SIX-DIMENSIONAL CONSIDERATIONS OF EINSTEIN ’ S CONNECTION FOR THE FIRST TWO CLASSES I . THE RECURRENCE RELATIONS IN 6g-UFT

Lower dimensional cases of Einstein’s connection were already investigated by many authors for n= 2,3,4,5. This paper is the first part of the following series of two papers, in which we obtain a surveyable tensorial representation of 6-dimensional Einstein’s connection in terms of the unified field tensor, with main emphasis on the derivation of powerful and useful recurrence relations which hold in 6-dimensional Einstein’s unified field theory (i.e., 6-g-UFT): I. The recurrence relations in 6-g-UFT. II. The Einstein’s connection in 6-g-UFT. All considerations in these papers are restricted to the first and second classes only, since the case of the third class, the simplest case, was already studied by many authors.

represent the general n-dimensional Einstein's connection in a surveyable tensorial form in terms of the unified field tensor g λµ .This is probably due to the complexity of the higher dimensions.
The purpose of the present paper, the first part of a series of two papers, is to derive powerful recurrence relations which hold in 6-g-UFT.In the second part, we prove a necessary and sufficient condition for the existence and uniqueness of the Einstein's connection in 6-g-UFT and establish a linear system of 43 equations for the solution of 6-dimensional Einstein's connection, employing the powerful recurrence relations obtained in Part I.
All considerations in this and subsequent papers are dealt with for the first and second classes only.

2.1.
n-dimensional g-unified field theory.The Einstein's n-dimensional unified field theory, denoted by n-g-UFT, is an n-dimensional generalization of the usual Einstein's 4-dimensional unified field theory in the space-time X 4 .It is based on the following three principles as indicated by Hlavatý [13].
Principle A. Let X n be an n-dimensional generalized Riemannian manifold referred to a real coordinate system x ν , which obeys the coordinate transformation x ν → x ν .(Throughout the present paper, Greek indices are used for the holonomic components of tensors, while Roman indices are used for the nonholonomic components of a tensor in X n .All indices take the values 1, 2,...,n, and follow the summation convention with the exception of nonholonomic indices x, y, z, t.) for which In n-g-UFT the manifold X n is endowed with a real nonsymmetric tensor g λµ , called the unified field tensor of X n .This tensor may be decomposed into its symmetric part h λµ and skew-symmetric part k λµ where We may define a unique tensor h λν = h νλ by In n-g-UFT the tensors h λµ and h λν will serve for raising and/or lowering indices of tensors in X n in the usual manner.
Principle B. The differential geometric structure on X n is imposed by the tensor g λµ by means of a connection Γ ν λµ defined by a system of equations Here D ω denotes the symbolic vector of the covariant derivative with respect to Γ ν λµ and S λµ ν is the torsion tensor of Γ ν λµ .The connection Γ ν λµ satisfying (2.4) is called the Einstein's connection.Under certain conditions the system (2.4) admits a unique solution Γ ν λµ .
Principle C. In order to obtain g λµ involved in the solution for Γ ν λµ certain conditions are imposed.These conditions may be condensed to where X λ is an arbitrary nonzero vector, and R ωµλ ν and R µλ are the curvature tensors of Γ ν λµ defined by (2.6)

Algebraic preliminaries.
In this subsection, notations, concepts, and several algebraic results in n-g-UFT are introduced.
(i) Notations.The following scalars, tensors, and notations are frequently used in our further considerations.
) (1) ) where ∇ ω is the symbolic vector of covariant derivative with respect to the Christoffel symbols ν λµ defined by h λµ .It has been shown that the scalars and tensors introduced in (2.7) satisfy Furthermore, we also use the following useful abbreviations, denoting an arbitrary tensor T ωµλ , skew-symmetric in the first two indices, by T : ) (2) of the second class with the jth category (j ≥ 1), if The solution of the system of equations (2.4) is most conveniently brought about in a nonholonomic frame of reference, which may be introduced by the projectivity (2.12) Definition 2.2.An eigenvector A ν of k λµ that satisfies (2.12) is called a basic vector in X n , and the corresponding eigenvalue M is termed a basic scalar.
It has been shown that the basic scalars M are solutions of the characteristic equation (2.13) (iii) Nonholonomic frame of reference.In the first and second classes, we have a set of n linearly independent basic vectors A i ν (i = 1,...,n) and a unique recip- With these two sets of vectors, we may construct a nonholonomic frame of reference as follows: An easy inspection shows that is the basic scalar corresponding to A x ν , then the nonholonomic components of (p) k λ ν are given by Without loss of generality we may choose the nonholonomic components of h λµ as where the index i 0 is taken so that det(h ij ) ≠ 0 when n is odd.

