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48
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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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.
The Submanifold Geometries associated to Grassmannian Systems
 735, viii + 95
"... There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order nonlinear system of partial di#erential equations in n variables, the so called U/Ksystem. Let G m,n denote the Gr ..."
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There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order nonlinear system of partial di#erential equations in n variables, the so called U/Ksystem. Let G m,n denote the Grassmannian of ndimensionallinear subspaces in R m+n , and G 1 m,n the Grassmannian of spacelike mdimensionallinear subspaces in the Lorentzian space R m+n,1 . In this paper, we use techniques from soliton theory to study submanifolds in space forms whose GaussCodazzi equations are gauge equivalent to the G m,n system or the G 1 m,n system. These include submanifolds with constant sectional curvatures, isothermic surfaces, and submanifolds admitting principal curvature coordinates. The dressing actions of simple elements on the space of solutions of the G m,n and G 1 m,n systems correspond toB acklund, Darboux and Ribaucour transformations for submanifolds. Tabl# of Contents 1.
Isothermic submanifolds of symmetric Rspaces
, 2009
"... We extend the classical theory of isothermic surfaces in conformal 3space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric Rspaces with essentially no loss of integrable structure. ..."
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We extend the classical theory of isothermic surfaces in conformal 3space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric Rspaces with essentially no loss of integrable structure.
ON THE MODULI OF CONSTANT MEAN CURVATURE CYLINDERS OF FINITE TYPE IN THE 3SPHERE
, 2008
"... We show that onesided Alexandrov embedded constant mean curvature cylinders of finite type in the 3sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3sphere are rotational. ..."
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We show that onesided Alexandrov embedded constant mean curvature cylinders of finite type in the 3sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3sphere are rotational.
Isothermic submanifolds of Euclidean space
, 2004
"... Abstract: We study the problem posed by F. Burstall of developing a theory of isothermic Euclidean submanifolds of dimension greater than or equal to three. As a natural extension of the definition in the surface case, we call a Euclidean submanifold isothermic if it is locally the image of a confor ..."
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Abstract: We study the problem posed by F. Burstall of developing a theory of isothermic Euclidean submanifolds of dimension greater than or equal to three. As a natural extension of the definition in the surface case, we call a Euclidean submanifold isothermic if it is locally the image of a conformal immersion of a Riemannian product of Riemannian manifolds whose second fundamental form is adapted to the product net of the manifold. Our main result is a complete classification of all such conformal immersions of Riemannian products of dimension greater than or equal to three. We derive several consequences of this result. We also study whether the classical characterizations of isothermic surfaces as solutions of Christoffel’s problem and as envelopes of nontrivial conformal sphere congruences extend to higher dimensions.
The Ribaucour transformation in Lie sphere geometry
 Differential Geom. Appl
"... Abstract. We discuss the Ribaucour transformation of Legendre maps in Lie sphere geometry. In this context, we give a simple conceptual proof of Bianchi’s original Permutability Theorem and its generalisation by Dajczer– Tojeiro. We go on to formulate and prove a higher dimensional version of the Pe ..."
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Abstract. We discuss the Ribaucour transformation of Legendre maps in Lie sphere geometry. In this context, we give a simple conceptual proof of Bianchi’s original Permutability Theorem and its generalisation by Dajczer– Tojeiro. We go on to formulate and prove a higher dimensional version of the Permutability Theorem. It is shown how these theorems descend to the corresponding results for submanifolds in space forms. 1.
Conformal de Rham decomposition of Riemannian manifolds
 Houston J. Math
"... Abstract: We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riem ..."
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Abstract: We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riemannian or a warped product. We also obtain other related de Rhamtype decomposition theorems. As an application, we study Riemannian manifolds that admit a Codazzi tensor with two distinct eigenvalues everywhere.
Curved flats, exterior differential systems, and conservation laws, Complex, contact and symmetric manifolds
 235–254, Progr. Math., 234, Birkhauser
, 2005
"... Abstract. Let σ be an involution of a real semisimple Lie group U, U0 the subgroup fixed by σ, and U/U0 the corresponding symmetric space. Ferus and Pedit called a submanifold M of a rank r symmetric space U/U0 a curved flat if TpM is tangent to an rdimensional flat of U/U0 at p for each p ∈ M. Th ..."
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Abstract. Let σ be an involution of a real semisimple Lie group U, U0 the subgroup fixed by σ, and U/U0 the corresponding symmetric space. Ferus and Pedit called a submanifold M of a rank r symmetric space U/U0 a curved flat if TpM is tangent to an rdimensional flat of U/U0 at p for each p ∈ M. They noted that the equation for curved flats is an integrable system. Bryant used the involution σ to construct an involutive exterior differential system Iσ such that integral submanifolds of Iσ are curved flats. Terng used r first flows in the U/U0hierarchy of commuting soliton equations to construct the U/U0system. She showed that the U/U0system and the curved flat system are gauge equivalent, used the inverse scattering theory to solve the Cauchy problem globally with smooth rapidly decaying initial data, used loop group factorization to construct infinitely many families of explicit solutions, and noted that many these systems occur as the GaussCodazzi equations for submanifolds in space forms. The main goals of this paper are: (i) give a review of these known results, (ii) use techniques from soliton theory to construct infinitely many integral submanifolds and conservation laws for the exterior differential system Iσ. 1.
Spacelike Willmore surfaces in 4dimensional Lorentzian space forms, Sci
 in China: Ser. A, Math
"... Spacelike Willmore surfaces in 4dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S 4, are studied in this paper. We define two kinds of transforms for such a surface, which produce the socalled left/right polar surfaces ..."
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Cited by 8 (7 self)
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Spacelike Willmore surfaces in 4dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S 4, are studied in this paper. We define two kinds of transforms for such a surface, which produce the socalled left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds interesting duality theorem. As an application spacelike Willmore 2spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.
The BianchiDarboux transform of Lisothermic surfaces
 Sch] [WP] W.K. Schief. Isothermic surfaces in spaces of
"... Abstract. We study an analogue of the classical Bäcklund transformation for Lisothermic surfaces in Laguerre geometry, the Bianchi–Darboux transformation. We show how to construct the Bianchi–Darboux transforms of an Lisothermic surface by solving an integrable linear differential system. We then ..."
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Abstract. We study an analogue of the classical Bäcklund transformation for Lisothermic surfaces in Laguerre geometry, the Bianchi–Darboux transformation. We show how to construct the Bianchi–Darboux transforms of an Lisothermic surface by solving an integrable linear differential system. We then establish a permutability theorem for iterated Bianchi–Darboux transforms. 1. Introduction. Certain types of integrable nonlinear PDEs (soliton equations) arise in differential geometry as compatibility conditions for the linear equations obeyed by frames adapted to surfaces in higher dimensional manifolds. In a number of situations, the construction of new solutions of the arising PDE relies on the existence of Bäcklund type