ON THE ME-MANIFOLD IN n-* g-UFT AND ITS CONFORMAL CHANGE

An Einstein’s connection which takes the form (3.1) is called an ME-connection. A generalized n-dimensional Riemannian manifold X, on which the differential geometric structure is imposed by a tensor field *gx through a unique ME-connection subject to the conditions of Agreement (4.1) is called "g-ME-manifold and we denote it by "g-MEX.. The purpose of the present paper is to introduce this new concept of *g-MEX. and investigate its properties. In this paper, we first prove a necessary and sufficient condition for the unique existence of ME-connection in X,, and derive a surveyable tensorial representation of the ME-connection. In the second, we investigate the conformal change of "g-MEX,, and present a useful tensorial representation of the conformal change of the ME-connection.

Recently, Ko  ([15], 1987) and Yoo ([2].lSS)introduced a new concept of ME-manifold in n-g-UFT, assigning to X,, a ME-connection which is similar to Yano and Imai's semi-symmetric metric connection, and investigated its curvature tensors and conformal change in n-g-UFT.The purpose to introduce this manifold is similar to Chung's purpose to introduce SE-manifold.
The purpose of the present paper is to introduce a new concept of the n-dimensional *g-MEmanifold (denoted by *g-MEX,, ), assigning an Einstein's connection of the form (3.1) to Xn, called a ME-connection in what follows, and investigate its properties.This paper consists of five sections.The second section introduces some preliminary notations, definitions, and results.The third section concerns with a necessary and sufficient condition for the existence of unique ME-connection in n-*g-UFT.
The fourth section deals with a precise tensorial representation of the ME-connection in terms of .gx.In the last section, we investigate the conformal change of *g-MEX,, with particular emphasis on the conformal invariants of *g-MEXn.In this section, we display a surveyable tensorial representation of the conformal change of the ME-connection 2. PRELIMINARIES.
This section is a brief collection of the basic concepts, notations, and results, which are needed in our further considerations in the present paper.It is based on the results and symbolisms of Hlavat]( [43],1957) and Chung([10], 1963; [13], 1981; [161,1985).
A. n-DIMENSIONAL *g-UNIFIED FIELD THEORY.
Hlavat characterized Einstein's 4-dimensional unified field theory (4-g-UFT) as a set of geometrical postulates in a space-time X4 for the first time and gave its mathematical foundation.
Generalizing this theory we may consider Einstein's n-dimensional unified field theory(n-g-UFT).
Similarly, our n-dimensional *g-unified field theory(n-*g-UFT), initiated by Chung(1963) and originally suggested by Hlavat:(1957), is based on the following three principles.
Principle A. Let X,, be a n-dimensional generalized Riemannian manifold referred to a real coordinate system x", which obeys coordinate transformation x" -x '' for which Det( # 0 (2.1) Let q, be a general real nonsymmetric tensor which may be decomposed into its symmetric part hxt, and skew-symmetric part where g Det(gx,) # O, I} Det(hx,) # 0 (2. 3) The algebraic structure on Xn is imposed by the basic real tensor *gX, uniquely defined by g,, It may also be decomposed into its symmetric part *h xu and skew-symmetric part *k x gX= .hX+ ,kx (2.5) (*) Throughout the present paper,Greek indices are used for the holonomic components of tensors in X,.
Since Det(" h x')# 0, we may define a unique tensor*hat, by *h A'" (2.6} *ha '" In n-*g-UFT we use both *h A'" and *h.xu as the tensors for raising and/or lowering indices of all tensors defined in X, in the usual manner.We then have Principle B. The differential geometrical structure on X, is imposed by the tensor *ga'" by means of a connection Fa'', defined by a system of equations, so called Einstein's equations D,"gX, _2S,.,,,,,,ga,,, (,) (2.8) Here D denotes the symbol of the covariant derivative with respect to FaV, and SA,'" is the torsion tensor of Fa'',.The connection Fav, satisfying (2.8) is called an Einstein's connection.
In the following Remark, we state the main differences between n-g-UFT and n-*g-UFT.n-g-UFT REMARK 2.1.In n-*g-UFT the algebraic structure is imposed on X, by the tensor {gx, { the tensor h x and gX and hX are used for raising and/or lowering the indices of the tensor and h n-g-UFT tensors in X, On the other hand, in the differential geometrical structure is imposed n-*g-UFT (*) It has been shown by Hiavat:#(1957) that the system (2.8) is equivalent to Dgx, 2Sagxa (2.8)' which is the original Einstein's equation.gxu { on X. by .gx,,through Fx", satisfying ( 2 .8 ) ( ' ) ' 8 ) ' gx,, in n-g-UFT it will be expressed in terms of .gX in n-*g-UFT admits a solution l"x",, B. SOME NOTATIONS AND RESULTS.
