Seong-A KIM and Toshiyuki SUGAWA

*Invariant differential operators associated with a conformal metric*

### Abstract

Peschl defined invariant higher-order derivatives of a holomorphic
or meromorphic function on the unit disk.
Here, the invariance is concerned with the hyperbolic metric of the
source domain and the canonical metric of the target domain.
Minda and Schippers extended Peschl's invariant derivatives
to the case of general conformal metrics.
We introduce similar invariant derivatives for smooth functions on a
Riemann surface and show a complete analogue of Faà di Bruno's formula
for the composition of a smooth function with a holomorphic map
with respect to the derivatives.
An interpretation of these derivatives in terms of intrinsic geometry
and some applications will be also given.

submission: 9 October, 2006

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