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313
Schwarzian derivatives and flows of surfaces
 Contemp. Math
"... Over the last decades it has been widely recognized that many completely integrable PDE’s from mathematical physics arise naturally in geometry. Their integrable character in the geometric context – usually associated with the presence of a Lax pair and a spectral deformation – is nothing but the fl ..."
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Cited by 27 (3 self)
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Over the last decades it has been widely recognized that many completely integrable PDE’s from mathematical physics arise naturally in geometry. Their integrable character in the geometric context – usually associated with the presence of a Lax pair and a spectral deformation – is nothing but the flatness condition of a naturally occurring connection. Examples of interest to geometers generally come from the integrability equations of special surfaces in various ambient spaces: among the best known examples is the sinhGordon equation describing constant mean curvature tori in 3space. In fact, most of the classically studied surfaces, such as isothermic surfaces, surfaces of constant curvature and Willmore surfaces, give rise to such completely integrable PDE. A common thread in many of these examples is the appearance of a harmonic map into some symmetric space, which is wellknown to admit a Lax representation with spectral parameter. Once the geometric problem is formulated this way algebrogeometric integration techniques give explicit parameterizations of the surfaces in question in terms of theta functions. One contribution of the geometric view of these integrable PDE is a much better understanding of the meaning of the hierarchy of flows associated to these equations. In mathematical physics these hierarchies are obtained by deformations of the Lax operators preserving their eigenvalues. This is rather unsatisfactory from the geometric viewpoint, where one wants to see these flows as geometric deformations on the surfaces.
A solution of a problem of Sophus Lie: Normal forms of 2dim metrics admitting two projective vector fields.
, 2007
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Bäcklund–Darboux transformations in Sato’s Grassmannian
, 1996
"... We define Bäcklund–Darboux transformations in Sato’s Grassmannian. They can be regarded as Darboux transformations on maximal algebras of commuting ordinary differential operators. We describe the action of these transformations on related objects: wave functions, taufunctions and spectral algebras ..."
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Cited by 24 (9 self)
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We define Bäcklund–Darboux transformations in Sato’s Grassmannian. They can be regarded as Darboux transformations on maximal algebras of commuting ordinary differential operators. We describe the action of these transformations on related objects: wave functions, taufunctions and spectral algebras.
Discretization of Surfaces and Integrable Systems
 In: Discrete Integrable Geometry and Physics; Eds.: A.I. Bobenko and
, 1998
"... this paper are analogues of the asymptotic or of the curvature lines on smooth surfaces, one can additionally assume that the number of edges meeting at each vertice is even. The case G = Z ..."
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Cited by 24 (8 self)
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this paper are analogues of the asymptotic or of the curvature lines on smooth surfaces, one can additionally assume that the number of edges meeting at each vertice is even. The case G = Z
A New Parameterization of the Attitude Kinematics
 Journal of the Astronautical Sciences
, 1995
"... We present a new method for describing the kinematics of the rotational motion of a rigid body. The new kinematic formulation provides a threedimensional parameterization of the rotation group using two perpendicular rotations; thus it complements the Eulerian angles (three rotations) and EulerRod ..."
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Cited by 24 (14 self)
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We present a new method for describing the kinematics of the rotational motion of a rigid body. The new kinematic formulation provides a threedimensional parameterization of the rotation group using two perpendicular rotations; thus it complements the Eulerian angles (three rotations) and EulerRodrigues parameters (one rotation). The differential equations can be described by two scalar equations. We show the connection of the new parameterization with the other classical parameterizations. The new kinematic formulation has potential applications in astrodynamics, attitude control, robotics and other fields. Introduction In recent years a considerable amount of effort has been devoted to the development of a comprehensive theory that will allow a better understanding of the complex dynamic behavior associated with the motion of rotating rigid bodies. A cornerstone in this effort is the development of alternative ways of describing the kinematics of this motion. As far as the dynamic...
The history of qcalculus and a new method
, 2000
"... 1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8 ..."
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Cited by 23 (10 self)
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1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8
On organizing principles of discrete differential geometry. Geometry of spheres
 RUSSIAN MATH. SURVEYS 62:1 1–43
, 2007
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differential geometry and integrability of systems of hydrodynamic type, Applications of analytic and geometric methods to nonlinear differential equations
 MR MR1261665 (94m:58129) 32 A. D. SMITH
, 1992
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Structure Theorems for Constant Mean Curvature Surfaces Bounded by a Planar Curve
 Indiana Univ. Math J
, 1991
"... Introduction A circle C in R 3 is the boundary of two spherical caps of constant mean curvature H for any positive number H, which is at most the radius of C. It is natural to ask whether spherical caps are the only possible examples. Some examples of constant mean curvature immersed tori by Went ..."
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Introduction A circle C in R 3 is the boundary of two spherical caps of constant mean curvature H for any positive number H, which is at most the radius of C. It is natural to ask whether spherical caps are the only possible examples. Some examples of constant mean curvature immersed tori by Wente [7] indicate that there are compact genusone immersed constant mean curvature surfaces with boundary C that are approximated by compact domains in Wente tori; however, this has not been proved. Still one has the conjecture: Conjecture 1 A compact constant mean curvature surface bounded by a circle is a spherical cap if either of the following conditions hold: 1. The surface has genus 0 and is immersed; 2. The surface is embedded. If M is a compact embedded constant mean curvature surface in R 3 with boundary C<F14