The following theorems have been proved already in a X, ([101, 1963; [13], 08).Here and in what follows, the index s is assumed to take the values 0,2,4,-in the specified range.THEOREM 2.2.The basic scalars M satisfy the following equation: where { x", are the Christoffel symbols defined by *hx, and (2.17) 3. THE ME-CONNECTION IN n-*g-UFT In the section, we introduce the concept of ME-connection in n-*g-UFT and devote mainly to the proof of a necessary and suffcient condition for the existence of ME-connection in a general Xn.
DEFINITION 3.1.An Einstein's connection I'x'v of the form (3.1) for a non-null vector Xx is called a ME-connection in n-*g-UFT, and Xx the corresponding ME-vector.
Besides the ME-connection F, given by (3.1) and (3.14), assume that there exists another ME-connection F"t, {x''t,} + 2,''X#-2*gxt,X,, Xx Xx (3.17) Applying the same method to derive (3.14) to this connection, we have X, C B X , which is a contradiction to our assumption (3.17).This proves the uniqueness of the ME- connection under the condition (3.13). 4. *g-ME-MANIFOLD AND A REPRESENTATION OF ITS CONNECTION.
In this section, we introduce the concei)t of *g-ME-manifold and display a surveyable ten- sorial representation of a unique ME-connection FxVu in terms of the tensor field "9 xv using two useful recurrence relations.
AGREEMENT4.1.In our further considerations in the present paper, we impose priori the following conditions to the tensor field (a) The quantity Odd 1-n( 0) (4.1) l+n is not a basic scalar of Xn(see (2.13)).
(b) The condition (3.13) is satisfied by the tensor field ,gX According to the condition (b), we note that there always exists a unique ME-connection Fx, in our n-*.g-UFT.In virtue of (3.1) and (3.14), this connection may be given by (4.2) An n-dimensional generalized Riemannian manifold X,, on which the differential geometric structure is imposed by the tensor *gX satisfying the conditions of Agreement (4.1) by means of the unique ME-connection given by (4.2), is called an n-dimensional *g-ME-manifold and denoted by *g-MEX,.
In our further considerations in the present paper, we use the following useful abbreviations for an arbitrary vector Vx, where p=2,3,4,...
Hence, in *g-MEX.the following relation always holds in virtue of (2.15) and (4.6): H,,_ 0 (4.17) We note that the relation (4.17) justifies the validity of the representation (4.10).Furthermore, we also note that the condition (4.17) is a necessary and sufficient condition for X, to admit a unique ME-vector Xa in n-*g-UFT.This is the reason why we imposed prior the condition of Agreement (4.1)(a).Now that we have obtained a representation of the ME-vector Xa in terms of *gau, it is possible for us to obtain a surveyable representation of the ME-connection of *g-MEX, in terms of *ga by simply substituting (4.10) into (3.1).Formally we state THEOREM 4.7.The ME-connection of *g-MEX.may be given by (4.18)   where the vectors CA and (')Qa are respectively defined by (3.15) and (4.12), the quantities H,,_ and by (4.4) and (4.11) respectively, and 5. CONFORMAL CHANGE OF *g-MEX,.
In this section, we investigate change of several geometrical quantities, particular emphasis on the change of the unique ME-vector and ME-connection, induced by a conformal change of the unified field tensor *g-MEX.
Let *g-MEX, be n-dimensional *g-ME-manifold, on which differential geometric structure is imposed by a unified field tensor field ,0a through the ME-connection Fa, given by (4.18) F,, given by (5.1) r. =* {. + a. ._c+ .("-')0 (5.1) (See Agreement (5.2) for , G a, ',, Ca, and 0a DEFINITION 5.1.Two manifolds *g-MEX, and *g MEX, are said to be conformal, if their basic tensor fields are related by *y"() -" *"()(*) (5.2) where Q fl(.r) is an arbitrary Mnction of position with at least two derivatives.This conformal change enforces a change of ME-vector and ME-connection, and an explicit tensorial representation of this change will be displayed in this section.AGREEMENT 5.2.Throughout this section, we agree that, if T is a fl,nction of *gxt,, then we denote by T the same function of *Ox,.In particular, if T is a tensor, so is T Furthermore, the indices of T( will be raised and/or lowered by means of *h xv (.xv) and/or *hxt, (*x,) The following two theorems are immediate consequences of Definition(5.1)and Agreement( 5.2 ).   the relation (5.5)a immediately follows in virtue of (5.6).Similarly, the change (5.5)b may be shown by substituting (5.3)b and (5.5)a into Making use of (3.15) and (5.3)a, the change (5.5)c for p 1 may be obtained from (5.5)b as in the following way: ,_----,- n-2 t2) x (5.7) Cx= h V, kx,,=Cx+