Event info

La Trobe-Kyushu Joint Seminar on Mathematics for Industry (10)

Date & Time : Dec. 1, 2015  14:00(Australian time) / 12:00(Japan time)
Venue: IMI Australia Branch in La Trobe University / C512 Middle Lecture Room, IMI, Kyushu University
Title: Totally geodesic hypersurfaces of homogeneous spaces
Speaker: Yuri Nikolayevsky, Department of Mathematics and Statistics, La Trobe University

Abstract:
We show that a simply connected Riemannian homogeneous space M which admits
a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped product of the Euclidean space and a homogeneous space, or (c)  the twisted product of the line and a homogeneous space (with the warping/twisting function in the last two cases given explicitly). In the first case, the hyper surface F by itself is also the Riemannian product; in the last two cases, it is a leaf of a totally geodesic homogeneous fibration. Case (c) can alternatively be characterised by the fact that M admits a Riemannian submersion onto the universal cover of the group SL(2) equipped with a particular left-invariant metric, and F is the preimage of a two-dimensional solvable totally geodesic subgroup of SL(2).