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 $\nabla^*$ and $\overline{\nabla}^*$ on manifold $\mathbf{B}$, we show that $\pi: (\mathbf{M},\nabla) \longrightarrow (\mathbf{B}, \nabla^{*}) $ is a conformal submersion with horizontal distribution if and only if $\pi: (\mathbf{M},\overline{\nabla}) \longrightarrow (\mathbf{B}, \overline{\nabla^{*}}) $ 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}) \rightarrow (\mathbf{B},\nabla^{*},g_{b})$ a conformal submersion with horizontal distribution.