The vectorial Ribaucour transformation for submanifolds and applications

Dajczer, M.; Florit, L.; Tojeiro, R.

doi:10.1090/S0002-9947-07-04211-0

The vectorial Ribaucour transformation for submanifolds and applications
HTML articles powered by AMS MathViewer

by M. Dajczer, L. A. Florit and R. Tojeiro PDF

Trans. Amer. Math. Soc. 359 (2007), 4977-4997 Request permission

Abstract:

In this paper we develop the vectorial Ribaucour transformation for Euclidean submanifolds. We prove a general decomposition theorem showing that under appropriate

conditions the composition of two or more vectorial Ribaucour transformations is again a vectorial Ribaucour transformation. An immediate consequence of this result is the classical permutability of Ribaucour transformations. Our main application is to provide an explicit local construction of an arbitrary Euclidean $n$-dimensional submanifold with flat normal bundle and codimension $m$ by means of a commuting family of $m$ Hessian matrices on an open subset of Euclidean space $\mathbb {R}^n$. Actually, this is a particular case of a more general result. Namely, we obtain a similar local construction of all Euclidean submanifolds carrying a parallel flat normal subbundle, in particular of all those that carry a parallel normal vector field. Finally, we describe all submanifolds carrying a Dupin principal curvature normal vector field with integrable conullity, a concept that has proven to be crucial in the study of reducibility of Dupin submanifolds.

References

Jürgen Berndt, Sergio Console, and Carlos Olmos, Submanifolds and holonomy, Chapman & Hall/CRC Research Notes in Mathematics, vol. 434, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1990032, DOI 10.1201/9780203499153
F.E. Burstall and U. Hertrich-Jeromin, The Ribaucour transformation in Lie sphere geometry, Preprint arXiv:math.DG/0407244 v1, 2004.
G. Darboux, Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars, Paris, 1910.
Marcos Dajczer and Ruy Tojeiro, An extension of the classical Ribaucour transformation, Proc. London Math. Soc. (3) 85 (2002), no. 1, 211–232. MR 1901374, DOI 10.1112/S0024611502013552
Marcos Dajczer and Ruy Tojeiro, Commuting Codazzi tensors and the Ribaucour transformation for submanifolds, Results Math. 44 (2003), no. 3-4, 258–278. MR 2028680, DOI 10.1007/BF03322986
Marcos Dajczer, Luis A. Florit, and Ruy Tojeiro, Reducibility of Dupin submanifolds, Illinois J. Math. 49 (2005), no. 3, 759–791. MR 2210258
I. Bivens, J.-P. Bourguignon, A. Derdziński, D. Ferus, O. Kowalski, T. Klotz Milnor, V. Oliker, U. Simon, W. Strübing, and K. Voss, Discussion on Codazzi-tensors, Global differential geometry and global analysis (Berlin, 1979) Lecture Notes in Math., vol. 838, Springer, Berlin-New York, 1981, pp. 243–299. MR 636289
E. V. Ferapontov, Surfaces with flat normal bundle: an explicit construction, Differential Geom. Appl. 14 (2001), no. 1, 15–37. MR 1811562, DOI 10.1016/S0926-2245(00)00042-5
E. V. Ferapontov, Hamiltonian systems of hydrodynamic type and their realizations on hypersurfaces of a pseudo-Euclidean space, Problems in geometry, Vol. 22 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, pp. 59–96, 219 (Russian). Translated in J. Soviet Math. 55 (1991), no. 5, 1970–1995. MR 1099220
E. V. Ferapontov, Dupin hypersurfaces and integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants, Differential Geom. Appl. 5 (1995), no. 2, 121–152. MR 1334839, DOI 10.1016/0926-2245(95)00011-R
E. I. Ganzha and S. P. Tsarev, An algebraic superposition formula and the completeness of Bäcklund transformations of $(2+1)$-dimensional integrable systems, Uspekhi Mat. Nauk 51 (1996), no. 6(312), 197–198 (Russian); English transl., Russian Math. Surveys 51 (1996), no. 6, 1200–1202. MR 1440161, DOI 10.1070/RM1996v051n06ABEH003007
Udo Hertrich-Jeromin, Introduction to Möbius differential geometry, London Mathematical Society Lecture Note Series, vol. 300, Cambridge University Press, Cambridge, 2003. MR 2004958, DOI 10.1017/CBO9780511546693
Q. P. Liu and Manuel Mañas, Vectorial Ribaucour transformations for the Lamé equations, J. Phys. A 31 (1998), no. 10, L193–L200. MR 1629398, DOI 10.1088/0305-4470/31/10/003