# Flat approximations of hypersurfaces along curves

## TL;DR

This paper investigates the existence and uniqueness of flat surface approximations along curves in higher-dimensional surfaces, providing explicit constructions based on classical characterizations of flat surfaces.

## Contribution

It introduces a method to construct flat surface approximations along curves in higher dimensions using explicit parametric formulas.

## Key findings

- Existence and uniqueness results for flat approximations along curves.
- Explicit parametric construction based on torses.
- Extension of classical flat surface characterizations to higher dimensions.

## Abstract

Given a smooth curve $\gamma$ in some $m$-dimensional surface $M$ in $\mathbb{R}^{m+1}$, we study existence and uniqueness of a flat surface $H$ having the same field of normal vectors as $M$ along $\gamma$, which we call a flat approximation of $M$ along $\gamma$. In particular, the well-known characterisation of flat surfaces as torses (ruled surfaces with tangent plane stable along the rulings) allows us to give an explicit parametric construction of such approximation.