Semi-Riemannian submersions from real and complex pseudo-hyperbolic spaces

Semi-invariant semi-Riemannian submersions

“…for any E; F 2 (T M ); where v and h are the vertical and horizontal projections (see [2], [8]). From (2.3) and (2.4), one can obtain…”

Para-contact product semi-Riemannian submersions

“…for any E; F 2 (T M ); where v and h are the vertical and horizontal projections (see [4], [10]). From (13) and (14), one can obtain…”

Classification of pseudo-Riemannian submersions with totally geodesic fibres from pseudo-hyperbolic spaces

“…The Hopf pseudo-Riemannian submersions are homogeneous, that is, of the form π : G/K → G/H with K ⊂ H closed Lie subgroups: (5,4)/Spin(4, 4)) 0 . By Harvey's book [28, p. 312], each of Spin(5, 4)/Spin (3,4) and Spin(5, 4)/Spin(4, 4) has two connected components: a pseudo-sphere and a pseudo-hyperbolic space. Here (·) 0 denotes the pseudo-hyperbolic component.…”

“…In early work, Escobales [15,16] and Ranjan [39] classified Riemannian submersions with connected totally geodesic fibres from a sphere, and with complex connected totally geodesic fibres from a complex projective space. Using a topological argument, Ucci [44] showed that there are no Riemannian submersions with fibres CP 3 from the complex projective space CP 7 onto S 8 (4), and with fibres HP 1 from the quaternionic projective space HP 3 onto S 8 (4). A major advance obtained by Gromoll and Grove [26] is that, up to equivalence, the only Riemannian submersions from spheres with connected fibres are the Hopf fibrations, except possibly for fibrations of the 15-sphere by homotopy 7-spheres.…”

“…In the pseudo-Riemannian set-up, the pioneering work is due to Magid [33], who proved that the pseudo-Riemannian submersions with connected totally geodesic fibres from an anti-de Sitter space onto a Riemannian manifold are equivalent to the Hopf pseudo-Riemannian submersions H 2m+1 1 → CH m . Generalizing Magid's result, Stere Ianuş and I showed that any pseudo-Riemannian submersion with connected totally geodesic fibres from a pseudo-hyperbolic space onto a Riemannian manifold is equivalent to one of the Hopf pseudo-Riemannian submersions: H 2m+1 1 → CH m , H 4m+3 3 → HH m or H 15 7 → H 8 (−4), and as a consequence we classified the pseudo-Riemannian submersions with connected complex totally geodesic fibres from a complex pseudo-hyperbolic space onto a Riemannian manifold (see [4]). In [3], I extended these results to the case of a pseudo-Riemannian base under the assumption that either (i) the base space is isotropic or (ii) the dimension of fibres is less than or equal to 3, and the metrics induced on the fibres are negative definite.…”