49 research outputs found
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry
Abstract. We provide a method of constructing families of hypersurfaces of a space form with zero Gauss-Kronecker curvature, from a given such hypersurface, based on Ribaucour transformations. Applications provide a 1-parameter family of complete, non-cylindrical hypersurfaces of R 4 , with zero Gauss-Kronecker curvature, a 5-parameter family of compact Dupin hypersurfaces of S 4 , with vanishing Gauss-Kronecker curvature, infinite families of hypersurfaces of R n+1 and of the hyperbolic space H 4 , with flat Gauss-Kronecker curvature
Curved Flats, Pluriharmonic Maps and Constant Curvature Immersions into Pseudo-Riemannian Space Forms
We study two aspects of the loop group formulation for isometric immersions
with flat normal bundle of space forms. The first aspect is to examine the loop
group maps along different ranges of the loop parameter. This leads to various
equivalences between global isometric immersion problems among different space
forms and pseudo-Riemannian space forms. As a corollary, we obtain a
non-immersibility theorem for spheres into certain pseudo-Riemannian spheres
and hyperbolic spaces.
The second aspect pursued is to clarify the relationship between the loop
group formulation of isometric immersions of space forms and that of
pluriharmonic maps into symmetric spaces. We show that the objects in the first
class are, in the real analytic case, extended pluriharmonic maps into certain
symmetric spaces which satisfy an extra reality condition along a totally real
submanifold. We show how to construct such pluriharmonic maps for general
symmetric spaces from curved flats, using a generalised DPW method.Comment: 21 Pages, reference adde
Scalar second order evolution equations possessing an irreducible sl-valued zero curvature representation
We find all scalar second order evolution equations possessing an
sl-valued zero curvature representation that is not reducible to a proper
subalgebra of sl. None of these zero-curvature representations admits a
parameter.Comment: 10 pages, requires nath.st
A geometric interpretation of the spectral parameter for surfaces of constant mean curvature
Considering the kinematics of the moving frame associated with a constant
mean curvature surface immersed in S^3 we derive a linear problem with the
spectral parameter corresponding to elliptic sinh-Gordon equation. The spectral
parameter is related to the radius R of the sphere S^3. The application of the
Sym formula to this linear problem yields constant mean curvature surfaces in
E^3. Independently, we show that the Sym formula itself can be derived by an
appropriate limiting process R -> infinity.Comment: 12 page
Multidimensional Toda type systems
On the base of Lie algebraic and differential geometry methods, a wide class
of multidimensional nonlinear systems is obtained, and the integration scheme
for such equations is proposed.Comment: 29 pages, LaTeX fil