# Intrinsic and extrinsic geometry of hypersurfaces in $\mathbb{S}^n \times \mathbb{R}$ and $\mathbb{H}^n \times \mathbb{R}$

## TL;DR

This paper provides a geometric classification of certain hypersurfaces in product spaces involving spheres and hyperbolic spaces, linking their intrinsic and extrinsic properties with special curvature conditions.

## Contribution

It introduces new classifications of conformally flat and radially flat hypersurfaces in these product spaces based on extrinsic geometry and shape operator conditions.

## Key findings

- Conformally flat hypersurfaces are classified as rotation hypersurfaces.
- Radially flat hypersurfaces are related to semi-parallel hypersurfaces.
- Descriptions of hypersurfaces with Einstein metrics, Ricci solitons, or constant scalar curvature are provided.

## Abstract

In this paper, geometric characterizations of conformally flat and radially flat hypersurfaces in $\mathbb{S}^n \times \mathbb{R}$ and $\mathbb{H}^n \times \mathbb{R}$ are given by means of their extrinsic geometry. Under suitable conditions on the shape operator, we classify conformally flat hypersurfaces in terms of rotation hypersurfaces. In addition, a close relation between radially flat hypersurfaces and semi-parallel hypersurfaces is established. These results lead to geometric descriptions of hypersurfaces with special intrinsic structures, such as Einstein metrics, Ricci solitons and hypersurfaces with constant scalar curvature.