Flat approximations of hypersurfaces along curves

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.

arXiv comment11 pages, no figures

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