Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces

被引:0
|
作者
Knieper, Gerhard [1 ]
Parker, John R. [2 ]
Peyerimhoff, Norbert [2 ]
机构
[1] Ruhr Univ Bochum, Dept Math, D-44780 Bochum, Germany
[2] Univ Durham, Dept Math Sci, Durham DH1 3LE, England
关键词
Damek-Ricci spaces; Harmonic manifolds; Minimal foliations; MANIFOLDS;
D O I
10.1016/j.difgeo.2020.101605
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
In this article we consider solvable hypersurfaces of the form N exp(RH) with induced metrics in the symmetric space M = SL(3, C)/SU(3), where Ha suitable unit length vector in the subgroup A of the Iwasawa decomposition SL(3, C) = NAK. Since Mis rank 2, A is 2-dimensional and we can parametrize these hypersurfaces via an angle alpha is an element of [-pi/2, pi/2] determining the direction of H. We show that one of the hypersurfaces (corresponding to alpha = 0) is minimally embedded and isometric to the non-symmetric 7-dimensional Damek-Ricci space. We also provide an explicit formula for the Ricci curvatures of these hypersurfaces and show that all hypersurfaces for alpha is an element of [-pi/2, 0) boolean AND (0, pi/2] admit planes of both negative and positive sectional curvature. Moreover, the symmetric space Madmits a minimal foliation with all leaves isometric to the non-symmetric 7-dimensional Damek-Ricci space. (C) 2020 Elsevier B.V. All rights reserved.
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页数:13
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