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Conformal Schwarzian derivatives and differential equations. (English) Zbl 1043.53015
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6--15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 271-283 (2003).
Some years ago Sasaki and Yoshida gave the fundamental system of linear equations, which is the key system connecting the theory of conformal connections and the uniformizing differential equations in the geometry of symmetric domains of type IV [cf. {\it T. Sasaki} and {\it M. Yoshida}, Tôhoku Math. J., II. Ser. 41, 321--348 (1989; Zbl 0661.35014)]. It is a system of equations with $n$ variables such that the maximal dimension of the solution space is $n+2$. The solutions naturally provide a map into the projective space whose image is contained in the hyperquadric, and accepts the conformal transformation group as its symmetry. Sasaki and Yoshida considered the equations as a higher dimensional analogue of the Gauss-Schwarz equation. In projective geometry of higher dimension, they defined Schwarzian derivatives as a difference of normal Cartan connections moved by a diffeomorphism and, using the Schwarzian derivatives, they got a system of linear equations such that the maximal dimension of the solution space is $n+1$ on $n$ variables. In the paper under review the authors investigate the fundamental system of equations in the theory of conformal geometry, whose coefficients are considered as the conformal Schwarzian derivative. They obtain the integrability condition of the system by a simple method, which allows to find a natural geometric structure on the solution space. From the solution spaces, using this geometric structure, the authors get a transformation whose Schwarzian derivative is equal to the given coefficients of the equation. For the entire collection see [Zbl 1008.00022].
53A55Differential invariants (local theory), geometric objects
53B20Local Riemannian geometry
53B30Lorentz metrics, indefinite metrics