The Riemann Surface of the Logarithm Constructed in a Geometrical Framework

Abstract

The logarithmic Riemann surface Sigma_{log} is a classical holomorphic 1-manifold. It lives into R^4 and induces a covering space of C - 0 defined by exp. This paper suggests a geometric construction of it, derived as the limit of a sequence of vector fields extending exp suitably to embeddings of C into R^3, which turn to be helicoid surfaces living into C x R. In the limit we obtain a bijective complex exponential on the covering space in question, and thus a well-defined complex logarithm. In addition, the helicoids are diffeomorphic (not bi-holomorphic) copies of Sigma_{log} as smooth realizations living into R^3, without obstruction. Our approach is purely geometrical and does not employ any tools provided by the complex structure, thus holomorphy is no longer necessary to obtain constructively this Riemann surface Sigma_{log}. Moreover, the differential geometric framework we adopt affords explicit generalization on submanifolds of C^m x \R^m and certain corollaries are derived.