Minimal surfaces in quaternionic symmetric spaces
نویسنده
چکیده
Theorem. Any compact Riemann surface may be minimally immersed in S. To prove this, Bryant considers the Penrose fibration π : CP 3 → S = HP . The perpendicular complement to the fibres (with respect to the Fubini-Study metric) furnishes CP 3 with a holomorphic distribution H ⊂ T CP 3 and it is well-known that a holomorphic curve in CP 3 tangent to H (a horizontal holomorphic curve) projects onto a minimal surface in S. Bryant gave explicit formulae for the horizontality condition on an affine chart which enabled him to integrate it and provide a “Weierstraß formula” for horizontal curves. Indeed, if f , g are meromorphic functions on a Riemann surface M then the curve Φ(f, g) : M → CP 3 given on an affine chart by Φ(f, g) = (f − 1 2 g(dg/df), g, 1 2 (df/dg))
منابع مشابه
Harmonic potentials for quaternionic symmetric σ-models
We construct N = 2 superspace Lagrangians for quaternionic symmetric σmodels G/H × Sp(1), or equivalently, quaternionic potentials for these symmetric spaces. They are homogeneous H invariant polynomials of order 4 which are similar to the quadratic Casimir operator of H. The construction is based on an identity for the structure constants specific for quaternionic symmetric spaces. † On leave ...
متن کاملHolomorphic Curves and Toda Systems *
Geometry of holomorphic curves from point of view of open Toda systems is discussed. Parametrization of curves related this way to non-exceptional simple Lie algebras is given. This gives rise to explicit formulas for minimal surfaces in real, complex and quaternionic projective spaces or complex quadrics. The paper generalizes the well known connection between minimal surfaces in E3, their Wei...
متن کاملTotally Geodesic Submanifolds of the Complex and the Quaternionic 2-grassmannians
In this article, the totally geodesic submanifolds in the complex 2Grassmannian G2(C) and in the quaternionic 2-Grassmannian G2(H) are classified. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank 2 published by Chen and Nagano (1978) is incomplete. For example, G2(H) with n ≥ 5 contains totally...
متن کاملCohomogeneity One Actions on Noncompact Symmetric Spaces of Rank One
We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces CHn, n ≥ 3. For the quaternionic hyperbolic spaces HHn, n ≥ 3, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classificat...
متن کامل2 4 M ay 2 00 5 Cohomogeneity one actions on noncompact symmetric spaces of rank one
We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces CH, n ≥ 3. For the quaternionic hyperbolic spaces HH, n ≥ 3, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classificatio...
متن کامل