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Almost contact metric 3-submersions. (English) Zbl 0583.53033

Considering an almost contact metric 3-submersion as a Riemannian submersion, \(\pi\), from an almost contact metric manifold \((M^{4m+3},(\phi_ i,\xi_ i,\eta_ i)^ 3_{i=1},g)\) onto an almost quaternionic manifold \((N^{4n},(J_ i)^ 3_{i=1},h)\) which commutes with the structure tensors of type (1,1) the author proves that for various restrictions on \(\nabla_ i\), (e.g., M is 3-Sasakian), there are corresponding limitations on the second fundamental form of the fibres and on the complete integrability of the horizontal distribution. Relations are derived between the Betti numbers of a compact total space and the base space. For instance, if M is 3-quasi-Sasakian \((d\Phi =0)\), then \(b_ 1(N)\leq b_ 1(M)\). The respective \(\phi_ i\)-holomorphic sectional and bisectional curvature tensors are studied and several interesting results are obtained. For 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: \(B\phi_ i(X,Y)=B'\!_{J'}(X_*,Y_*)-2.\) Applications to the real differential geometry of the Yang-Mills 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.
Reviewer: A.M.Naveira

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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