TITLE:
The Lie Group SU(2) Hopf Fibration and the Fourier Equation
AUTHORS:
Adelin Mulenda Mbuto, Lucien Zihindula Biguru, Jean Masudi Kalongama, Joseph Cimbela Kabongo, Albert Kabasele Yenga-Yenga
KEYWORDS:
Fiber, Hopf Fiber, Geotherm, Quaternion, Geomagnetic Field and Potential Energy
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.8 No.7,
July
25,
2020
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.