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

Knieper, Gerhard; Parker, John R; Peyerimhoff, Norbert

Abstract

In this article we consider solvable hypersurfaces of the form with induced metrics in the symmetric space , where H a suitable unit length vector in the subgroup A of the Iwasawa decomposition . Since M is rank 2, A is 2-dimensional and we can parametrize these hypersurfaces via an angle determining the direction of H. We show that one of the hypersurfaces (corresponding to ) 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 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.

Citation

Knieper, G., Parker, J. R., & Peyerimhoff, N. (2020). Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces. Differential Geometry and its Applications, 69, Article 101605. https://doi.org/10.1016/j.difgeo.2020.101605

• Journal Article Type: Article
• Acceptance Date: Jan 28, 2020
• Online Publication Date: Feb 7, 2020
• Publication Date: Apr 30, 2020
• Deposit Date: Jan 28, 2020
• Publicly Available Date: Feb 7, 2021
• Journal: Differential Geometry and its Applications
• Print ISSN: 0926-2245
• Electronic ISSN: 1872-6984
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 69
• Article Number: 101605
• DOI: https://doi.org/10.1016/j.difgeo.2020.101605
• Public URL: https://durham-repository.worktribe.com/output/1272883