Almost contact metric 3‐submersions

Abstract

An almost contact metric 3‐submersion is a Riemannian submersion, π from an almost contact metric manifold onto an almost quaternionic manifold which commutes with the structure tensors of type (1, 1);i.e., π*φi = Jiπ*, for i = 1, 2, 3. For various restrictions on ∇φi, (e.g., M is 3‐Sasakian), we show corresponding limitations on the second fundamental form of the fibres and on the complete integrability of the horizontal distribution. Concommitantly, relations are derived between the Betti numbers of a compact total space and the base space. For instance, if M is 3‐quasi‐Saskian (dΦ = 0), then b1(N) ≤ b1(M). The respective φi‐holomorphic sectional and bisectional curvature tensors are studied and several unexpected results are obtained. As an example, if X and Y are orthogonal horizontal vector fields on the 3‐contact (a relatively weak structure) total space of such a submersion, then the respective holomorphic bisectional curvatures satisfy: . Applications to the real differential geometry of Yarg‐Milis field equations are indicated based on the fact that a principal SU(2)‐bundle over a compactified realized space‐time can be given the structure of an almost contact metric 3‐submersion.