An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves

IIT Madras Course , Prof. T.E. Venkata Balaji

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Lecture 21: Characterizing Moebius Transformations with Two Fixed Points I

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        An Introduction to Riemann Surfaces and Algebraic Curves Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit httpwww.nptel.iitm.ac.insyllabus111106044Goals To analyze Moebius transformations with more than one fixed point in the extended complex plane To continue with the classification of Moebius transformations begun in the previous lecture by defining the notions of loxodromic, elliptic and hyperbolic Moebius transformations using the values of the square of the trace of the transformation To characterize geometrically the loxodromic, elliptic and hyperbolic Moebius transformations by showing that they can be conjugated by suitable Moebius transformations to multiplication by a complex number To show that the elliptic Moebius transformations are precisely those that are conjugate to a rotation about the origin To show that the hyperbolic Moebius transformations are precisely those that are conjugate to a real scalingKeywords Parabolic, elliptic, hyperbolic and loxodromic Moebius transformations, fixed point of a Moebius transformation, square of the trace of a Moebius transformation, translation, conjugation by a Moebius transformation, special linear group, projective special linear group

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