Classification of $$\big (\widetilde{\mathrm {Sp}}(n,\mathbb {R})\times \widetilde{\mathrm {Sp}}(1,\mathbb {R})\big )$$ ( Sp ~ ( n , R ) × Sp ~ ( 1 , R ) ) -manifolds

Abstract

Let M be an analytic complete finite volume pseudo-Riemannian manifold. We characterize the structure of the manifold M assuming that the Lie group \(\widetilde{\mathrm {Sp}}(n,\mathbb {R})\times \widetilde{\mathrm {Sp}}(1,\mathbb {R})\) acts isometrically with a dense orbit on M, where the \(\widetilde{\mathrm {Sp}}(1,\mathbb {R})\)-orbits are non-degenerated and its dimension satisfies \(\dim (M)\le (n+1)(2n+3)\).