Discrete and smooth orthogonal systems: $C^\infty$-approximation

Details Discrete and smooth orthogonal systems: $C^\infty$-approximation Discrete and smooth orthogonal systems: $C^\infty$-approximation

2003-03-26

arxiv.org/abs/math/0303333v1

Intern. Math. Research Notices, 2003, Nr. 45, p. 2415-2459

Mathematics

Differential Geometry

Scientific paper

Discrete conjugate systems are quadrilateral nets with all planar faces. Discrete orthogonal systems are defined by the additional property of all faces being concircular. Their geometric properties allow one to consider them as proper discretization of conjugate, resp. orthogonal coordinate systems of classical differential geometry. We develop techniques that allow us to extend this known qualitative analogy to rigorous convergence results. In particular, we prove the $C^\infty$-convergence of discrete conjugate/orthogonal coordinate systems to smooth ones. We also show how to construct the approximating discrete nets. Coordinate systems and their transformations are treated on an equal footing, and the approximation results hold for transformations as well.

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