Semi-Riemannian submersions with totally geodesic fibres

Abstract

We classify semi-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to three. Also, we obtain the classification of semi-Riemannian submersions with connected complex totally geodesic fibres from a complex pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to two. We prove that there are no semi-Riemannian submersions with connected quaternionic fibres from a quaternionic pseudo-hyperbolic space onto a Riemannian manifold.

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.…”

“…A key ingredient for understanding the geometry of the base and of the fibres is the construction of a special orthonormal basis B of H along a fibre, which we recall from [3]. First, we state the following lemma, which provides useful properties of O'Neill's integrability tensor for a constant curvature total space.…”

“…. , Y r be the horizontal lifts along the fibre π −1 (π(p)) of [3]), which can be restated as the following lemma. Set L 0 = X.…”

“…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. I also proved that condition (ii) implies (i) (see [3]).…”

“…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. I also proved that condition (ii) implies (i) (see [3]). In this paper, we drop these assumptions and we prove the following main result.…”