Transversal Jacobi Operators in Almost Contact Manifolds

Jong Taek Cho,Makoto Kimura

doi:10.3390/math9010031

Abstract

Along a transversal geodesic γ whose tangent belongs to the contact distribution D, we define the transversal Jacobi operator Rγ=R(·,γ˙)γ˙ on an almost contact Riemannian manifold M. Then, using the transversal Jacobi operator Rγ, we give a new characterization of the Sasakian sphere. In the second part, we characterize the complete ruled real hypersurfaces in complex hyperbolic space.

Highlights

A smooth manifold M2n−1 is an almost contact manifold if its structure group of the linear frame bundle is reducible to U(n − 1) × {1} [1]
We prove that a complete and connected K-contact manifold M is isometric to the unit sphere if and only if the contact distribution D is invariant by the transversal Jacobi operator Rγ (Rγ(D) ⊂ D) and, at the same time, it is invariant by the covariant derivative Rγ = (∇γ R)(·, γ )γ (Rγ(D) ⊂ D) for any transversal geodesic γ (Corollary 1)
We treat real hypersurfaces of a complex hyperbolic space, and we prove a complete real hypersurface M in a non-flat complex space form Mn(c), c = 0 is congruent to a complete ruled real hypersurface in complex hyperbolic space HnC if and only if the contact ditribution D is invariant by the transversal Jacobi operator Rγ and at the same time it is invariant by the covariant derivative Rγ for any transversal geodesic γ (Corollary 2)

Summary

Introduction

A smooth manifold M2n−1 is an almost contact manifold if its structure group of the linear frame bundle is reducible to U(n − 1) × {1} [1]. In Riemannian geometry the Jacobi operator Rγ = R(·, γ )γalong geodesics γ plays an important role, where R denotes the Riemannian curvature tensor. In these circumstances, it is very interesting to study the behavior of Jacobi operators on almost contact Riemannian manifolds in connection with their associated almost CR-structure. We prove that a complete and connected K-contact manifold M is isometric to the unit sphere if and only if the contact distribution D is invariant by the transversal Jacobi operator Rγ (Rγ(D) ⊂ D) and, at the same time, it is invariant by the covariant derivative Rγ = (∇γ R)(·, γ )γ (Rγ(D) ⊂ D) for any transversal geodesic γ (Corollary 1).

Preliminaries

A Characterization of the Unit Sphere