Almost contact metric 3-submersions - Bill Watson (International Journal of Mathematics and Mathematical Sciences, 1984)
An almost contact metric 3-submersion is a Riemannian submersion, π from an almost contact metric manifold (M4m+3,(φi,ξi,ηi)i=13,g) onto an almost quaternionic manifold (N4n,(Ji)i=13,h) 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: Bφi(X,Y)=B′J′i(X*,Y*)−2. 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.