tailieunhanh - Real hypersurfaces in complex two-plane Grassmannians whose shape operator is recurrent for the generalized Tanaka-Webster connection

We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose shape operator A is generalized Tanaka–Webster recurrent if the principal curvature of the structure vector field is not equal to trace(A). | Turk J Math (2015) 39: 313 – 321 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Real hypersurfaces in complex two-plane Grassmannians whose shape operator is recurrent for the generalized Tanaka–Webster connection 1,∗ ´ Juan de Dios PEREZ , Young Jin SUH2 , Changhwa WOO2 Department of Geometry and Topology, University of Granada, Granada, Spain 2 Department of Mathematics, Kyungpook National University, Daegu, Republic of Korea 1 Received: • Accepted/Published Online: • Printed: Abstract: We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose shape operator A is generalized Tanaka–Webster recurrent if the principal curvature of the structure vector field is not equal to trace(A). Key words: Real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, generalized Tanaka–Webster connection, recurrent shape operator 1. Introduction The generalized Tanaka–Webster connection (from now on, g-Tanaka–Webster connection) for contact metric manifolds was introduced by Tanno ([13]) as a generalization of the connection defined by Tanaka in [12] and, independently, by Webster in [14]. This connection coincides with the Tanaka–Webster connection if the associated CR-structure is integrable. The Tanaka–Webster connection is defined as a canonical affine connection on a non-degenerate, pseudo-Hermitian CR-manifold. A real hypersurface M in a K¨ahler manifold has an (integrable) CR-structure associated with the almost contact structure (ϕ, ξ, η, g) induced on M by the K¨ahler structure, but, in general, this CR-structure is not guaranteed to be pseudo-Hermitian. Cho [4] and Tanno [13] defined the g-Tanaka–Webster connection for a real hypersurface of a K¨ahler manifold by ˆ (k) Y = ∇X Y + g(ϕAX, Y )ξ − η(Y )ϕAX − kη(X)ϕY ∇ X () for any X, Y tangent to M , where ∇ denotes the Levi-Civita connection on M ,