Semi-invariant submersions whose total manifolds are locally product Riemannian

Abstract

We study Riemannian and semi-invariant submersions whose total man-ifolds are locally product Riemannian. The necessary and sufficient conditions for the integrability and totally geodesicness of all distributions which are introduced in the denition of the semi-invariant submersion are obtained. We also give a charac- terization theorem for the proper semi-invariant submersions with totally umbilical fibers and find some results for such submersions with parallel canonical structures. Moreover, we define first variational formula on the fibers of a semi-invariant sub-mersion and by the virtue of that we prove a new theorem which has a condition for the harmonicity of a semi-invariant submersion.Mathematics Subject Classication (2010): Primary 53C15, 53B20.Key words: Riemannian submersion, semi-invariant submersions, horizontal distribution fiber, locally product Riemannian manifold, rst variational formula.