Annales Academię Scientiarum Fennicę
Mathematica
Volumen 32, 2007, 497-521
University of Manitoba, Machray Hall, Department of Mathematics
Winnipeg, Manitoba, R3T 2N2 Canada; eric_schippers 'at' umanitoba.ca
Abstract. Minda and Peschl invented a kind of derivative of maps between Riemann surfaces, which depends on the choice of conformal metric. We give explicit formulas relating the Minda-Peschl derivatives to the Levi-Civita connection, which express the difference between the two in terms of the curvature of the metric. Furthermore, we exhibit a geometric interpretation of the derivatives in terms of a decomposition of the space of symmetric complex differentials. Finally, this decomposition is used to give simple formulas for parallel transport of complex differentials which hold for conformal metrics on a Riemann surface.
2000 Mathematics Subject Classification: Primary 30C99, 30F45, 30F30, 53B05, 53A30.
Key words: Conformal metrics, Minda-Peschl derivatives, Levi-Civita connection.
Reference to this article: E. Schippers: The calculus of conformal metrics. Ann. Acad. Sci. Fenn. Math. 32 (2007), 497-521.
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