Homogeneous Spaces with Sections

On homogeneous manifolds whose isotropy actions are polar

“…Since actions of cohomogeneity one are polar, our main result can be viewed as a generalization of the well-known fact that two-point homogeneous spaces are rank-one symmetric spaces in the special case of homogeneous spaces of semisimple compact Lie groups. Other similar results were proved in [11], [15], [16] and [20]. In [7], invariant Finsler metrics on homogeneous spaces with polar isotropy representations were studied.…”

“…In both cases, the isotropy group at the identity e is given by the group of automorphisms of M whose differential at e is an orthonormal map [2, §3.1.13 and §4.1.12]. Then, as follows from [19] (see also [15]), H, the connected component of the identity of the isotropy group of M, is given by Spin(m) • K, where If M is a generalized Heisenberg group, the isotropy representation is the action of H = Spin(m) • K on v ⊕ z given by the direct sum of the action of H on v described above and the standard representation Spin(m) → SO(m) on z = R m , whereas if M is a nonsymmetric Damek-Ricci space, the isotropy representation is the action of H = Spin(m)• K on a⊕v⊕z given by the direct sum of the trivial action on a, the action of H on v described above, and the standard representation on z = R m .…”

Manifolds with large isotropy groups

“…Since we consider K-representations where the group K acts on m 1 ⊕ m 2 as the automorphism group of a Clifford module of the Clifford algebra C n (cf. [13] or Propositions 3.2 and 3.3 of [11]), we may assume that m 2 carries the structure of a Clifford module over C n such that the action of k 0 on m 2 is given by Clifford multiplication. A standard argument from the representation theory shows that Hom K (m 1 ⊗ m 2 , m 2 ) has real dimension at most one in the case of representations (3) and (5) and complex dimension at most one in case of representations (2) and (4).…”

“…Since we consider K-representations where the group K acts on m 1 ⊕m 2 as the automorphism group of a Clifford module of the Clifford algebra Cℓ n (cf. [9] or Propositions 3.2 and 3.3 of [8]), we may assume that m 2 carries the structure of a Clifford module over Cℓ n such that the action of k 0 on m 2 is given by Clifford multiplication, i.e.…”

“…In case n = 6, 7 it follows that g is simple, immediately leading to a contradiction in case n = 6, since there is no simple real Lie algebra of dimension 29; if n = 7 then g is a simple real Lie algebra of dimension 36 containing spin (7) as a subalgebra and it follows that g is isomorphic to either spin(9), spin (8,1) or spin(7, 2); however, only in the first two cases the isotropy representation can be equivalent to (5), hence M = Spin(9)/Spin (7) or M = Spin(8, 1)/Spin (7). Now consider the case of representations (2) and (3).…”