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Projective Differential Geometrical Structure of the Painlevé Equations

✍ Scribed by M.V. Babich; L.A. Bordag

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
250 KB
Volume
157
Category
Article
ISSN
0022-0396

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✦ Synopsis

The necessary and sufficient conditions that an equation of the form y"= f (x, y, y$) can be reduced to one of the Painleve equations under a general point transformation are obtained. A procedure to check these conditions is found. The theory of invariants plays a leading role in this investigation. The reduction of all six Painleve equations to the form y"= f (x, y) is obtained. The structure of equivalence classes is investigated for all the Painleve equations. Following Cartan the space of the normal projective connection which is uniquely associated with any class of equivalent equations is considered. The specific structure of the spaces under investigation allows us to immerse them into RP 3 . Each immersion generates a triple of two-dimensional manifolds in RP 3 . The surfaces corresponding to all the Painleve equations are presented.

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