Clairaut Riemannian maps whose total manifolds admit a Ricci soliton

In this paper, we study Clairaut Riemannian maps whose total manifolds admit a Ricci soliton and give a nontrivial example of such Clairaut Riemannian maps. First, we calculate Ricci tensors and scalar curvature of total manifolds of Clairaut Riemannian maps. Then we obtain necessary conditions for the fibers of such Clairaut Riemannian maps to be Einstein and almost Ricci solitons. We also obtain a necessary condition for vector field [Formula: see text] to be conformal, where [Formula: see text] is a geodesic curve on total manifold of Clairaut Riemannian map. Further, we show that if total manifolds of Clairaut Riemannian maps admit a Ricci soliton with the potential mean curvature vector field of [Formula: see text] then the total manifolds of Clairaut Riemannian maps also admit a gradient Ricci soliton and obtain a necessary and sufficient condition for such maps to be harmonic by solving Poisson equation.

A Survey on Differential Geometry of Riemannian Maps Between

The main aim of this paper is to state recent results in Riemannian geometry obtained by the existence of a Riemannian map between Riemannian manifolds and to introduce certain geometric objects along such maps which allow one to use the techniques of submanifolds or Riemannian submersions for Riemannian maps. The paper also contains several open problems related to the research area.

SEMI-INVARIANT RIEMANNIAN MAPS TO KÄHLER MANIFOLDS

This paper has two aims. First, we show that the usual notion of umbilical maps between Riemannian manifolds does not work for Riemannian maps. Then we introduce a new notion of umbilical Riemannian maps between Riemannian manifolds and give a method on how to construct examples of umbilical Riemannian maps. In the second part, as a generalization of CR-submanifolds, holomorphic submersions, anti-invariant submersions, invariant Riemannian maps and anti-invariant Riemannian maps, we introduce semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds, give examples and investigate the geometry of distributions which are arisen from definition. We also obtain a decomposition theorem and give necessary and sufficient conditions for a semi-invariant Riemannian map to be totally geodesic. Then we study the geometry of umbilical semi-invariant Riemannian maps and obtain a classification theorem for such Riemannian maps.

INVARIANT AND ANTI-INVARIANT RIEMANNIAN MAPS TO KÄHLER MANIFOLDS

As a generalization of isometric immersions and Riemannian submersions, Riemannian maps were introduced by Fischer [Riemannian maps between Riemannian manifolds, Contemp. Math.132 (1992) 331–366]. It is known that a real valued Riemannian map satisfies the eikonal equation which provides a bridge between physical optics and geometrical optics. In this paper, we introduce invariant and anti-invariant Riemannian maps between Riemannian manifolds and almost Hermitian manifolds as a generalization of invariant immersions and totally real immersions, respectively. Then we give examples, present a characterization and obtain a geometric characterization of harmonic invariant Riemannian maps in terms of the distributions which are involved in the definition of such maps. We also give a decomposition theorem by using the existence of invariant Riemannian maps to Kähler manifolds. Moreover, we study anti-invariant Riemannian maps, give examples and obtain a classification theorem for umbilical anti-invariant Riemannian maps.