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Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces
Differential Geometry and its Applications ( IF 0.6 ) Pub Date : 2020-02-07 , DOI: 10.1016/j.difgeo.2020.101605
Gerhard Knieper , John R. Parker , Norbert Peyerimhoff

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[π/2,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 curvatures of these hypersurfaces and show that all hypersurfaces for α[π2,0)(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.



中文翻译:

Damek-Ricci空间的最小余维一对称性

在本文中,我们考虑以下形式的可解超曲面 ñ经验值[RH 在对称空间中引入度量 中号=小号大号3C/小号ü3,其中H是Iwasawa分解的子组A中合适的单位长度向量小号大号3C=ñ一种ķ。由于M为2级,A为二维,我们可以通过一个角度对这些超曲面进行参数化α[-π/2π/2]确定H的方向。我们表明,超曲面之一(对应于α=0)最小嵌入并且与非对称7维Damek-Ricci空间等距。我们还为这些超曲面的Ricci曲率提供了一个明确的公式,并表明所有超曲面α[-π200π2]允许平面具有负和正的截面曲率。此外,对称空间M允许最小的叶面,所有叶子都与非对称7维Damek-Ricci空间等距。

更新日期:2020-02-07
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