  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 given explicitly). In the first case, F is also a 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 the
two-dimensional solvable totally geodesic subgroup.
