# Weingarten map of the hypersurface in 4-dimensional Euclidean space and its applications

## TL;DR

This paper develops a practical method for computing the Weingarten map of hypersurfaces in 4-dimensional Euclidean space, enabling deeper analysis of their geometric properties similar to surface theory in 3D.

## Contribution

It introduces an efficient computational approach for the Weingarten map of hypersurfaces in E^4, extending surface theory concepts to higher dimensions.

## Key findings

- Method for calculating Weingarten map in E^4
- Introduction of Gaussian and mean curvatures for hypersurfaces
- Analysis of fundamental forms and Dupin indicatrix in E^4

## Abstract

In this paper, by taking into account the beginning of the hypersurface theory in Euclidean space $E^4$, a practical method for the matrix of the Weingarten map (or the shape operator) of an oriented hypersurface $M^3$ in $E^4$ is obtained. By taking this efficient method, it is possible to study of the hypersurface theory in $E^4$ which is analog the surface theory in $E^3$. Furthermore, the Gaussian curvature, mean curvature, fundamental forms and Dupin indicatrix of $M^3$ is introduced.