The vectorial Ribaucour transformation for submanifolds and applications. (English) Zbl 1124.53009
By extending the transformation in [Q. P. Liu; M. Manas, J. Phys. A, Math. Gen. 31, No. 10, L193–L200 (1998; Zbl 0922.58088)], a vectorial Ribaucour transformation for Euclidean submanifolds is developed here for orthogonal coordinate systems. A general decomposition theorem gives the conditions under which the composition of two or more vectorial Ribaucour transformations is again a Ribaucour transformation. The main application of this is an explicit local construction of an arbitrary Euclidean \(n\)-dimensional submanifold with flat normal bundle and codimension \(m\), by using a commuting family of \(m\) Hessian matrices on an open subset of the Euclidean space \(\mathbb{R}^n\). More general, a local construction of all Euclidean submanifolds carrying a parallel flat normal bundle is obtained. In this context, see J. Berndt, S. Console and C. Olmos [Submanifolds and holonomy. Boca Raton, FL: Chapman and Hall/CRC (2003; Zbl 1043.53044)].
The authors obtain an explicit construction in terms of the vectorial Ribaucour transformations of all Euclidean submanifolds that carry a Dupin principal curvature normal vector field with integrable conullity, which is a crucial concept in the study of reducibility of Dupin submanifolds [M. Dajczer, L. Florit and R. Tojeiro, Ill. J. Math. 49, 759–791 (2005; Zbl 1090.53024)].
The authors obtain an explicit construction in terms of the vectorial Ribaucour transformations of all Euclidean submanifolds that carry a Dupin principal curvature normal vector field with integrable conullity, which is a crucial concept in the study of reducibility of Dupin submanifolds [M. Dajczer, L. Florit and R. Tojeiro, Ill. J. Math. 49, 759–791 (2005; Zbl 1090.53024)].
Reviewer: Cornelia-Livia Bejan (Iaşi)
MSC:
| 53B25 | Local submanifolds |
| 58J72 | Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds |
References:
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