# On Riemannian submersions

## TL;DR

This paper proves that the image of a real analytic Riemannian manifold under a smooth Riemannian submersion is necessarily real analytic, establishing a key property of such mappings.

## Contribution

It establishes that the image of a real analytic Riemannian manifold under a smooth Riemannian submersion is always real analytic, a new result in differential geometry.

## Key findings

- The image of a real analytic Riemannian manifold under a smooth Riemannian submersion is real analytic.
- The result bridges properties of the manifold and its image under submersion.
- Provides foundational insight into the structure of Riemannian submersions.

## Abstract

We prove that the image of a real analytic Riemannian manifold under a smooth Riemannian submersion is necessarily real analytic.