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INTEGRABLE FOUR-COMPONENT SYSTEMS OF CONSERVATION LAWS AND LINEAR CONGRUENCES IN ${\mathbb P}^5$

Published online by Cambridge University Press:  14 July 2005

S. I. AGAFONOV  and
E. V. FERAPONTOV
S. I. AGAFONOV
Affiliation:
Fachbereich Mathematik und Informatik, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle (Saale), Germany e-mail: Agafonov@mathematik.uni-halle.de
E. V. FERAPONTOV
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom e-mail: E.V.Ferapontov@lboro.ac.uk
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Abstract

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We propose a differential-geometric classification of the four-component hyperbolic systems of conservation laws which satisfy the following properties: (a) they do not possess Riemann invariants; (b) they are linearly degenerate; (c) their rarefaction curves are rectilinear; (d) the cross-ratio of the four characteristic speeds is harmonic. This turns out to provide a classification of projective congruences in ${\mathbb P}^5$ whose developable surfaces are planar pencils of lines, each of these lines cutting the focal variety at points forming a harmonic quadruplet. Symmetry properties and the connection of these congruences to Cartan's isoparametric hypersurfaces are discussed.

Keywords

53A2553B5035L65
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 47 , Issue A , June 2005 , pp. 17 - 32
Copyright
2005 Glasgow Mathematical Journal Trust