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

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 S (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.

( ) , q q a a ′ ′ + = + + u v (1) with + u v the vector addition in where ⋅ u v and ∧ u v are respectively the scalar product and the vector product in 3 .
 We verify that ( ) 4 , , + *  is a commutative body, called the quaternion body. We mark it H, in honor of Hamilton who built it first.
When θ is a multiple of π, we immediately have the result stated by the theorem, whatever the pure quaternion considered.

The S 2 and S 3 Spheres A) Definitions
The unit sphere n S , of dimension n, of , with , , , and (  )   2   : , with , , , We find ( ) ( ) This method, of stereographic projection, makes it possible to obtain a differentiable structure for any sphere S n . For n = 3, the stereographic projections give for S 3 the atlas and ( ) , , , , B) Link between H and SU (2) 1) Definitions .
Let us use relations (2), (9), (21) and (22) to determine the angle of rotation q R and his axis.
According to (9), we have ( ) cos sin cos sin q q p p Which allows to deduce: As p is invariant by q R , then p is a vector of the axis of rotation of q R b) If p is perpendicular to m, then 0 p m ⋅ =. It then comes that By comparing this relation to (9), we conclude that q R is a rotation of angle 2θ, and of axis p.
Theorem 5 is therefore proven.
We therefore observe that a quaternion q is also a rotation of angle θ, and of axis p.
2) Matrix of a rotation R q of an element q of SU(2) or S 3 Consider the following matrices in SU (2) (2), and properties of the Pauli matrices, we find With u unit vector of 4  along the axis p of the relation (29). We also have in 3  the following unit vectors ( ) with i imaginary unity. The relation (31) indicates to us that a quaternion is a rotation, in the space of 1/2-spinner. It is a rotation in space of rank 1 spinners.

Euler Angles of a Quaternion
Any matrix q of SU(2) or S 3 can be decomposed, using Euler angles, as below [2] [4] [6]: The approach formulated by relations (31) and (32) is used in Quantum Mechanics, in the theory of angular momentum.

Prelude
The total space P = SU(2) from the topological point of view is identifiable with S 3 . Consider a group U(1), and perform a class decomposition of SU(2) with respect to U(1). For Let . We consider the set . Then we consider an element k of SU(2), which is neither in U(1), nor in g . We build the set We continue the process of class construction in SU(2)\U(1). In order to account, we succeed in writing SU(2) as an infinite union of classes of the type ( ) is considered to be an infinite union of circles [Courses and Seminars (1)] ( Figure 1).
The structural group U(1) acts by multiplication on the right, on the total space SU(2). We have chosen U(1) as a distinguished subgroup of SU(2). This fibration in circles of ( ) is called Hopf fibration of ( )

Hopf Fiber of a Lie G Group 1) Algebraic formulation
Let G be a Lie group, and H a Lie subgroup of G. We assume that H is closed in G, so that the topology of the quotient group G/H is separated. We consider the equivalence relation R defining the classes to the left of H. Thus two elements g and k of G are in relation R, if and only if k g g H ∈ = * . Consider the application h; defined as below [2] [13]: The application h defines a main fibration. Any group G is thus a principal fibered space above G/H. The structural group being H. The quotient set G/H, which is the basis of the bundle, is only a group if H is distinguished in G.
We use the notation H → G → G/H, to characterize a fibration h of G above G/H. In the previous paragraph, we then introduced this fibration U(1) → SU(2) → SU(2)/U(1) (Figure 3).

( )
: is a differential variety Consider ; ; where : is a differential variety the projection of onto We say that P is a vector bundled space of base M, of projection π and of fi- ; ∃ a neighborhood U of m and a diffeomorphism Consider (P; M; π; F; G), where P, M and F are differential varieties. π: the projection of P on M, a differential manifold. G: a diffeomorphism group from F to F. We will say that P is a fibered space of base M, of projection π, of fiber type F and of structural group G if: ; ∃ an open neighborhood U of m in M and a diffeomorphism ( )   . Let us look for q such that We then find ( ) ( ) The relation (37), allows us to affirm that the relation (39) is given by quaternions q describing a circle of 3 S . Indeed; let 1 q and 2 q in 3 S , such that ( ) ( ) ( )

Local Trivialization of Fibration h
The local trivialization of the h fibration is obtained from relations (39) allows us to obtain local trivialization (U; ψ).

Stereographic Projection Visualizing the Fibration of HOPF from the Lie SU(2) Group
Stereographic projection is a method of representing a sphere, deprived of a which is the x-axis of the plane (x, y) or (x; z) of ( )

