Publikationsdatum:
1984-01-01
Beschreibung:
An almost contact metric3-submersion is a Riemannian submersion,πfrom 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*, fori=1,2,3. For various restrictions on∇φi, (e.g.,Mis3-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, ifMis3-quasi-Saskian(dΦ=0), thenb1(N)≤b1(M). The respectiveφi-holomorphic sectional and bisectional curvature tensors are studied and several unexpected results are obtained. As an example, ifXandYare orthogonal horizontal vector fields on the3-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 principalSU(2)-bundle over a compactified realized space-time can be given the structure of an almost contact metric3-submersion.
Print ISSN:
0161-1712
Digitale ISSN:
1687-0425
Thema:
Mathematik
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