Riemann Surface Structure for a Curved Surface with Punctured Features

Authors

  • Ramya Deepak Shetty Ramaiah University of Applied Sciences, University House, Gnanagangothri Campus, New BEL Road, M S R Nagar, Bangalore, Karnataka, INDIA - 560 054
  • Indira Narayana Swamy Ramaiah University of Applied Sciences, University House, Gnanagangothri Campus, New BEL Road, M S R Nagar, Bangalore, Karnataka, INDIA - 560 054
  • Govind R Kadambi Ramaiah University of Applied Sciences, University House, Gnanagangothri Campus, New BEL Road, M S R Nagar, Bangalore, Karnataka, INDIA - 560 054

DOI:

https://doi.org/10.3126/njmathsci.v2i1.36504

Keywords:

Riemann surface structure, Holomorphic function, Transition functions, Canonical curved surface

Abstract

In this paper, a generic procedure for the development and subsequent validation of the Riemann surface structure (RSS) for a punctured curved surface lying on a Riemann surface is discussed. The proposed procedure differs from the existing methods involving triangular meshes and rectangular grids that rely on induced patches on surfaces. This procedure can be applied to non-punctured surfaces as well as to surfaces with irregularly located punctures. Further, by defining appropriate transition functions, the proposed procedure eliminates the requirement for smooth transitions across the boundaries of adjacent patches. The analytic formulations of the RSS for an ellipsoid and a sphere are elaborated using the proposed procedure. Moreover, the RSS of a sphere defined through a family of conformal unit discs is proven equivalent to that defined by an existing method based on stereographic projection. This study proves that a smooth projection between the surface and (a subset of) the complex plane  , can be remapped to the original surface.

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Published

2021-04-30

How to Cite

Shetty, R. D., Swamy, I. N., & Kadambi, G. R. (2021). Riemann Surface Structure for a Curved Surface with Punctured Features. Nepal Journal of Mathematical Sciences, 2(1), 7–16. https://doi.org/10.3126/njmathsci.v2i1.36504

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Articles