Abstract
We are dealing with domains of the complex plane which are not symmetric in common sense, but support fixed point free antianalytic involutions. They are fundamental domains of different classes of analytic functions and the respective involutions are obtained by composing their canonical projections onto the complex plane with the simplest antianalytic involution of the Riemann sphere. What we obtain are hidden symmetries of the complex plane. The list given here of these domains is far from exhaustive.
Highlights
The word symmetry comes from the Greek συμμετρια, which means “agreement in dimensions, due proportion, arrangement”
We are dealing with domains of the complex plane which are not symmetric in common sense, but support fixed point free antianalytic involutions
They are fundamental domains of different classes of analytic functions and the respective involutions are obtained by composing their canonical projections onto the complex plane with the simplest antianalytic involution of the Riemann sphere
Summary
The word symmetry comes from the Greek συμμετρια (symmetria), which means “agreement in dimensions, due proportion, arrangement”. In complex analysis objects remaining invariant to some very simple groups of transformations are legitimately called symmetric, without appearing necessarily visually symmetric These are the hidden symmetries of complex analysis. A symmetric Riemann surface S is a (bordered, or border free) Riemann surface endowed with a fixed point free antianalytic involution k This means that for every φ1,φ2 ∈ Φ , φ2 k φ1−1 is a mapping of φ1 (V1 ) onto φ2 (V2 ) whose complex conjugate is conformal. This group is denoted by k and the respective symmetric Riemann surface is denoted by (S, k ) This is a symmetry in the sense of Klein (see [2]) and in this paper we will be talking only about this kind of symmetries. When β ( X ) ≠ ∅ we have a bordered Klein surface and when β ( X ) ≠ ∅ we have a border free Klein surface
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