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Totally geodesic hypersurfaces of homogeneous spaces

Hold Date 2015-12-01 12:00~2015-12-01 13:00

Place Lecture Room M W1-C-512, West Zone 1, Ito campus, Kyushu University

Object person  

Speaker Yuri Nikolayevsky (Department of Mathematics and Statistics, La Trobe University)

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).