On Surfaces in R^n via Gauss Map, Caustics, Duality and Pseudo Euclidean Geometry of Quadratic Forms
Résumé
We get new results (and rederive some know ones) on smooth surfaces in 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 R^2 . A key result (Th. 3.3.1): for a surface M in R^n the intersection of its caustic with the normal space NpM is the polar dual hypersurface (in NpM ) of the curvature ellipse at p. Moreover, all local objects X (cf. the invariants and their relations) have a "paired" version X * (with X * * = X) -this provides new results on the original objects.
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