The Lie Group SU(2) Hoft Fibration and the Fourier Equation

Abstract

The Fourier equation explains the dynamics of heat transfer. But bringing this phenomenon closer to the notion of fibration seems difficult to achieve. This study then aims to find the solution of the one-dimensional Fourier equation and to interpret it in terms of bundle. And then apply the results obtained at the Kankule site in Katana in South Kivu. To do this work, we resorted to geometric or topological analysis of the Hopf fibration of the unit sphere S3 (identifiable in SU(2)). We had taken the temperatures of the thermal waters and the soil of Kankule, from 2010 to 2014, in situ. And laboratory analyses had allowed us to know the physical and chemical properties of the soil and water at each of our 14 study sites in Kankule. The data of the geomagnetic field of each site, were taken in on the site NOAA, for our period of study. We then determined the integral curve (geotherm) of the Fourier equation and wrote it as a unit quaternion which is a bundle. The constants intervened in the geotherm, for each site of Kankule, we had obtained them statistically. We have found that the geotherm of each Kankule site is a bundle. We have compared this model to the bundle model of the geomagnetic field. From there we realized that to determine the energy potential of Kankule, we should consider the thermal springs separately. We were able to find a connection between the fibration of the geomagnetic field and the heat field for the Kankule site.