On Surfaces in R^n via Gauss Map, Caustics, Duality and Pseudo Euclidean Geometry of Quadratic Forms

Abstract

We get new results (and rederive some know ones) on smooth surfaces in ${\mathbb{R}}^n$ by unifying several view points into a coherent general view. Namely, we show and use new relations of the evolute (caustic) with the curvature ellipse, the Gauss map and the pseudo-Euclidean geometry of the $3$-space of quadratic forms on ${\mathbb{R}}^2$. A key result (Th.3.3.1): for a surface $M$ in ${\mathbb{R}}^n$ the intersection of its caustic with the normal space $N_{p}M$ is the polar dual hypersurface (in $N_{p}M$) of the curvature ellipse at $p$. Moreover, all local objects $X$ (cf. the invariants and their relations) have a "paired" version $X^{\ast }$ (with ${X^{\ast }}^{\ast }=X$) -- this provides new results on the original objects.