Orthogonal nets and Clifford algebras

Details Orthogonal nets and Clifford algebras Orthogonal nets and Clifford algebras

1998-02-27

arxiv.org/abs/math/9802126v1

Mathematics

Differential Geometry

Plain TeX, 16 pages, 4 pictures

Scientific paper

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.

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