Circular trajectories on real hypersurfaces in a nonflat complex space form

Abstract

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.Mathematics Subject Classification (2010). Primary 53C40; Secondary 53C22.

Magnetic curves in Sasakian manifolds

“…In a different approach, the Sasakian space forms may be realized as particular homogeneous real hypersurfaces in non-flat complex space forms CM n (c). Adachi and Bao [6] showed that the fundamental 2-form of an orientable real hypersurface in a Kähler manifold is closed. They called a magnetic field on real hypersurfaces given by constant multiple of the fundamental 2-form a Sasakian magnetic field and they studied the magnetic trajectories on real hypersurfaces of type A in CM n (c) in [6], and of type B in the hyperbolic complex space CH n (c) in [7,8].…”

Structure torsions of trajectories on real hypersurfaces of exceptional type in complex projective space

“…We define a 2-form F φ on M by F φ (v, w) = v, φ(w) . One can easily check that it is a closed 2-form, hence is a magnetic field (see [2]). We say a constant multiple F κ = κF φ a Sasakian magnetic field.…”

“…It is needless to say that Riemannian geometry was developed by investigations of geodesics. In this context, Adachi and the author [2] field on real hypersurfaces of type (A) in a nonflat complex space forms. They showed that the condition for trajectories to be circles, and called them circular trajectories.…”

Extrinsic Circular Trajectories on Geodesic Spheres in a Complex Projective Space

“…We define a 2-form F φ associated with this structure by F φ (v, w) = v, φ(w) . One can easily find that it is a closed form (see [7]). Generally, a closed 2-form on a Riemannian manifold is said to be a magnetic field because it can be regarded as a generalization of a static magnetic field on a Euclidean 3-space R 3 (see [15], for example).…”