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

Abstract

We classify pseudo-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a pseudo-Riemannian manifold. Also, we obtain the classification of the pseudo-Riemannian submersions with (para-)complex connected totally geodesic fibres from a (para-)complex pseudo-hyperbolic space onto a pseudo-Riemannian manifold.

The Horizontal Heat Kernel on the Quaternionic Anti-De Sitter Spaces and Related Twistor Spaces

“…The Lie algebra of SU (2) Denote p := i · su(2) and P = SL(2, C)/SU (2). To geometrically describe P , we recall the complexification of SU(2) ∼ = S 3 .…”

“…Hence the push forward of ∆ SU(2) through Wick's rotations gives −∆ P where ∆ P := I 2 + J 2 + K 2 is the Laplacian on the symmetric space P = H 3 = SL(2, C)/SU (2). Formula (3.15) indicates then that the 4n + 3 real dimensional real hyperbolic space is dual to AdS 4n+3 (H) by complexification of the fibers.…”

“…Moreover, it shows that AdS 4n+3 (H) is simply connected since both the base space and the fiber are so. We refer to [2] for a description of this submersion, and more generally for a description of totally geodesic submersions from pseudo-hyperbolic spaces.…”

“…This operator is obtained from the horizontal Laplacian on the quaternionic anti-de Sitter space by complexification of the fiber SU (2). Indeed, the spectrum of the 'fiber' part is purely continuous: Using the identity tanh 2 (r) = 1 − 1/ cosh 2 (r), then we are led to ∂ 2 ∂r 2 + ((4n − 1) coth r + 3 tanh r) ∂ ∂r 2 − λ 2 + 1 cosh 2 r + (λ 2 + 1) v λ (t, r) = ∂ t v λ (t, r).…”

“…The fibration (1.1) therefore inherits a large symmetry group. As a consequence, the heat kernel of the horizontal Laplacian on AdS 4n+3 (H) only depends on two variables: a variable r which is the radial coordinate on the base space HH n and a variable η which is the radial coordinate on the fiber SU (2). We prove that in these coordinates, the horizontal Laplacian of the fibration writes ∂ 2 ∂r 2 + ((4n − 1) coth r + 3 tanh r) ∂ ∂r + tanh 2 r ∂ 2 ∂η 2 + 2 cot η ∂ ∂η .…”