Conformal Submersion with Horizontal Distribution

TL;DR

This paper introduces the concept of conformal submersion with horizontal distribution in Riemannian manifolds, providing conditions for their existence and characterizing when geodesics are preserved under such mappings.

Contribution

It generalizes affine submersions to conformal submersions with horizontal distribution and establishes necessary and sufficient conditions for their existence and properties.

Findings

Necessary condition for conformal submersion with horizontal distribution.

Equivalence of conformal submersion conditions under dual connections.

Characterization of geodesic preservation under conformal submersions.

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.