# Para Blaschke isoparametric spacelike hypersurfaces in Lorentzian space forms

## TL;DR

This paper classifies para-Blaschke isoparametric spacelike hypersurfaces in Lorentzian space forms, focusing on those with constant eigenvalues of a combined conformal tensor, under conformal transformations.

## Contribution

It provides a classification of para-Blaschke isoparametric spacelike hypersurfaces in Lorentzian space forms, extending the understanding of conformal invariants in Lorentzian geometry.

## Key findings

- Classification of para-Blaschke isoparametric hypersurfaces achieved
- Characterization of hypersurfaces with constant eigenvalues of the para-Blaschke tensor
- Identification of geometric conditions under conformal transformations

## Abstract

Let $M^n$ be an $n$-dimensional umbilic-free hypersurface in the $(n+1)$-dimensional Lorentzian space form $M^{n+1}_1(c)$. Three basic invariants of $M^n$ under the conformal transformation group of $M^{n+1}_1(c)$ are a $1$-form $C$, called conformal $1$-form, a symmetric $(0,2)$ tensor $B$, called conformal second fundamental form, and a symmetric $(0,2)$ tensor $A$, called Blaschke tensor. The so-called para-Blaschke tensor $D^{\lambda}=A+\lambda B$, the linear combination of $A$ and $B$, is still a symmetric $(0,2)$ tensor. A spacelike hypersurface is called a para-Blaschke isoparametric spacelike hypersurface, if the conform $1$-form vanishes and the eigenvalues of the para-Blaschke tensor are constant. In this paper, we classify the para-Blaschke isoparametric spacelike hypersurfaces under the conformal group of $M^{n+1}_1(c)$.