RIEMANN SURFACES
ASHAY DHARWADKER DISTINGUISHED PROFESSOR OF
INSTITUTE OF MATHEMATICS


Riemann surfaces were first studied by Bernhard Riemann in his Inauguraldissertation
at Göttingen in 1851. We explicitly show the construction of the surfaces w = z^{n}. 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. Google Scholar Citations © 2002



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.


Consider the function from the complex plane to itself given by
w = f(z) = z^{n}, where n is at least 2. The zplane
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 onetoone correspondence between each sector and the whole
wplane, 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
zplane,
take n identical copies of the wplane 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 selfintersection but the idealized model shall be free of this
discrepancy. The result of the construction is a Riemann surface
whose points are in onetoone correspondence with the points of the zplane.
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.

Keywords: Riemann Surface; Algebraic Function
MSC2000 Classification: 30F99 Zentralblatt No: 05264891 
REFERENCES 

Copyright © 2002 by Ashay Dharwadker. All rights reserved.
