Abstract
We present a revised definition of a Ribaucour transformation for submanifolds of space forms, with flat normal bundle, motivated by the classical definition and by more recent extensions. The new definition provides a precise treatment of the geometric aspect of such transformations preserving lines of curvature and it can be applied to submanifolds whose principal curvatures have multiplicity bigger than one. Ribaucour transformations are applied as a method of obtaining linear Weingarten surfaces contained in Euclidean space, from a given such surface. Examples are included for minimal surfaces, constant mean curvature and linear Weingarten surfaces. The examples show the existence of complete hyperbolic linear Weingarten surfaces in Euclidean space.
Highlights
This is an expository article which presents a new definition of a Ribaucour transformation and includes some of its applications
The new definition provides a precise treatment of the geometric aspect of such transformations preserving lines of curvature and it extends Ribaucour transformations to submanifolds whose principal curvatures have multiplicity bigger than one
Whenever we say that a submanifold Mis locally associated by a Ribaucour transformation to M with respect to ei, we are assuming that ei are orthonormal principal direction vector fields on M and there are functions i, and Wγ, locally defined, satisfying the system (7)-(10)
Summary
This is an expository article which presents a new definition of a Ribaucour transformation and includes some of its applications. KETI TENENBLAT transformations can be applied as a method of constructing linear Weingarten surfaces contained in R3, in particular minimal and constant mean curvature (cmc) surfaces, from a given such surface. The method, when applied to the cylinder, produces complete n-bubble cmc and linear Weingarten surfaces. By Hilbert’s theorem, that there are no complete surfaces of constant negative curvature immersed in R3 Such surfaces and hyperbolic linear Weingarten surfaces correspond to solutions of the sine-Gordon equation, the examples constructed in (Corro et al 2001) show that there exist infinitely many complete hyperbolic linear Weingarten surfaces in R3
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