3
; ; x y z =  ; that is to say a circle of infinite radius. 2) which is a circle of the plane (y; z) of ( ) where a is a constant characteristic of each physical situation and ξ is a quantity corresponding to the particular phenomenon of transport studied like the diffusion of the molecules, the viscosity and the conduction of heat. In this work, we will focus on the last case of these types of transport phenomena whose mathematical solutions relate to a temperature distribution. The phenomenon of energy and/or material transport is of great importance in statistical physics. This phenomenon for heat is described by the Fourier equation which we will solve for a spatial dimension before modeling this solution for the Kankule site. The solution of this equation is called the geotherm 2) One-dimensional Fourier equation The Fourier equation for the conduction of heat in the ground, when this energy varies as a function of time t and depth z is:  Introduce the values of (50) in (49), we find: As a result ( ) d: is called the depth of penetration of the waves (or also the depth of damping). Starting from (49), according to Wu and Nofziger [22], taking into account the boundary conditions, the previous solution can be written: Using solution of (49), we looked for the gradient, for each site, with the relation ( ) T z f z ∂ ∂ = and deduce the depth of exploration to the Kankule site studies 3) Exploitation of geothermy The Geoscience review of March 16, 2013, shows the operating conditions of geothermal energy to produce electricity. Having active tectonic or volcanic zones in which a surface heat source makes it possible to have a geothermal fluid at a temperature sufficient to produce electricity [23]. On page 14, Figure 8 [23], already shows that with a temperature of 50˚C at 1 km deep, we can consider the production of electricity. It will then be necessary to dig for this to more than 3 km deep, in order to trap the geothermal fluid at more than 150˚C. The same article presents a world map where high energy geothermal energy is likely to be exploited. Eastern DR. Congo is there, as in all the countries of the Albertine Rift (or Valley). From the ground temperature down to 90˚C, electricity can already be produced [24].
The latest study distinguishes between surface and deep geothermal energy as follows: between 25˚C -30˚C and 150˚C and >500 m: deep geothermal energy. This takes over the low and medium temperatures (enthalpy).
Still the same study, specifies that if the geothermal reservoir reaches a temperature above 150˚C and is encountered at shallow depths, the resource must therefore be in a region where the geothermal gradient is normal (~30˚C/km) or beyond normal. This is for the production of electricity. This classification is consistent with that suggested by Lindal for France [25].
The exploitation of high energy geothermal energy is favorable in areas which knows geodynamic contexts, showing some form of volcanism. At these points, a significant part of the internal energy of the globe is released. The history of mountain chain formations and terrestrial magmatism allows us to understand the genesis of magma chambers. The latter are foci, zones with strong thermal gradient [26].

Hopf Fiber of the Heat Transfer Phenomenon
Relations (50) or (56) determine solutions of Equation (49).
cos ;sin ;0;0 ; with 0; 2 , By comparing to relation (38), we observe that q is a fiber above the north pole. Our q being a solution of (50) or (56); we conclude that each solution of these equations is associated with a fiber in For the three-dimensional Fourier equation  This model is analogous to that obtained by T.M. and Dirk Bouwmeester [27], for the modeling of lights (electromagnetic waves of the visible spectrum) with Hopf fibration. Indeed, each fiber appears as a line of Earth's magnetic field.
And since two distinct lines of fields don't join, so do two fibers in circles that don't cut. And since two distinct lines of fields don't join, so do two fibers in circles that don't cut. As a result each geomagnetic field line is representative of a heat fiber. By exploiting Maxwell's equations of the Terrestrial geomagnetic field, we find that which is a solution of the same type as (50).

Methodology
To do this work, we used the following methodological approach:

Analytic
To do this we sought: Let us relate it between Quaternions units and the Lie group SU(2); determining the coefficients of the geotherm; The links between the Geotherm of the one-dimensional Fourier equation with the Hopf fibration of the sphere 3 S ; See how to visualize Hopf fibration using stereographic projection; The significance in terms of bundle of the geotherm.

Comparative
To do this, we compared our results from the ground geotherm with various accepted results, in order to see if each thermal source can be compared to a bundle.
a function that varies with depth z.   Table 3. Thermal water electrolysis data and associated thermal field.  Let's determine the maximum values of this function, using differential calculus. It comes at a given depth Journal of Applied Mathematics and Physics This is also the result of the work of William T.M. and Dirk Bouwmeester [27].
And reciprocally, the linear links between B and T, allows us to corroborate the notion of fibration of the field T, for the site of Kankule I.
For the other Kankule sites, we had obtained similar results. Even if the null hypothesis is rejected, as in Kankule IV. There is always a certain linearity, as weak as it is, between the magnetic field of the site and the geomagnetic field.
This implies that there is an interaction between the magnetic field of the site and the soil of the corresponding site, however small. Indeed, the ionization of thermal waters observed with the results of Table 3, allows the creation of the electromagnetic field in place [16]. And this field in thermal waters interacts with the geomagnetic field of the place. In view of the above, we can say that the geomagnetic field allows us to detect the state of electrification at any Kankule geothermal site. There is therefore parallelism, in terms of interaction, between the B and T fields of a given Kankule site.

Which Geothermy Could Be Exploited in Kankule?
The average Kankule thermal gradient of 0.117˚C/m differs from the global average thermal gradient of 0.033˚C/m. This Kankule gradient is 3.5 times greater than the global average thermal gradient. With this average Kankule thermal gradient, we should be able to exploit different types of geothermal energy as indicated [24]. The different types of geothermal energy that can be used in Kankule are:  <30˚C and <500 m: surface geothermal energy. This takes up the very low temperatures (enthalpy). Because at 257 m deep, we will have temperatures of at most 30˚C.  between 25˚C -30˚C and 150˚C and >500 m: deep geothermal energy. Because at a depth of 500 m, we will already have a temperature of 58.5˚C. In order to generate electricity, it would have to be over 100˚C. By digging up to 1500 m, or 1.5 km, deep, we will have temperatures reaching 175.5˚C. Going up to 2000 m deep, we will have a temperature of 234˚C. The production of electricity is then possible from 50˚C as already pointed out [23].  For Kankule IV, with a gradient of 125, 471, 195˚C/km. You can reach desired temperatures by drawing water more than 2 km deep, reaching temper-

Conclusions
In the present work, we have found that: Each thermal water site is represented by the temperatures of the soil or of its waters; Each site is a heat beam taken as a fibrous equivalent to a line of the magnetic field of the place; There is interaction between the bundle of the site's geomagnetic field and the bundle of heat.