On the standard nondegenerate almost CR structure of tangent hyperquadric bundles

Abstract

The tangent bundle TM of a semi-Riemannian manifold (M, g) admits a natural indefinite almost Kaehler structure (J, G) and the unit tangent hyperquadric bundle $T_{\varepsilon } (M,g)$, ${\varepsilon }=\pm 1$, is an orientable nondegenerate hypersurface of (TM, J, G), then there is a canonical way to define a standard contact semi-Riemannian structure, equivalently a standard nondegenerate almost CR structure, on $T_{\varepsilon } (M,g)$. In this paper we study the geometry of this natural nondegenerate almost CR structure on $T_\varepsilon M$. More precisely, we investigate the integrability, the pseudohermitian torsion, the H-contact and $\eta$-Einstein structures, and Sasaki–Einstein structures on $T_{\varepsilon } (M,g)$. In particular, the tangent pseudo-sphere bundle $T_{1} (\mathbb S^{n}_{\nu })$ and the tangent pseudo-hyperbolic bundle $T_{-1} (\mathbb H^{n}_{\nu })$ admit a Sasaki–Einstein semi-Riemannian structure.