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

Avik De,Tee-How Loo,Changhwa Woo

doi:10.33044/revuma.v60n2a15

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 non-existence of real hypersurfaces in complex Grassmannians of rank two with recurrent structure Jacobi operator is proved.

Highlights

Let Mm(c) be the compact complex Grassmannian SUm+2/S(U2Um) of rank two (resp. noncompact complex Grassmannian SU2,m/S(U2Um) of rank two) for c > 0, where c = max K /8 is a scaling factor for the Riemannian metric g and K is the sectional curvature for Mm(c)
It is an irreducible Riemannian symmetric space equipped with a Kahler structure J and a quaternionic Kahler structure J not containing J
Let M be a Hopf hypersurface in SUm+2/S(U2Um), m ≥ 3, with Reeb parallel structure Jacobi operator

Summary

Introduction

We proved the non-existence of real hypersurfaces in SUm+2/S(U2Um), m ≥ 3, with pseudo-parallel normal Jacobi operator [5]. Let M be a Hopf hypersurface in SUm+2/S(U2Um), m ≥ 3, with Reeb parallel structure Jacobi operator. Machado et al proved the non-existence of Hopf real hypersurfaces in SUm+2/S(U2Um), m ≥ 3, with commuting structure Jacobi operator under certain conditions [14]. There does not exist any Hopf hypersurface in SUm+2/S(U2Um), m ≥ 3, with semi-parallel structure Jacobi operator if the smooth function α = g(Aξ, ξ) is constant along the direction of ξ. The non-existence of real hypersurfaces with semi-parallel structure Jacobi operator in a non flat complex space form has been proved in [3, 7].

The equations of Gauss and Codazzi are respectively given by

It is clear that