Sahin, Bayram2024-08-042024-08-0420100219-8878https://doi.org/10.1142/S0219887810004324https://hdl.handle.net/11616/95040As 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 Kahler manifolds. Moreover, we study anti-invariant Riemannian maps, give examples and obtain a classification theorem for umbilical anti-invariant Riemannian maps.eninfo:eu-repo/semantics/closedAccessKahler manifoldRiemannian mapsinvariant Riemannian mapanti-invariant Riemannian mapharmonic mapdecomposition theoremArticle7333735510.1142/S02198878100043242-s2.0-77952646986Q3WOS:000277810000001Q3