Almost contact metric 3 -submersions

An almost contact metric 3 -submersion is a Riemannian submersion, π from an almost contact metric manifold ( M 4 m + 3 , ( φ i , ξ i , η i ) i = 1 3 , g ) onto an almost quaternionic manifold ( N 4 n , ( J i ) i = 1 3 , h ) which commutes with the structure tensors of type ( 1 , 1 ) ;i.e., π * φ i = J i π * , for i = 1 , 2 , 3 . For various restrictions on ∇ φ i , (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 ( d Φ = 0 ) , then b 1 ( N ) ≤ b 1 ( M ) . The respective φ i -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: B φ i ( 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.