Prime order automorphisms of Klein surfaces representable by rotations of the euclidean space

TL;DR

This paper characterizes prime order automorphisms of bordered orientable Klein surfaces that can be realized as rotations in Euclidean space, providing conditions for their topological and conformal equivalence.

Contribution

It establishes criteria for when such automorphisms are equivalent to rotations in Euclidean space, including explicit representations for notable automorphisms.

Findings

Conditions for conformal equivalence to Euclidean rotations

Representation of automorphisms of Klein quartic and Wiman surface as rotations

Explicit geometric realizations in R^4 and S^4

Abstract

Let S be a bordered orientable Klein surface and p a prime. Assume that f is an order p automorphism of S. In this work we obtain the conditions on the topological type of (S,f) to be conformally equivalent to (S',f') where S' is a bordered orientable Klein surface embedded in the Euclidean space and f' is the restriction to S' of a prime order rotation. We represent two famous automorphisms using rotations of R^4 and S^4 : the order seven automorphisms of the Klein quartic and the Wiman surface.