# Principal configurations around umbilics of spacelike surfaces in null hypersurfaces of $\mathbb{R}_1^4$

## TL;DR

This paper analyzes the principal configurations around umbilic points on spacelike surfaces within null hypersurfaces of Minkowski space, revealing geometric structures and special configurations in specific null hypersurfaces.

## Contribution

It characterizes principal configurations near umbilic points on spacelike surfaces in null hypersurfaces of Minkowski space, including special cases like null rotation hypersurfaces.

## Key findings

- Description of principal configurations around umbilic points
- Recovery of Darbouxian configurations in null rotation hypersurfaces
- Insights into geometric structures of spacelike surfaces in null hypersurfaces

## Abstract

We study the principal configurations around an isolated $\eta$-umbilical point on a generic spacelike surface $S$ immersed in a null hypersurface $M$ of Minkowski space $\mathbb{R}_1^4$ relative to a well-defined null vector field $\eta$ orthogonal to the surface $S$. In the particular case of $M$ being a null rotation hypersurface of $\mathbb{R}_1^4$ we also recover the local Darbouxian principal configurations around that kind of $\eta$-umbilical points.