Bounded isometries and homogeneous Riemannian quotient manifolds (Q1076352)

Let M be a connected homogeneous Riemannian manifold admitting a properly discontinuous group action \(\Gamma\) of isometries. In a variety of cases, it is well-known that the conditions: ''\(\Gamma\) is a group of Clifford translation'' and ''\(\bar M=\Gamma/M\) is a homogeneous Riemannian manifold'' are equivalent. Let G' be a connected semisimple Lie group without compact local factors and let G' act transitively and effectively on M by isometries. Regarding this \(G'\) as a subgroup of the isometry group \(I(M)\), let B be the centralizer of G' in \(I(M)\). Then B is the set of all bounded isometries of M. In particular, every bounded isometry of M is a Clifford translation. Next, if G is the closure of \(G'\) in \(I(M)\), then G is a connected reductive Lie subgroup of \(I(M)\) and \(G'\) is its derived group. For the isotropy subgroup \(H=\{g\in G;\) \(gx_ 0=x_ 0,\) \(x_ 0\in M\},\) let its normalizer in G be denoted by \(N_ G(H)\), and set \(R(U):\{gH\to gu^{-1}H,\) \(u\in N_ G(H)\}.\) Then \(R(U)\) is the set of all bounded isometries of M. On the basis of these two facts the author proves that the two conditions introduced above are equivalent if M admits a transitive semisimple group of isometries that has no compact local factor.