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Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces | | Gerhard Knieper
; John R. Parker
; Norbert Peyerimhoff
; | | Date: |
15 Apr 2019 | | Abstract: | In this article we consider solvable hypersurfaces of the form $N exp(R H)$
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 $alpha in [0,pi/2]$ 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 curvature of these hypersurfaces and show that all hypersurfaces for
$alpha in (0,frac{pi}{2}]$ 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. | | Source: | arXiv, 1904.7288 | | Services: | Forum | Review | PDF | Favorites | |
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