Real hypersurfaces in complex Grassmannians of rank two with semi-parallel structure Jacobi operator

Abstract

We prove that there does not exist any real hypersurface in complex Grassmannians of rank two with semi-parallel structure Jacobi operator. With this result, the nonexistence of real hypersurface in complex Grassmannians of rank two with recurrent structure Jacobi operator is proved.2010 Mathematics Subject Classification. Primary 53C25, 53C15; Secondary 53B20.Key words and phrases. complex Grassmannians of rank two. semi-parallel structure Jacobi operator. vanishing structure Jacobi operator.Let M be a connected, oriented real hypersurface isometrically immersed inM m (c), m ≥ 2, and N be a unit normal vector field on M . Denote by the same g the Riemannian metric on M . The Reeb vector field ξ is defined by ξ = −JN , and we define ξ a = −J a N , a ∈ {1, 2, 3}, where {J 1 , J 2 , J 3 } is a canonical local basis of J. Denote by D ⊥ (resp. D ⊥ ) the distribution on M spanned by ξ (resp. {ξ 1 , ξ 2 , ξ 3 }). A real hypersurface M in a Kähler manifold is said to be Hopf if the Reeb vector field is principal, that is, Aξ = αξ.The study of real hypersurfaces inM m (c) was initiated by Berndt and Suh in [1,2]. They considered the invariance of D ⊥ under the shape operator A of Hopf hypersurfaces M inM m (c) and proved a classification of such Hopf hypersurfaces inM m (c).The structures J and J of the ambient space impose several restrictions on the geometry of its real hypersurfaces, for example, there does not exist any semi-parallel real hypersurface in SU m+2 /S(U 2 U m ) [13] while the non-existence problem of Hopf hypersurfaces inM m (c) with parallel Ricci tensor were studied in [15,16].Besides the shape operator and the Ricci tensor, there are particularly two operators on a real hypersurface M which draw much attention, namely the normal Jacobi operator R N and the structure Jacobi operator R ξ . Denote byR and R the curvature tensor onM m (c) and that induced on M respectively. We define R N (X) =R(X, N )N and R ξ (X) = R(X, ξ)ξ for any vector field X tangent to M .A (1, s)-tensor field P is said to be semi-parallel if R · P = 0, where the curvature tensor R acts on P as a derivation. More precisely, (R(X, Y ) · P )(X 1 , · · · , X s ) = R(X, Y )P (X 1 , · · · , X s ) − s j=1 P (X 1 , · · · , R(X, Y )X j , · · · , X s ).The tensor field P is said to be recurrent if there exists an 1-form ω on M such that (∇ X P )(X 1 , · · · , X s ) = ω(X)P (X 1 , · · · , X s ).Clearly, a vanishing ω leads to parallelism of P .Recently, we proved the non-existence of real hypersurface in SU m+2 /S(U 2 U m ), m ≥ 3, with pseudo-parallel normal Jacobi operator [5]. On the other hand, related to the structure Jacobi operator R ξ , Jeong, et al. proved that there does not exist any Hopf hypersurface in SU m+2 /S(U 2 U m ), m ≥ 3, with parallel structure Jacobi operator [10]. Also, the non-existence of Hopf hypersurfaces with D ⊥ -parallel structure Jacobi operator is obtained under certain conditions [9]. Jeong, et al. considered Reeb-parallel structure Jacobi operator and proved the following: Theorem 1.1 ([8]). Let M be a Hopf ...