Invariant Schwarzian derivatives of higher order

TL;DR

This paper explores the relationships between higher-order Schwarzian derivatives and Aharonov invariants, proposing recursive formulas and a new invariant Schwarzian derivative for holomorphic maps between Riemann surfaces.

Contribution

It introduces a new definition of invariant Schwarzian derivatives for holomorphic maps between Riemann surfaces with conformal metrics and provides recursive formulas for these derivatives.

Findings

Established relations between Aharonov invariants and Tamanoi's Schwarzians

Derived recursion formulas for Tamanoi's Schwarzians

Proposed a new invariant Schwarzian derivative for holomorphic maps

Abstract

We argue relations between the Aharonov invariants and Tamanoi's Schwarzian derivatives of higher order and give a recursion formula for Tamanoi's Schwarzians. Then we propose a definition of invariant Schwarzian derivatives of a nonconstant holomorphic map between Riemann surfaces with conformal metrics. We show a recursion fomula also for our invariant Schwarzians.