2025/04/18 by Uribe-Vargas, Ricardo
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.14104
We get new results (and rederive some know ones) on smooth surfaces in ℝn 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 ℝ2. A key result (Th.3.3.1): for a surface M in ℝn 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.