CONFORMAL CHANGE OF THE TENSOR U ν λµ FOR THE SECOND CATEGORY IN 6-DIMENSIONAL g-uft. Chung Hyun Cho

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1 Kangweon-Kyungki Math. Jour. 5 (1997), No. 2, pp CONFORMAL CHANGE OF THE TENSOR U ν λµ FOR THE SECOND CATEGORY IN 6-DIMENSIONAL g-uft Chung Hyun Cho Abstract. We investigate chnage of the tensor U ν λµ induced by the conformal change in 6-dimensional g-unified field theory. These topics will be studied for the second class with the second category in 6-dimensional case. 1. Introduction The conformal change in a generalied 4-dimensional Riemannian space connected by an Einstein s connection was primarily studied by HLAVATÝ([9], 1957), CHUNG ([7], 1968) also investigated the same topic in 4-dimensional g-unified field theory. The Einstein s connection induced by the conformal change for all classes in 3-dimensional case and for the second and third classes in 5-dimensional case and for the first class in 5-dimensional case and for the second class with the first category in 6-dimensional case were investigated by CHO([1],1992, [2],1994, [3],1995, [4],1996). In the present paper, we investigate change of the tensor U ν λµ induced by the conformal change in 6-dimensional g-unified field theory. These topics will be studied for the second class with the second category in 6-dimensional case. 2. Preliminaries This chapter is a brief collection of basic concepts, notations, theorems, and results needed in our further considerations. They may Received June 23, Mathematics Subject Classification: 53A45, 83E99. Key words and phrases: Einstein connection, conformal change. Supported by the Institute for Basic Science of Inha University, 1996.

2 142 Chung Hyun Cho be refferd to CHUNG([5],1982; [3],1988), CHO([1],1992; [2],1994; [3],1995; [4],1996) n-dimensional g-unified field theory The n-dimensional g-unified field theory (n-g-uft hereafter) was originally suggested by HLAVATÝ([9],1957) and systematically introduced by CHUNG([8],1963). Let X 1 n be an n-dimensional generalized Riemannian manifold, referred to a real coordinate system x ν obeying coordinate transformations x ν x ν, for which (( )) x (2.1) Det x 0. In the usual Einstein s n-dimensional unified field theory, the manifold X n is endowed with a general real nonsymmetric tensor g λµ which may be split into its symmetric part h λµ and skew-symmetric part k λµ 2 : (2.2) g λµ = h λµ + k λµ (2.3) Det((g λµ )) 0, Det((h λµ )) 0. Therefore we may define a unique tensor h λν = h νλ by (2.4) h λµ h λν = δ ν µ. In our n-g-uft, the tensors h λµ and h λν will serve for raising and/or lowering indices of the tensors in X n in the usual manner. The manifold X n is connected by a general real connection Γ ν ωµ with the following transformation rule : (2.5) Γ ν ω µ = xν x α ( x β x ω xγ x µ Γ α βγ + 2 x α x ω x µ 1 Throughout the present paper, we assumed that n 2. 2 Throughout this paper, Greek indices are used for holonomic components of tensors. In X n all indices take the values 1,, n and follow the summation convertion. )

3 Conformal change of the tensor U ν λµ for the second category 143 and satisfies the system of Einstein s equations (2.6) D w g λµ = 2S wµ α g λα D w denotes the covariant derivative with respect to Γ ν λµ and (2.7) S λµ ν = Γ ν [λµ] is the torsion tensor of Γ ν λµ. The connection Γν λµ satisfying (2.6) is called the Einstein s connection. In our further considerations, the following scalars, tensors, abbreviations, and notations for p = 0, 1, 2, are frequently used : (2.8)a g = Det((g λµ )) 0, h = Det((h λµ )) 0, k = Det((k λµ )), (2.8)b g = g h, k = k h, (2.8)c K p = k [α1 α1 k αp ] α p, (p = 0, 1, 2, ) (2.8)d (0) k λ ν = δ ν λ, (1) k λ ν = k λ ν, (p) k λ ν = (p 1) k λ α k α ν, (2.8)e K ωµν = ν k ωµ + ω k νµ + µ k ων, (2.8)f σ = { 1 if n is odd 0 if n is even. ω is the symbolic vector of the convariant derivative with respect to the Christoffel symbols { ν λµ } defined by h λµ. The scalars and vectors introduced in (2.8) satisfy (2.9)a K 0 = 1; K n = k if n is even; K p = 0 if p is odd,

