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Conformal Riemannian maps from almost Hermitian manifolds

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Conformal Riemannian maps from almost Hermitian manifolds to Riemannian manifolds, namely conformal invariant Riemannian maps, holomorphic conformal Riemannian maps, and conformal antiinvariant Riemannian maps, are introduced. | Turk J Math (2018) 42: 2436 – 2451 © TÜBİTAK doi:10.3906/mat-1711-34 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Conformal Riemannian maps from almost Hermitian manifolds Bayram ŞAHİN1,∗,, Şener YANAN2 Department of Mathematics, Faculty of Science, Ege University, İzmir, Turkey 2 Department of Mathematics, Faculty of Science and Arts, Adıyaman University, Adıyaman, Turkey 1 Received: 08.11.2017 • Accepted/Published Online: 09.07.2018 • Final Version: 27.09.2018 Abstract: Conformal Riemannian maps from almost Hermitian manifolds to Riemannian manifolds, namely conformal invariant Riemannian maps, holomorphic conformal Riemannian maps, and conformal antiinvariant Riemannian maps, are introduced. We mainly focus on conformal antiinvariant Riemannian maps from Kaehlerian manifolds. We give proper examples of conformal antiinvariant Riemannian maps, obtain the integrability of certain distributions, and investigate the geometry of leaves of these distributions. We also obtain various conditions for such maps to be horizontally homothetic maps. Key words: Riemannian submersion, Riemannian map, Kaehlerian manifold, conformal holomorphic Riemannian map, conformal antiinvariant Riemannian map 1. Introduction As a generalization of the notions of isometric immersions and Riemannian submersions, Riemannian maps between Riemannian manifolds were introduced by Fischer [5]; see also [3, 4, 6, 7, 11, 20]. Let Φ : (M1 , g1 ) −→ (M2 , g2 ) be a smooth map between Riemannian manifolds such that 0 < rank Φ ≤ min{m, n} , where dimM1 = m and dimM2 = n. In that case, we state the kernel space of Φ∗ by ker Φ∗p1 at p1 ∈ M1 and take into consideration the orthogonal complementary space H = (ker Φ∗p1 )⊥ to ker Φ∗p1 . Thus, the tangent space of M1 at p1 has the following decomposition: Tp1 M1 = Hp1 ⊕ ker Φ∗p1 . Denote the range of Φ∗p1 by range Φ∗p1 and consider the orthogonal complementary space of range Φ∗ by (range Φ∗p1 )⊥ in TΦ(p1 ) M2 .