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The Riemann Surface of the Logarithm Constructed in a Geometrical Framework

By Nikolaos I. Katzourakis


The logarithmic Riemann surface Σlog is a classical holomorphic 1-manifold. It lives into R4 and induces a covering space of C � {0} defined by expC. This paper suggests a geometric construction of it, derived as the limit of a sequence of vector fields extending expC suitably to embeddings of C into R3, which turn to be helicoid surfaces living into C×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 Σlog as C ∞-realizations living into R3, 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 Σlog. Moreover, the differential geometric framework we adopt affords explicit generalization on submanifolds of Cm × Rm and certain corollaries are derived

Topics: Riemann Surface of the Logarithm, surfaces and submanifolds of Euclidean space
Year: 2005
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