4 144 Chung Hyun Cho (2.9)b g = 1 + K K n σ, (2.9)c (p) k λµ = ( 1) p(p) k µλ, (p) k λν = ( 1) p(p) k νλ. Furthermore, we also use the following useful abbreviations, denoting an arbitrary tensor T ωµν, skew-symmetric in the first two indices, by T : (2.10)a pqr T = pqr T ωµν = T αβγ (p) k ω α(q) k µ β(r) k ν γ, (2.10)b T = T ωµν = 000 T, (2.10)c 2 pqr T ω[λµ] = pqr T ωλµ pqr T ωµλ, (2.10)d We then have (2.11) 2 (pq)r T ωλµ = pqr T ωλµ + qpr T ωλµ. pqr T ωλµ = qpr T λωµ. If the system (2.6) admits Γ ν λµ, using the above abbreviations it was shown that the connection is of the form (2.12) Γ ν ωµ = { ν ωµ} + S ωµ ν + U ν ωµ (2.13) U νωµ = 100 S (ωµ)ν. The above two relations show that our problem of determining Γ ν ωµ in terms of g λµ is reduced to that of studying the tensor S ωµ ν. On the other hand, it has also been shown that the tensor S ωµ ν satisfies (2.14) S = B 3 (110) S (2.15) 2B ωµν = K ωµν + 3K α[µβ k ω] α k ν β.

5 Conformal change of the tensor U ν λµ for the second category Some results in 6-g-UFT In this section, we introduce some results 6-g-UFT without proof, which are needed in our subsequent considerations. They may be referred to CHO([5],1993). Definition 2.1. In 6-g-UFT, the tensor g λµ (k λµ ) is said to be the second class with the second categoty, if K 4 0, K 6 = 0. Theorem 2.2. (Main recurrence relations) For the second class with the second category in 6-UFT, the following recurrence relation hold (2.16) (p+4) k λ ν = K 2 (p+2) k λ ν K 4 (p) k λ ν, (p = 0, 1, 2, ) Theorem 2.3. (For the second class with the second category in 6-g-UFT). A necessary and sufficient condition for the existence of the solution of (2.5) is (2.17) (1 + K 2 + K 4 )[(1 K 2 + 5K 4 ) 2 4K 4 (2 K 2 ) 2 ] Conformal change of the 6-dimensional tensor U ν λµ for the second class with the second category. In this final chapter we investigate the change U ν λµ U ν λµ of the tensor induced by the conformal change of the tensor g λµ, using the recurrence relations and theorems introduced in the preceding chapter. We say that X n and X n are conformal if and only if (3.1) g λµ (x) = e Ω g λµ (x) Ω = Ω(x) is an at least twice differentiable function. This conformal change enforces a change of the tensor U ν λµ. An explicit representation of the change of 6-dimensional tensor U ν λµ for the second class with the second category will be exhibited in this chapter.

6 146 Chung Hyun Cho Agreement 3.1. Throughout this section, we agree that, if T is a function of g λµ, then we denote T the same function of g λµ. In particular, if T is a tensor, so is T. Furthermore, the indices of T (T ) will be raised and/or lowered by means of h λν (h λν ) and/or h λµ (h λµ ). The results in the following theorems needed in our further considerations. They may be referred to CHO([1],1992, [2],1994, [3],1995, [4],1996). Theorem 3.2. In n-g-uft, the conformal change (3.1) induces the following changes : (3.2)a (p) k λµ = e Ω(p) k λµ, (p) k λν = e Ω(p) k λν (p) k λ ν = (p) k λ ν, (3.2)b g = g, K p = K p, (p = 1, 2, ). Now, we are ready to derive representations of the changes U ν ωµ U ν ωµ in 6-g-UFT for the second class with the second category induced by the conformal change (3.1). Theorem 3.3. The change S ν ωµ S ν ωµ induced by conformal change (3.1) may be represented by S ωµ ν =S ωµ ν + 1 C [a 1k ωµ Ω ν + a 2 k ν [ωω µ] + a 3 h ν [ωk µ] δ Ω δ + a 4 δ ν [ωk µ] (3.3) + a 5 k ν [ω (2) k µ] δ Ω δ + a 6 (2) k ν [ωk µ] δ Ω δ + a 7 k ωµ (2) k νδ Ω δ + a 8 (3) k ν [ωω µ] + a 9 (3) k ν [ωω µ] + a 10 δ ν [ω (3) k µ] δ Ω δ + 2a 11 (3) k ν [ω (2) k µ] δ Ω δ + 2a 12 (2) k ν [ω (3) k µ] δ Ω δ + a 13 (3) k ωµ (2) k νδ Ω δ],

