EINSTEIN S CONNECTION IN 5-DIMENSIONAL ES-MANIFOLD. In Ho Hwang

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1 Korean J. ath. 2 (2017), No. 1, pp EINTEIN CONNECTION IN -DIENIONAL E-ANIFOLD In Ho Hwang Abstract. The manifold g EX n is a generalied n-dimensional Riemannian manifold on which the differential geometric structure is imposed b the unified field tensor g λν through the E-connection which is both Einstein and semi-smmetric. The purpose of the present paper is to prove a necessar and sufficient condition for a unique Einstein s connection to exist in -dimensional g EX and to displa a surveable tnesorial representation of -dimensional Einstein s connection in terms of the unified field tensor, emploing the powerful recurrence relations in the first class. 1. Preliminaries This paper is a direct continuation of our previous paper [1], which will be denoted b I in the present paper. All considerations in this paper are based on the results and smbolism of I. Whenever necessar, the will be quoted in the present paper. In this section, we introduce a brief collection of basic concepts, notations, and results of I, which are frequentl used in the present paper([2],[3],[4]). (a) n-simensional g-unified field theor Received Februar 27, Revised arch 21, Accepted arch 29, athematics ubject Classification: 83E0, 83C0, 8A0. Ke words and phrases: E-manifold, Einstein s connection. This work was supported b Incheon National Universit Research Grant, c The Kangwon-Kungki athematical ociet, This is an Open Access article distributed under the terms of the Creative commons Attribution Non-Commercial License ( -nc/3.0/) which permits unrestricted non-commercial use, distribution and reproduction in an medium, provided the original work is properl cited.

2 128 In Ho Hwang Let X n be an n-dimensional generalied Riemannian manifold referred to a real coordinate sstem x ν, which obes the coordinate transformations x ν x ν for which (1.1) det( x x ) 0 In n g UF T the manifold X n is endowed with a real nonsmmetric tensor g λµ, which ma be decomposed into its smmetric part h λµ and skew-smmetric part k λµ : (1.2) (1.3) g λµ = h λµ + k λµ g = det(g λµ ) 0, h = det(h λµ ) 0, k = det(k λµ ) In n g UF T the algebraic structure on X n is imposed b the basic real tensor g λν defined b (1.4) g λµ g λν = g µλ g νλ = δ ν µ and skew- It ma be also decomposed into its smmetric part h λν smmetric part k λν : (1.) g λν = h λν + k λν ince det( h λν ) 0, we ma define a unique tensor h λµ b (1.6) h λµ h λν = δ ν µ In n g-uft we use both h λν and h λµ as tensors for raising and/or lowering indices of all tensors in X n in the usual manner. We then have (1.7) so that (1.8) k λµ = k ρσ h λρ h µσ, g λµ = h λµ + k λµ g λµ = g ρσ h λρ h µσ The differential geometric structure on X n is imposed b the tensor g λν b means of a connection Γ λ ν µ defined b a sstem of equations (1.9) D ω g λν = 2 ωα ν g λα D ω denotes the smbol of the covariant derivative with respect to Γ λ ν µ and λµ ν is the torsion tensor of Γ λ ν µ. Under certain conditions the sstem (1.9) admits a unique solutions Γ λ ν µ.

3 Einstein s connection in -dimensional E-manifold 129 ν It has been shown in [] that if the sstem (1.9) admits Γ λ µ, it must be of the form { } ν ν (1.10) Γ λ µ = + U λµ ν λµ + ν λµ. (1.11) U νλµ = 100 (λµ)ν + 2 (10)0 ν(λµ) (b) ome notations and results The following quantities are frequentl used in our further considerations: (1.12) g = det( g λµ ), h = det( h λµ ), k = det( k λµ ) (1.13) (1.14) g = g h, k = k h. K p = k [α1 α 1 k α2 α2 k αp] αp, (p = 0, 1, 2, ). (1.1) (0) k λ ν = δ ν λ, (p) k λ ν = k λ α (p 1) k α ν (p = 1, 2, ). (1.16) K ωµν = ν k ωµ + ω k νµ + µ k ων ω is the smbolic vector { of } the covariant derivative with respect ν to the christoffel smbols defined b λµ h λµ in the usual wa. In X n it was proved in [] that (1.17) K 0 = 1, K n = k if n is even, and K n = 0 if n is odd. (1.18) g = 1 + K K n σ. (1.19) n σ K (n s) s k ν λ = 0 (p = 0, 1, 2, ). s=0 We also use the following useful abbreviations, denoting an arbitrar tensor T ωµν skew-smmetric in the first two indices b T : (1.20) pqr T = pqr T ωµλ = T (p) αβγ k α(q) ω k β(r) γ µ k λ

4 130 In Ho Hwang and for an arbitrar tensor T for p = 1, 2, 3, : (1.21) (p) T ν = (p 1) k ν α T α. On the other hand, it has shown in [6] that the tensor λµ ν satisfies (1.22) (1.23) = B 3 (110) 2B ωµν = K ωµν + 3K α[µβ k ω ] α k ν β In our subsequent chapter, we start with the relation (1.22) to solve the sstem (1.9). Furthermore, for the first class, the nonholonomic solution of (1.22) ma be given b (1.24) x x = B x or equivalentl (1.2) 4 x x = (2 + x + )K x + ( x + )K x + ( + )K x (1.26) x = 1 + x + + x Therefore, in virtue of (1.24), we see that a necessar and sufficient condition for the sstem (1.9) to have a unique solution in the first class is (1.27) 0 for all x,, x Definition 1.1. A connection Γ λ ν µ is said to be semi-smmetric if its torsion tensor λµ ν is of the form (1.28) λµ ν = 2δ ν [λx µ]. for an arbitrar non-null vector X µ.

