Hildebrand classified all semi-homogeneous cones in R 3 and computed their corresponding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand's new examples. After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hildebrand's examples. The generic member of these affine spheres is given by Weierstrass P, ζ and σ functions. In general any regular convex cone in R 3 has a natural associated S 1 -family of such cones, which deserve further studies.
Abstract. We associate a natural λ-family (λ ∈ R \ {0}) of flat Lagrangian immersions in C n with non-degenerate normal bundle to any given one. We prove that the structure equations for such immersions admit the same Lax pair as the first order integrable system associated to the symmetric spaceAn interesting observation is that the family degenerates to an Egoroff net on R n when λ → 0. We construct an action of a rational loop group on such immersions by identifying its generators and computing their dressing actions. The action of the generator with one simple pole gives the geometric Ribaucour transformation and we provide the permutability formula for such transformations. The action of the generator with two poles and the action of a rational loop in the translation subgroup produce new transformations. The corresponding results for flat Lagrangian submanifolds in CP n−1 and ∂-invariant Egoroff nets follow nicely via a spherical restriction and Hopf fibration.
Let σ be an involution of a real semi-simple 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 r-dimensional 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/U0-hierarchy of commuting soliton equations to construct the U/U0-system. She showed that the U/U0-system 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 Gauss-Codazzi 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.Let U be the fixed point set of τ , i.e., a real form of G. We will still use τ, σ to denote dτ e and dσ e respectively. Let G, U denote the Lie algebras of G and U respectively. Since σ and τ commute, σ(U ) ⊂ U . So σ|U is an involution of U . Let U 0 , U 1 denote the +1, −1 eigenspaces of σ on U . ThenThe quotient space U/U 0 is a symmetric space, and the eigen-decomposition U = U 0 + U 1 is called a Cartan decomposition.Ferus and Pedit ([8]) called a submanifold M of a rank r symmetric space U/U 0 a curved flat if T p M is tangent to an r-dimensional flat of U/U 0 at p for each p ∈ M . They noted that the equation for curved flats is an integrable system. Bryant ([6]) used the involution σ to construct a natural *
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