sasakian magnetic fields on real hypersurfaces of type (B) in a complex hyperbolic space

On a real hypersurface in a Kahler manifold we can consider a natural closed 2-form associated with the almost contact metric structure induced by Kahler structure. Contrary to real hypersurfaces of type (A), on real hypersurfaces of type (B) in a complex hyperbolic space we show that non-geodesic trajectories under Sasakian magnetic fields, which are constant multiples of the natural closed 2-form, are not curves of order 2.

Trajectories on real hypersurfaces of type (B) in a complex hyperbolic space are not of order 2

We study trajectories for Sasakian magnetic fields which are also circles of positive geodesic curvature on geodesic spheres in a complex projective space. Investigating their extrinsic shapes we give a condition for them to be closed. By use of information on lengths of circles on a complex projective space, we give their lengths, and estimate the bottom of the length spectrum of circular trajectories.

/pdf/lengths-of-circular-trajectories-on-geodesic-spheres-in-a-4qlbziw6ef.pdf

Tuya Bao

Papers

Congruence classes of minimal ruled real hypersurfaces in a nonflat complex space form

Circular trajectories on real hypersurfaces in a nonflat complex space form

Trajectories for Sasakian magnetic fields on real hypersurfaces of type (B) in a complex hyperbolic space

Trajectories on real hypersurfaces of type (B) in a complex hyperbolic space are not of order 2

Lengths of circular trajectories on geodesic spheres in a complex projective space