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RIEMANN SURFACES


ASHAY DHARWADKER

DISTINGUISHED PROFESSOR OF
MATHEMATICS & NATURAL SCIENCES

ENDOWED CHAIR

INSTITUTE OF MATHEMATICS
H-501 PALAM VIHAR
DISTRICT  GURGAON
HARYANA  1 2 2 0 1 7
INDIA

ashay@dharwadker.org


ABSTRACT
Riemann surfaces were first studied by Bernhard Riemann in his Inauguraldissertation at Göttingen in 1851. We explicitly show the construction of the surfaces w = zn. This construction played a major role in the proof of the Four Colour Theorem in 2000 and the Grand Unification of the Standard Model with Quantum Gravity in 2008.
ACKNOWLEDGEMENTS
Thanks to Michael Joswig and Konrad Polthier, Managing Editors of Electronic Geometry Models for their help in setting up the Java applets to display the Riemann surface construction. The Endowed Chair of the Institute of Mathematics was bestowed upon Distinguished Professor Ashay Dharwadker in 2012 to honour his fundamental contributions to Mathematics and Natural Sciences.
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Consider the function from the complex plane to itself given by w = f(z) = zn, where n is at least 2. The z-plane may be divided into n sectors given by arg z lying between (k - 1)(2π/n) and k(2π/n) for k = 1, ..., n. There is a one-to-one correspondence between each sector and the whole w-plane, except for the positive real axis. The image of each sector is obtained by performing a cut along the positive real axis; this cut has an upper and a lower edge. Corresponding to the n sectors in the z-plane, take n identical copies of the w-plane with the cut. These will be the sheets of the Riemann surface and are distinguished by a label k which serves to identify the corresponding sector. For k = 1, ..., n -1 attach the lower edge of the sheet labeled k with the upper edge of the sheet labeled k + 1. To complete the cycle, attach the lower edge of the sheet labeled n to the upper edge of the sheet labeled 1. In a physical sense, this is not possible without self-intersection but the idealized model shall be free of this discrepancy. The result of the construction is a Riemann surface whose points are in one-to-one correspondence with the points of the z-plane. 
 
 
Figure 1. Riemann surface of two sheets

This correspondence is continuous in the following sense. When z moves in its plane the corresponding point w is free to move on the Riemann surface. The point w = 0 connects all the sheets and is called the branch point. A curve must wind n times around the branch point before it closes.

Figure 2. Riemann surface of three sheets
Now consider the n - valued relation z = nth root(w). To each nonzero w, there correspond n values of z. If the w-plane is replaced by the Riemann surface just constructed, then each complex nonzero w is represented by n points of the Riemann surface at superposed positions. Let the point on the uppermost sheet represent the principal value and the other n - 1 points represent the other values. Then z = nth root(w) becomes a single-valued, continuous, one-to-one correspondence of the points of the Riemann surface with the points of the z-plane. The Riemann surface is orientable, since every orientation of a sheet is carried over to the sheet next to it.

 
Keywords:  Riemann Surface; Algebraic Function
MSC-2000 Classification:  30F99
Zentralblatt No:  05264891

References

  • B. Riemann: Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, Inauguraldissertation, Göttingen (1851), http://www.emis.de/classics/Riemann/.
  • D. Hilbert and S. Cohn-Vossen: Anschauliche Geometrie, English Translation by Chelsea Publishing Company (1932).


ISBN 1466265302

Copyright © 2002 by Ashay Dharwadker. All rights reserved.