Differential geometric preliminaries.
In this subsection, we present several useful results involving Einstein's connection.These results are needed in our subsequent considerations for the solution of (2.4).
If the system (2.4) admits a solution Γ ν λµ , it must be of the form where

.19)
The above two relations show that our problem of determining Γ ν λµ in terms of g λµ is reduced to that of studying the tensor S λµ ν .On the other hand, it has been shown that the tensor S λµ ν satisfies where Therefore, the Einstein's connection Γ ν λµ satisfying (2.4) may be determined if the solution S λµ ν of the system (2.20) is found.The main purpose of the present paper is to find a device to solve the system (2.20) when n = 6.Furthermore, for the first two classes, the nonholonomic solution of (2.20) is given by where (2.23) Therefore, in virtue of (2.22), we see that a necessary and sufficient nonholonomic conditions for the system (2.4) to have a unique solution in the first two classes is M xyz ≠ 0 for all x, y, z. (2.24) 3. The recurrence relations of the first kind in n-g-UFT.This section is devoted to the derivation of the recurrence relations of the first kind and two other useful relations which hold in n-g-UFT.All considerations in this section are also dealt with for a general n > 1.
The recurrence relations of the first kind in n-g-UFT are those which are satisfied by the tensors (p) k λ ν .These relations will be proved in the following theorem.
Theorem 3.1 (The recurrence relations of the first kind in n-g-UFT).The tensors (p) K λ ν satisfy the following recurrence relations: For the first class.
which may be condensed to For the second class with the jth category.
which may be condensed to Proof of the case of the first class.Let M x be a basic scalar.Then, in virtue of (2.13), we have Multiplying δ i x to both sides of (3.3) and making use of (2.16), we have The relation (3.1) immediately follows by multiplying (P ) k α ν to both sides of (3.4b).
Proof of the case of the second class with the jth category.When g λµ belongs to the second class with the jth category, the characteristic equation (2.13) is reduced to x is a root of (3.5a), it satisfies In virtue of (2.16), multiplication of δ i x to both sides of (3.5b) gives The holonomic form of (3.6a) is Consequently, the relation (3.2) follows by multiplying (p−1) k α ν to both sides of (3.6b).
Remark 3.2.When g λµ belongs to the second class with the first category, the relation (3.2) is reduced to In the following two theorems we prove two useful relations.
Theorem 3.3 ( For the first and second classes).In the first two classes, a tensor T ωµν , skew-symmetric in the first two indices, satisfies Proof.Making use of (2.15b) and (2.17), the relation (3.8a) may be proved as in the following way: (3.9) The second relation can be proved similarly.
Theorem 3.4 (For all cases).The tensor B ωµν , given by (2.21), satisfies where Proof.In virtue of (2.9) and (2.20), the relation (3.10) may be shown as in the following way: ( After a lengthy calculation, we note that the right-hand side of the above equation is equal to (3.10).The relation (3.11) may be proved similarly.

4.
The recurrence relations of the second and third kinds in 6-g-UFT.This section is specially concerned with the 6-dimensional case; that is with 6-g-UFT.In this section, we first investigate the basic scalars and some relations satisfied by them.In order to obtain a tensorial representation of the 6-dimensional Einstein's connection Γ ν λµ in terms of g λµ , we need powerful recurrence relations of the third kind which are satisfied by an arbitrary tensor T ωλµ , skew-symmetric in the first two indices.Therefore, we finally derive these relations, after introducing the recurrence relations of the second kind which are satisfied by the basic scalars.All considerations in this section are restricted to n = 6.
In 6-g-UFT there are four cases; that is, the unified field tensor g λµ belongs to (1) the first class, if K 6 ≠ 0, (2) the second class with the first category, if the second class with the second category, if K 4 ≠ 0, K 6 = 0, (4) the third class, if K 2 = K 4 = K 6 = 0.In this section we investigate the first three cases.
Before we start investigations about the basic scalars, we first note that in 6-g-UFT the relation (2.8b) is reduced to and formally state in the following theorem the recurrence relations of the first kind when n = 6, which are direct consequences of (3.1) and (3.2).
The second class with the second category The second class with the first category where Proof.Since the characteristic equation (2.13) for the first class in 6-g-UFT is reduced to equation (4.3a) follows by the method of Cardano, using the notations given by (4.4a).
In this case we note that all basic scalars are not zero in virtue of (4.6d).The other cases may be shown similarly.The first class

.6d)
The second class with the second category

.7c)
The second class with the first category (x, y = b, c) Here, the indices a, b, c are assumed to take values as a = 1, 2; b = 3, 4; c = 5, 6.
For the first class .Proof.We first note that the terms in the right-hand side of (3.8a) vanish identically when x = y.Therefore, whenever we use (3.8a), it suffices to consider the terms corresponding to the cases x ≠ y only.The proof of the above relations follow from (3.8a), using (4.12) for the proof of (4.16), (4.13) for the proof of (4.17), and (4.14) for the proof of (4.18), respectively.For example, the relation (4.16b) may be proved in the following way:

2 . 1 .
The tensor g λµ (or k λµ ) is said to be (1) of the first class, if K n−σ ≠ 0;

Theorem 4 . 3 .
The basic scalars M x in 6-g-UFT satisfy the following relations:Classes and CategoryRelations between the basic scalars