Chapter 6 - Riemannian Maps To Almost Hermitian Manifolds

Abstract

In this chapter, we study Riemannian maps from Riemannian manifolds to almost Hermitian manifolds. In section 1, we study invariant Riemannian maps, that is, the image of derivative map is invariant under the almost complex structure of the base manifold. We give examples, investigate the geometry of foliations, and find new conditions for the harmonicity of such maps. In section 2, we study anti-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds. We give several examples, and obtain a method to find anti-invariant Riemannian maps. We also obtain a characterization for umbilical anti-invariant Riemannian maps. In section 3, we study semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of CR-submanifolds. We give examples, obtain the totally geodesicity of such maps, and relate it with PHWC maps. In section 4, we investigate generic Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of generic submanifolds. We give examples and obtain necessary and sufficient conditions for such maps to be totally geodesic and harmonic. In section 5, we introduce slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of slant submanifolds. We obtain characterizations for such maps and investigate the geometry of maps. In section 6, we study semi-slant Riemannian maps as a generalization of semi-slant submanifolds, give examples, and obtain new characterizations. In section 7, we define hemi-slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of hemi-slant submanifolds and slant Riemannian maps to almost Hermitian manifolds. We give examples, obtain a decomposition theorem and find necessary and sufficient conditions to be a totally geodesic map.