On low-dimensional manifolds with isometric \(\widetilde{\mathrm{U}} (p, q)\)-actions (Q1693283)

This paper considers a version of ``Zimmer's program'', the study of manifolds \(M\) of finite volume with a volume-preserving ergodic action by a simple Lie group \(G\). It is expected that such an \(M\) is a double coset space of a Lie group by a lattice and a compact subgroup that centralizes \(G\). Here, \(G\) is the reductive group \(\tilde{\text U}(p,q)\), the universal covering group of the pseudo-unitary group for index \((p,q)\), acting on a connected complete analytic pseudo-Riemannian manifold \(M\). It is further assumed that (i) the simple part \(\tilde{\text{SU}}(p,q)\) of \(G\) has dense orbit in \(M\), (ii) the center \(Z\) of \(\tilde{\text U}(p,q)\) acts non-trivially on \(M\), and (iii) \(\dim M\leq n(n+2)\), where \(n=p+q\). The description of the universal covering \(\tilde{M}\) is the content of three main theorems (Theorem 1.1, 1.2, 1.3), the precise statements of which are too long to be repeated here. Essentially, their content is that there exists a diffeomorphism \(\varphi:N\to\tilde{M}\) for a manifold \(N\) that is either (i) a product of \(\tilde{\text{SU}}(p,q)\) by a simply connected manifold, or (ii) \(N=\tilde{\text{SU}}(p+1,q)\) or \(N=\tilde{\text{SU}}(p,q+1)\), or (iii) \(N=K\backslash S \times \mathbb{R}\), where \(S\) is one of the groups in (ii). For each of these cases, the metric on \(M\) can be modified such that \(\varphi\) becomes an isometry. Moreover, the space \(M\) is obtained from a symmetric pair locally equivalent to \((S,G)\), with \(S\) one of the above groups. If \(M\) is weakly irreducible \(M\) and if \(N=S\) for one of the above groups, then there exists an equivariant diffeomorphism \(\phi:N\to\tilde{M}\), which becomes an equivariant isometry after modification of the metric on \(M\). In any of the three cases for \(N\) above, the fundamental group \(\pi_1(M)\) of \(M\) is isomorphic to a discrete subgroup of \(\text{Iso}(N)\).