Conformal Submersion with Horizontal Distribution

Abstract

In this article, conformal submersion with horizontal distribution of Riemannian manifolds is defined which is a generalization of the affine submersion with horizontal distribution. Then, a necessary condition is obtained for the existence of a conformal submersion with horizontal distribution. For the dual connections $\nabla$ and $\overline{\nabla }$ on manifold $\mathbf{M}$ and ${\nabla }^{\ast }$ and ${\overline{\nabla }}^{\ast }$ on manifold $\mathbf{B}$, we show that $\pi :(\mathbf{M},\nabla )⟶(\mathbf{B},{\nabla }^{\ast })$ is a conformal submersion with horizontal distribution if and only if $\pi :(\mathbf{M},\overline{\nabla })⟶(\mathbf{B},\overline{{\nabla }^{\ast }})$ is a conformal submersion with horizontal distribution. Also, we obtained a necessary and sufficient condition for $\pi \circ \sigma$ to become a geodesic of $\mathbf{B}$ if $\sigma$ is a geodesic of $\mathbf{M}$ for $\pi :(\mathbf{M},\nabla ,g_m)\to (\mathbf{B},{\nabla }^{\ast },g_b)$ a conformal submersion with horizontal distribution.