Advances in Theoretical and Mathematical Physics

Volume 21 (2017)

Number 1

Klein foams as families of real forms of Riemann surfaces

Pages: 231 – 241



Sabir M. Gusein-Zade (Faculty of Mechanics and Mathematics, Moscow State Lomonosov University, Moscow, Russia)

Sergey M. Natanzon (National Research University Higher School of Economics, Moscow, Russia)


Klein foams are analogues of Riemann surfaces for surfaces with one-dimensional singularities. They first appeared in mathematical physics (string theory etc.). By definition a Klein foam is constructed from Klein surfaces by gluing segments on their boundaries. We show that, a Klein foam is equivalent to a family of real forms of a complex algebraic curve with some structures. This correspondence reduces investigations of Klein foams to investigations of real forms of Riemann surfaces. We use known properties of real forms of Riemann surfaces to describe some topological and analytic properties of Klein foams.

2010 Mathematics Subject Classification

14H30, 14H37, 30F50, 57M20

Published 6 April 2017