Geometrical wave equation and the cauchy-like theorem for octonions

• Main Author: Borges Neto, Manoel Ferreira [UNESP]
• Publication Date: 2012
• Other Authors: Marão, José Antônio Pires Ferreira
• Format: Article
• Language: eng
• Source: Repositório Institucional da UNESP
• Download full: http://www.ijpam.eu/contents/2012-79-3/6/6.pdf; http://hdl.handle.net/11449/73658
• topic: Cauchy integral; Hypercomplex; Quaternions

Riemann surfaces, cohomology and homology groups, Cartan's spinors and triality, octonionic projective geometry, are all well supported by Complex Structures [1], [2], [3], [4]. Furthermore, in Theoretical Physics, mainly in General Relativity, Supersymmetry and Particle Physics, Complex Theory Plays a Key Role [5], [6], [7], [8]. In this context it is expected that generalizations of concepts and main results from the Classical Complex Theory, like conformal and quasiconformal mappings [9], [10] in both quaternionic and octonionic algebra, may be useful for other fields of research, as for graphical computing enviromment [11]. In this Note, following recent works by the autors [12], [13], the Cauchy Theorem will be extended for Octonions in an analogous way that it has recentely been made for quaternions [14]. Finally, will be given an octonionic treatment of the wave equation, which means a wave produced by a hyper-string with initial conditions similar to the one-dimensional case.