Real hypersurfaces in complex hyperbolic two‐plane Grassmannians with parallel Ricci tensor

Abstract

In this paper we prove that there does not exist any Hopf real hypersurface in complex hyperbolic two-plane Grassmannians SU 2,m /S(U 2 ·U m ) with parallel Ricci tensor.

Real Hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting shape operator

“…In this section we derive some basic formulas and the Codazzi equation for a real hypersurface in S U 2;m =S.U 2 U m / (see [1], [2], [8], and [9]). Let M be a real hypersurface in complex hyperbolic two-plane Grassmannian S U 2;m =S.U 2 U m /, that is, a hypersurface in S U 2;m =S.U 2 U m / with real codimension one.…”

“…Throughout this paper, we use some references [2], [7], [8], and [9] to recall the Riemannian geometry of SU 2;m =S.U 2 U m / and some fundamental formulas including the Codazzi and Gauss equations for a real hypersurface in S U 2;m =S.U 2 U m /.…”

Real hypersurfaces in the complex quadric with parallel Ricci tensor

“…Suh [12] strengthened this result to hypersurfaces in G 2 (C m+2 ) with parallel Ricci tensor. Moreover, Suh and Woo [15] studied another non-existence property for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians SU 2,m /S(U 2 U m ) with parallel Ricci tensor.…”

Real hypersurfaces in the complex quadric with commuting and parallel Ricci tensor

“…In the study of complex two-plane Grassmannian G 2 (C m+2 ) or complex hyperbolic two-plane Grassmannian SU 2,m /S(U 2 ·U m ) we studied hypersurfaces with parallel Ricci tensor and gave non-existence properties respectively (see [13] and [20]). In [18] we also considered the notion of parallel Ricci tensor ∇Ric = 0 for hypersurfaces M in Q m .…”