We introduce conformal anti-invariant submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations that arose from the definition of a conformal submersion, and find necessary and sufficient conditions for a conformal anti-invariant submersion to be totally geodesic. | Turk J Math (2016) 40: 43 – 70 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Conformal anti-invariant submersions from almost Hermitian manifolds ˙ 2,∗ Mehmet Akif AKYOL1 , Bayram S ¸ AHIN Department of Mathematics, Faculty of Science and Arts, Bing¨ ol University, Bing¨ ol, Turkey 2 ˙ on¨ Department of Mathematics, Faculty of Science and Arts, In¨ u University, Malatya, Turkey 1 Received: • Accepted/Published Online: • Final Version: Abstract: We introduce conformal anti-invariant submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations that arose from the definition of a conformal submersion, and find necessary and sufficient conditions for a conformal anti-invariant submersion to be totally geodesic. We also check the harmonicity of such submersions and show that the total space has certain product structures. Moreover, we obtain curvature relations between the base space and the total space, and find geometric implications of these relations. Key words: Riemannian submersion, anti-invariant submersion, conformal submersion, conformal anti-invariant submersion 1. Introduction One of the main methods to compare two manifolds and transfer certain structures from a manifold to another manifold is to define appropriate smooth maps between them. Given two manifolds, if the rank of a differential map is equal to the dimension of the source manifold, then such maps are called immersions and if the rank of a differential map is equal to the target manifold, then such maps are called submersions. Moreover, if these maps are isometric between manifolds, then the immersion is called isometric immersion (Riemannian submanifold) and the submersion is called Riemannian submersion. Riemannian submersions between Riemannian manifolds were studied by O’Neill [18] and Gray [10]; for recent .
