# On the real-analyticity of rigid spherical hypersurfaces in ${\mathbb C}^2$

## TL;DR

This paper proves that all smooth rigid spherical hypersurfaces in complex two-dimensional space are inherently real-analytic, extending existing classifications from the real-analytic to the smooth category.

## Contribution

It establishes the real-analyticity of smooth rigid spherical hypersurfaces in ${f C}^2$, enabling the classification results to be applied beyond the real-analytic setting.

## Key findings

- Smooth rigid spherical hypersurfaces are real-analytic
- Classification results extend to smooth cases
- Supports broader application of existing classifications

## Abstract

We prove that every smooth rigid spherical hypersurface in ${\mathbb C}^2$ is in fact real-analytic. As an application of this result, it follows that the classification of real-analytic rigid spherical hypersurfaces in ${\mathbb C}^2$ found by V. Ezhov and G. Schmalz applies in the smooth case.