International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 4, Pages 667-688
Almost contact metric -submersions
Department of Mathematics, St. John's University, Jamaica 11439, New York, USA
Received 14 July 1982
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.
An almost contact metric -submersion is a Riemannian submersion, from an almost contact metric manifold onto an almost quaternionic manifold which commutes with the structure tensors of type ;i.e., , for . For various restrictions on , (e.g., is -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 is -quasi-Saskian , then . The respective -holomorphic sectional and bisectional curvature tensors are studied and several unexpected results are obtained. As an example, if and are orthogonal horizontal vector fields on the -contact (a relatively weak structure) total space of such a submersion, then the respective holomorphic bisectional curvatures satisfy: . Applications to the real differential geometry of Yarg-Milis field equations are indicated based on the fact that a principal -bundle over a compactified realized space-time can be given the structure of an almost contact metric -submersion.