Semi-invariant submersions whose total manifolds are locally product Riemannian

We study Riemannian and semi-invariant submersions whose total manifolds are locally product Riemannian. The necessary and sufficient conditions for the integrability and totally geodesicness of all distributions which are introduced in the definition of the semi-invariant submersion are obtained. We also give a characterization 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 submersion and by the virtue of that we prove a new theorem which has a condition for the harmonicity of a semi-invariant submersion.