Minimal Codimension One Foliation of a Symmetric Space by Damek-Ricci Spaces

Abstract

In this article we consider solvable hypersurfaces of the form Nexp(RH) with induced metrics in the symmetric space M=SL(3,C)/SU(3), where H a suitable unit length vector in the subgroup A of the Iwasawa decomposition SL(3,C)=NAK. Since M is rank 2, A is 2-dimensional and we can parametrize these hypersurfaces via an angle α∈[0,π/2] determining the direction of H. We show that one of the hypersurfaces (corresponding to α=0) is minimally embedded and isometric to the non-symmetric 7-dimensional Damek-Ricci space. We also provide an explicit formula for the Ricci curvature of these hypersurfaces and show that all hypersurfaces for α∈(0,π2] admit planes of both negative and positive sectional curvature. Moreover, the symmetric space M admits a minimal foliation with all leaves isometric to the non-symmetric 7-dimensional Damek-Ricci space.