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.
Highlights
As a generalization of the notions of isometric immersions and Riemannian submersions, Riemannian maps between Riemannian manifolds were introduced by Fischer [5]; see [3, 4, 6, 7, 11, 20]
As a generalization of antiinvariant submersions and holomorphic submersions, we introduce both conformal antiinvariant Riemannian maps and holomorphic conformal Riemannian maps from complex manifolds to Riemannian manifolds
We study conformal antiinvariant Riemannian maps, provide examples, and investigate the geometry of leaves arising from such maps
Summary
As a generalization of the notions of isometric immersions and Riemannian submersions, Riemannian maps between Riemannian manifolds were introduced by Fischer [5]; see [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. One can see that Riemannian maps with ker Φ∗ = {0} (respectively, (range Φ∗)⊥ = {0} ) are isometric immersions (respectively, Riemannian submersions). Φ is a conformal Riemannian map at p ∈ M if 0 < rank Φ∗p ≤ min{m, n} and Φ∗p maps H(p) = ((ker (Φ∗p)⊥) conformally onto range (Φ∗p) , i.e. there exists a number λ2(p) ̸= 0 such that gN (Φ∗p(X), Φ∗p(Y )) = λ2(p)gM (X, Y ). Where (∇Φ∗)⊥(Y, Z) is the component of (∇Φ∗)(Y, Z) on (range Φ∗)⊥
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