Bäcklund's Theorem for n-Dimensional Submanifolds of R 2n - 1

Geometric aspects of Bäcklund transformations of Weingarten submanifolds

“…= r n -0, the theorems specialize to results due to Tenenblat and Terng [5]. Theorem 2 is noticeably simpler in the isoclinic case, where a* is simply the Riemann curvature tensor times the reciprocal of the sine of the isoclinic angle.…”

“…Following [5], let E and E' be m planes in V, and let P: V -> E be the orthogonal projection. We can define a symmetric bilinear form on E 1 by (vi,v 2 …”

“…Since a surface of constant negative curvature is equivalent to a solution of the Sine-Gordon equation, the Backlund transformation of surfaces corresponds to a Backlund transformation of the solution to the Sine-Gordon equation. In 1980, Tenenblat and Terng [5] returned to the geometric viewpoint and showed that the Backlund transformation could be extended to n-dimensional submanifolds of 2n -1-dimensional Euclidean space. Under the conditions that the connecting tangent line segment have constant length, that the normal affine spaces meet at an isoclinic, constant angle, and that the normal bundle be flat, they showed that the submanifolds have to have constant negative curvature.…”

Bäcklund Transformations for Integrable Geometric Curve Flows

“…In [4], Wahlquist and Estabrook [4] provides a systematic method to construct Bäcklund transformations of integrable systems by using the prolongation structure approach. Other effective methods to construct Bäcaklund transformations of integrable systems were also proposed in a number of literatures, see for example [2,[5][6][7][8][9][10][11][12] and many more references.…”

Riccati-Type Equations, Generalised WZNW Equations, and Multidimensional Toda Systems

“…For a systematic account of the development in the theory of the matrix differential Riccati equation up to seventies see, for example, the survey [1]. More recently there appeared papers where this equation was considered as a Bäcklund-type transformation for some integrable systems of differential geometry, in particular, for the Lamé and the Bourlet equations, and a relevant superposition principle for the equation has been studied on the basis of the theory of Lie algebras, see, for example, [2] and references therein. The matrix Riccati equation also arises as equation of motion on Grassmann manifolds and on homogeneous spaces attached to the Hartree-Fock-Bogoliubov problem, see, for example, [3] and references therein; and in some other subjects of applied mathematics and physics such as optimal control theory, plasma, etc., see, for example, [4].…”