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Hyperbolic Weingarten surfaces

Published online by Cambridge University Press:  24 October 2008

B. Van-Brunt  and
K. Grant
B. Van-Brunt
Affiliation:
Department of Mathematics, Massey University, Palmerston North, New Zealand
K. Grant
Affiliation:
Department of Mathematics, Massey University, Palmerston North, New Zealand
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Abstract

Weingarten surfaces which can be represented locally as solutions to second order hyperbolic partial differential equations are examined in this paper. In particular, the geometry of the families of curves corresponding to characteristics on these surfaces is investigated and the relationships of these curves with other curves on the surface such as asymptotic lines and lines of curvature are explored. It is shown that singularities in the lines of curvature, i.e. umbilic points, correspond to singularities in the families of characteristics, and that lines of curvature are non-characteristic curves. If there is a linear relation between the Gaussian and mean curvatures and real characteristics exist, then the characteristics form a Tchebychef net on the corresponding Weingarten surface.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 3 , November 1994 , pp. 489 - 504
Copyright
Copyright © Cambridge Philosophical Society 1994

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