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Early Investigations in Conformal and Differential Geometry
  • Raymond T Walter, University of Arkansas, Fayetteville
The present [undergraduate] honors thesis seeks to introduce fundamental notions of conformal and differ- ential geometry, especially when such notions are useful in various mathematical physics applications. Its primary achievement is a nontraditional proof of the classic result of Liouville that the only conformal transformations in Euclidean space of dimension greater than three are Möbius transformations. The proof is nontraditional in the sense that it is based on a representation of Möbius transformations using 2x2 matrices of Clifford numbers as well as the standard Dirac operator on Euclidean space. Clifford algebras and the Dirac operator are important in other important areas of pure mathematics and mathematical physics, e.g., the Atiyah-Singer Index theorem and the Dirac equation in relativistic quantum mechanics. Therefore, after a brief introduction, we first develop an intuitive idea of a Clifford algebra. The Clifford group, or Lipschitz group, is introduced and related to representations of orthogonal transformations composed with dilations. This exhausts Chapter 2. Chapter 3 develops various important notions of differen- tial geometry. Chapter 4 reiterates some points of differential geometry, introduces the Ahlfors-Vahlen representation of Mo ̈bius transformations (using 2x2 matrices of Clifford numbers), explains what are conformal mappings, and finally proves our main result. 
  • Clifford analysis,
  • Möbius transformation,
  • Conformal Geometry,
  • Differential Geometry
Publication Date
Spring May, 2013
Mathematical Sciences
John Ryan
Citation Information
Raymond T Walter. "Early Investigations in Conformal and Differential Geometry" (2013)
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Creative Commons License
This work is licensed under a Creative Commons CC_BY International License.