The present article introduces fundamental notions of conformal and differential geometry, especially where such notions are useful in 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 two are Möbius transformations. The proof is nontraditional in the sense that it uses the standard Dirac operator on Euclidean space and is based on a representation of Möbius transformations using 2x2 matrices over a Clifford algebra. Clifford algebras and the Dirac operator are important in other applications of pure mathematics and mathematical physics, such as the Atiyah-Singer Index Theorem and the Dirac equation in relativistic quantum mechanics. Therefore, after a brief introduction, the intuitive idea of a Clifford algebra is developed. The Clifford group, or Lipschitz group, is introduced and related to representations of orthogonal transformations composed with dilations; this exhausts Section 2. Differentiation and differentiable manifolds are discussed in Section 3. In Section 4 some points of differential geometry are reiterated, the Ahlfors-Vahlen representation of Möbius transformations using 2x2 matrices over a Clifford algebra is introduced, conformal mappings are explained, and the main result is proved.
Walter, R. T. (2014). Early Investigations in Conformal and Differential Geometry. Inquiry: The University of Arkansas Undergraduate Research Journal, 16(1). Retrieved from https://scholarworks.uark.edu/inquiry/vol16/iss1/6