Trajectories on Geodesic Spheres in a Non-Flat Complex Space Form

Abstract

In this paper we study some basic properties of trajectories for canonical magnetic fields induced by structure tensor on real hypersurfaces of types A0 and A1 in a complex space form. On each such real hypersurface, there are infinitely many canonical magnetic fields whose trajectories with null structure torsion are closed, and also infinitely many canonical magnetic fields whose trajectories with null structure torsion are open. We give a condition dividing canonical magnetic fields into these two classes and investigate lengths of closed trajectories. Our study also provide us information on the moduli space of Killing helices of proper order 4 on complex space forms. Mathematics Subject Classification (2000). 53C35, 57R30.

Sasakian Curves on Hypersurfaces in a Nonflat Complex Space Form

“…Suppose M is locally congruent to a complex space form M n (c). When c = 0, as geodesic spheres of radius smaller than the injectivity radius of M n (c) are of type (A), we find Conditions (2) and (3) hold. When c = 0, a geodesic sphere G(r) in C n is a standard sphere which is a totally umbilic hypersurface with parallel shape operator.…”

“…(2) and (3) in Theorem 1 hold. (2) The equivalency of constancy of structure torsions and the condition that shape operator A and characteristic tensor φ are simultaneously diagonalizable holds for a general real hypersurface in a Kähler manifold (see [2]). (3) When κ = 0 (i.e., the case of geodesics), the equivalency of Conditions (1) and (3) in Theorem 1 was proved in [10].…”

“…That is, we call two curves γ 1 , γ 2 on N strongly congruent to each other if there is an isometry ϕ of N with γ 2 (s) = (ϕ • γ 1 )(s) for all s. Trivially, a Riemannian manifold N is either a Euclidean space or a Riemannian symmetric space of rank one if and only if for every pair of geodesics γ 1 , γ 2 on N they are strongly congruent to each other. For Sasakian curves on a hypersurface of type either (A 0 ) or (A 1 ), the following is obtained in [2]. i) |ρ γ1 | = |ρ γ2 | = 1; ii) ρ γ1 = ρ γ2 = 0 and |κ 1 | = |κ 2 |; iii) 0 < |ρ γ1 | = |ρ γ2 | < 1 and κ 1 ρ γ1 = κ 2 ρ γ2 .…”

“…(2) For each Sasakian curve γ on G(r) the curve ι • γ is a homogeneous curve on totally geodesic CP 2 (c) of CP n (c) (see [2]). Hence every Sasakian curve on G(r) is an orbit of a one-parameter subgroup of SU (3).…”

“…On a real hypersurface N in a Kähler manifold ( M, J) a smooth curve γ is said to be a Sasakian curve if it satisfies ∇γγ = κφγ with some constant κ, where φ is the characteristic tensor induced by J. Sasakian curves on a manifold admitting an almost contact metric structure can be considered as correspondences of Kähler circles on Kähler manifolds (cf. [2,9]). On a hypersurface M of type (A) in M n (c), every geodesic is a homogeneous curve, that is, it is an orbit of a one-parameter subgroup of the isometry group I(M ) of M (see [12]).…”