7 Conformal change of the tensor U ν λµ for the second category 147 a 1 = α 2 β(1 + 4β) 2αβ(1 + β + 2β 2 ) + β(1 13β 2 ) C, a 2 = 2α 3 β α 2 β(1 2β) + 2αβ 2 (1 2β) + β 2 (3β 4) + C, a 3 = β 2 (2α 2 5α 9β + 7) C, a 4 = 2α 3 β + α 2 β(1 + 12β) 9αβ 2 β(3 + 5β + 18β 2 ), a 5 = 2α 4 α 3 (2β + 3) α 2 (1 + 9β + 4β 2 ) + α(2 10β β 2 + 8β 3 ) + β(6 + 13β + 19β 2 ), a 6 = 2α 4 + α 3 (1 + 18β) + 2α 2 β(1 8β) α(2 + 16β + 59β 2 + 8β 3 ) + β(27β 2 58β 10) 1 + 2C, a 7 = α 2 β(1 + 4β) + 2αβ(1 + β) + β(13β 2 + 4αβ 2 1) + C, a 8 = 3α 3 + α 2 (5β + 8β 2 4) α(2 + 36β + 5β 2 ) + 7β(2 6β) 3β 2 ) + 3, a 9 = α 2 (1 8β) 2α(1 6β 2 ) + β(8β β 12) + 1, a 10 = 2α 2 β( 5 + 2β) + 2αβ(3 6β + 4β 2 ) + 4β(1 + 2β 2β 2 ), a 11 = 2α 4 α 3 (1 + 3β) 4α 2 β 2 + α(1 + 7β + 4β 2 ) β(3 7α 4αβ) 2, a 12 = 2α 4 + α 3 (2β 15) + α 2 (22 19β + 4β 2 ) + α( β 6β 2 ) 3β + 1, a 13 = 4α 4 α 3 (1 8β) + 11α 2 β α(8 16β + 21β 2 ) + β(5β 2 + 2β 10) 3, α = K 2, β = K 4, (3.4) C = (1 + α + β)[(1 α + 5β) 2 4β(2 α) 2 ]. Theorem 3.4. The change U ν ωµ U ν ωµ induced by the con-

8 148 Chung Hyun Cho formal change (3.1) may be represented by (3.5) U ν ωµ =U ν ωµ + 1 C [b 1δ ν (ωω µ) + b 2 (2) k ν (ωω µ) + (b 3 δ ν (ω (2) k µ) δ + b 4 k ν (ωk µ) δ + b 5 (2) k ν (ωk µ) δ + b 6 k ν (ω (3) k µ) δ + b 7 (2) k ν (ω (2) k µ) δ + b 8 (3) k ν (ωk µ) δ + b 9 (2) k ν (ω (3) k µ) δ + b 10 (3) k ν (ω (3) k µ) δ )Ω δ ], b 1 = β[α 2 (8β 1) 2α(6β 2 1) β(8β β 12) 1], b 2 = α[α 2 (11β 1) α(10β 2 + β 2) 12β 3 33β β 1] + β 3 (3β 4) + C, b 3 = αβ[ 2α 3 + α 2 (1 + 3β) + 4αβ β 8β 2 ] + 2β, b 4 = β[ 2α 2 (α + β + 3) 2αβ(2β + 3) 29β 2 4β 2] 3C, b 5 = β[ 2α 3 + α 2 (12β + 1) 9αβ 18β 2 5β 3], b 6 = β[ 4α 2 (6β 4) + 2α(14β 1) + 34β 2 8β 6] + 2C, b 7 = α[ 2α 4 + 3α 3 (β + 1) + 4α 2 β 2 α(12β β + 2) + 8β 3 β 2 7β + 4] + β(19β β + 6), b 8 = α[2α 3 18α 2 β + 5α αβ 2 + 8αβ 8α + 8β β 2 56β 2] β(27β β β 38) + 1 2C, b 9 = 2β[α 2 (2β 5) + α(4β 2 6β + 3) + 2( 2β 2 + 2β + 1)], b 10 = α[ 10α 3 + α 2 (14β + 13) + 41αβ 2α(2β + 11) 35β 2 3β 8] + β(10β 2 + 4β 17) 7, α = K 2, β = K 4. Proof. In virtue of (2.13) and Agreement (3.1), we have (3.6) U νωµ = 100 S (ωµ)ν, The relation (3.5) follows by substituting (3.3), (2.10), Definition (2.1), (2.16), (3.2) into (3.6).

9 Conformal change of the tensor U ν λµ for the second category 149 References [1] C.H. Cho, Conformal change of the connection in 3- and 5-dimensional g λν - Unified Field Theory, BIBS, Inha Univ. 13 (1992). [2] C.H. Cho, Conformal change of the connection for the first class in 5-dimensional g λν -unified Field Theory, Kangweon-Kyungki Math. Jour. 2 (1994), [3] C.H. Cho, Conformal change of the torsion tensor in 6-dimensional g-unified field theory,, Kangweon-Kyungki Math. Jour. 3, No.1 (1995), [4] C.H. cho, conformal change of the tensor S ν λµ for the second category in 6- dimensional g-uft, Kangweon-Kyungki Math. Jour. 4, No.2 (1996), [5] C.H. Cho and G.T. Yang, On the recurrence relations in X 6, BIBS, Inha Univ. 14 (1993), [6] K.T. Chung, Three- and Five-dimensional considerations of the geometry of Einstein s g-unified field theory, International Journal of Theoretical Physics 27(9) (1988). [7] K.T. Chung, Conformal change in Einstein s g λν -unified field theory, Nuove Cimento (X) 58B (1968). [8] K.T. Chung, Einstein s connection in terms of g λν, Nuovo Cimento (X) 27 (1963). [9] V. HLAVATÝ, Geometry of Einstein s unified field theory, P. Noordhoff Ltd. (1957). Department of Mathematics Inha University Incheon , Korea