Conformal Submersion with Horizontal Distribution

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 $\mathrm{\nabla }$ and $\bar{\mathrm{\nabla }}$ on manifold $\mathbf{M}$ and ${\mathrm{\nabla }}^{\ast }$ and ${\bar{\mathrm{\nabla }}}^{\ast }$ 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.