Bill Watson

Copyright © 1984 Bill Watson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.