K-isothermic Hypersurfaces

Abstract

In this paper, we consider Hypersurfaces of dimension n in the Euclidean space and introduce the k-isothermic hypersurfaces, with k<n, as hypersurfaces that locally admit orthogonal parametrization by curvature lines with k coefficients of the first quadratic form distinct. Transformations that preserve k-isothermic hypersurface are isometries, dilations and invertions. We prove that there are no k-isothermic hypersurface dimension n with distinct principal curvatures for n ? k + 3. We introduced two ways to generate a (k + 1)-isothermic Hypersurface from a k-isothermic hypersurface, which we will call 2-reducible. Moreover, we provide a local characterization of Dupin 2-isothermic hypersurface and include explicit examples of Dupin 2-isothermic hypersurface 2-irreducible.