681 research outputs found
Orthogonal nets and Clifford algebras
A Clifford algebra model for M"obius geometry is presented. The notion of
Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced,
and the structure equations for adapted frames are derived. These equations are
discretized and the geometry of the occuring discrete nets and sphere
congruences is discussed in a conformal setting. This way, the notions of
``discrete Ribaucour congruences'' and ``discrete Ribaucour pairs of orthogonal
systems'' are obtained --- the latter as a generalization of discrete
orthogonal systems in Euclidean space. The relation of a Cauchy problem for
discrete orthogonal nets and a permutability theorem for the Ribaucour
transformation of smooth orthogonal systems is discussed.Comment: Plain TeX, 16 pages, 4 picture
The Ribaucour transformation in Lie sphere geometry
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.Comment: v2: Introduction expanded and references added. 20 pages, 4
Postscript figure
On the Birkhoff factorization problem for the Heisenberg magnet and nonlinear Schroedinger equations
A geometrical description of the Heisenberg magnet (HM) equation with
classical spins is given in terms of flows on the quotient space where
is an infinite dimensional Lie group and is a subgroup of . It is
shown that the HM flows are induced by an action of on ,
and that the HM equation can be integrated by solving a Birkhoff factorization
problem for . For the HM flows which are Laurent polynomials in the spectral
variable we derive an algebraic transformation between solutions of the
nonlinear Schroedinger (NLS) and Heisenberg magnet equations. The Birkhoff
factorization for is treated in terms of the geometry of the Segal-Wilson
Grassmannian . The solution of the problem is given in terms of a pair
of Baker functions for special subspaces of . The Baker functions are
constructed explicitly for subspaces which yield multisoliton solutions of NLS
and HM equations.Comment: To appear in Journal of Mathematical Physic
Incircular nets and confocal conics
We consider congruences of straight lines in a plane with the combinatorics
of the square grid, with all elementary quadrilaterals possessing an incircle.
It is shown that all the vertices of such nets (we call them incircular or
IC-nets) lie on confocal conics.
Our main new results are on checkerboard IC-nets in the plane. These are
congruences of straight lines in the plane with the combinatorics of the square
grid, combinatorially colored as a checkerboard, such that all black coordinate
quadrilaterals possess inscribed circles. We show how this larger class of
IC-nets appears quite naturally in Laguerre geometry of oriented planes and
spheres, and leads to new remarkable incidence theorems. Most of our results
are valid in hyperbolic and spherical geometries as well. We present also
generalizations in spaces of higher dimension, called checkerboard IS-nets. The
construction of these nets is based on a new 9 inspheres incidence theorem.Comment: 33 pages, 24 Figure
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