Chapter 3 - Riemannian submersions From Almost Hermitian Manifolds

Abstract

In this chapter, we introduce various new Riemannian submersions from almost Hermitian manifolds on to Riemannian manifolds. In section 1, we first review almost Hermitian manifolds and their submanifolds, and give brief information about holomorphic submersions and invariant Riemannian submersions. In this section, we also present three important maps, namely almost complex or holomorphic maps, pseudo-horizontally weakly conformal maps, and pluriharmonic maps, defined between almost complex manifolds and Riemannian manifolds. Then, in section 2, we introduce anti-invariant Riemannian submersions from almost Hermitian manifolds to Riemannian manifolds, and show that such submersions are useful to investigate the geometry of total space. In sections 3 and 4, as a generalization of anti-invariant submersions and invariant submersions, we also introduce semi-invariant submersions and slant submersions. In section 5, as a generalization of slant submersions we consider point-wise slant submersions and show that such submersions have rich geometric properties. In section 6, we define and study generic submersions as a generalization of semi-invariant submersions. In sections 7 and 8, we introduce semi-slant submersions and hemi-slant submersions as generalizations of semi-invariant submersions and slant submersions. In section 9, we check the Einstein conditions of the base space for anti-invariant submersions. In the last section, we investigate various submersions in this chapter in terms of Clairaut’s relation.