Riemannian Maps

In this chapter, we study Riemannian maps between Riemannian manifolds. In section 1, we define Riemannian maps and give the main properties of such maps. In section 2, we obtain Gauss-Weingarten-like formulas and then we obtain Gauss, Codazzi, and Ricci equations along Riemannian maps. In section 3, we find necessary and sufficient conditions for Riemannian maps to be totally geodesic by using the Bochner identity and generalized divergence theorem. In section 4, we introduce umbilical Riemannian maps and pseudoumbilical Riemannian maps, and obtain characterizations of such Riemannian maps. In section 5, we discuss the harmonicity and biharmonicity of Riemannian maps. In section 6, we define Clairaut Riemannian maps, give an example, and obtain a characterization. In section 6, we extend the result of Nomizu-Yano to the Riemannian maps. In section 7, we obtain Chen’s inequality for Riemannian maps. In the last section, we investigate necessary and sufficient conditions for Riemannian maps to have the Einstein property.