5 Einstein s connection in -dimensional E-manifold 131 A connection which is both semi-smmetric and Einstein is called an E connection. An n-dimensional generalied Riemannian manifold X n, on which the differential geometric structure is imposed b g λν b means of an E connection, is called an n-dimensional g E manifold. We denote this manifold b g EX n in our further considerations. In g EX, the following theorems were proved in I. Theorem 1.2. The basic scalars in g EX ma be given b 1 = 2 = L K 0 (1.29) (1.30) 3 = 4 = L K 0, = 0 K = K 2 2, L = ( K 2 2 )2 K 4 Theorem 1.3. The main recurrence relation in the first class is (1.31) (p+) k λ ν = K 2 (p+3) k λ ν K 4 (p+1) k λ ν, (p = 0, 1, 2, ) Theorem 1.4. The basic scalars x satisf (1.32) (1.33) (1.34) = = 0 1 = 2 = 3 = 4 = = = = = K 4 (1.3) = = = = K 2 In virtue of the above theorem, we have Theorem 1.. In the first class, the following identities hold for all values of x and when x (1.36) x (4 1) = x (3 2) K 2 x (2 1) (1.37) x (4 3) = K 4 x (2 1)

6 132 In Ho Hwang (1.38) (1.39) x 4 4 = K 4 2 x K 2 x K 4 x (3 1) 2 x (4 2) = x 3 3 K 2 x K 4 x Theorem 1.6. (Recurrence relations in the first class) If T ωµν is a tensor skew-smmetric in the first two indices, then the following recurrence relations hold in the first class of g EX : (1.40) (41)r T = (32)r T (21)r K 2 T (1.41) (1.42) (1.43) (43)r T (21)r = K 4 T 44r 22r 33r (31)r T = K 4 T + K 2 T + 2K 4 T 2 (42)r T = 33r 22r 11r T K 2 T + K 4 T 2. Einstein s connection Γ λ ν µ in the first class In this section, we shall derive surveable representations of Γ λ ν µ in terms of g λν, emploing the recurrence relations. In the following theorem, we shall prove two relations in X n. These relations will be used in our subsequent theorem when we are concerned with the solution of (1.9). (2.1) Theorem 2.1. We have (pq)r B = (pq)r + (p q )r + (p q)r + (pq )r (2.2) 2 (pq)r B ωµν = (pq)r K ωµν + r (pq) K ν[ωµ] ((pq )r K ωµν + (p q)r K ωµν + r p q K ν[ωµ] + r q p K ν[ωµ] ) (2.3) p = p + 1, q = q + 1, r = r + 1, r = r + 2

7 Einstein s connection in -dimensional E-manifold 133 Proof. In virtue of (1.22) and (1.20), the first relation (2.1) is obtained as in the following wa: (2.4) (pq)r B = (pq)r B ωµν = 1 2 B ωβγ( (p) k ω α(q) k µ β + (q) k ω α(p) k µ β ) (r) k ν γ = 1 2 ( αβγ + ɛηγ k α ɛ k β η + ɛβη k α ɛ k γ η + αɛη k β ɛ k γ η ) ( (p) k ω α(q) k µ β + (q) k ω α(p) k µ β ) (r) k ν γ After a length calculation, we note that the right-hand side of the above equation is equal to that of (2.1). imilarl, we verif (2.2) using (1.20) and (1.23). Theorem 2.2. A necessar and sufficient condition for the sstem (1.9) to admit a unique solution Γ λ ν µ in the first class is that (2.) (2.6) gab(c 2 4K 4 D 2 ) 0 A = 1 K 2 + K 4, B = 1 K 4 C = 1 K 2 + K 4, D = K 2 2 Proof. In virtue of (1.29) and (1.30), the smmetric scalars x defined b (1.26) takes values as in the following 3 cases: If two of the indices x,, are 1, 2 or 3, 4, then (2.7) = 1 + K + L, x 1 + K L If at least one of x,, is and no two take the values 1, 2 nor 3, 4, then (2.8) x = 1 K + L, 1 K L, 1 + K 4, 1 K 4, 1 In the remaining cases, (2.9) x = 1 K L 2 K 4, 1 K + L 2 K 4 1 K L + 2 K 4, 1 K + L + 2 K 4 It ma easil verified that the product of two factors in the right of (2.7) is g, that of five factors in the right of (2.8) is (1 K 2 +K 4 )(1 K 4 ), and that of four factors in the right of (2.9) is (1 K 2 +K 4 ) 2 4K 4 (K 2 2) 2. Hence we have proved our assertion (2.) in virtue of (1.27) and (2.6).

8 134 In Ho Hwang Theorem 2.3. The sstem of equations (1.22) in the first class is reduced to the following 2 equations: (2.10) B = (10)1 (10)1 B = (10)1 + (21)1 + (20) (12)1 B = (12)1 + (23) (13)2 (20)2 B = (20)2 + (31)2 + (30)3 + (21)3 2 (23)1 B = 2 (23)1 (21)1 + 2K K 4 K 2 2 (13)2 B = 2 (13) K 4 K (21)3 2K 2 (30)3 B = (30)3 (21)3 K 2 (32)3 + (40)4 + (31)4 (21)3 B = (21)3 + (32)3 + (31) (32)3 B = 2 (32)3 (21) K 4 + K 4 K (40)4 B = (40)4 (31)4 114 (30)3 (10)3 (30)1 K 2 K 4 + K 2 + K 2 K 4 + K 2 K 4 + (10)1 (21)3 (21)1 (32)3 (32)1 +K 4 + K 2 + K 2 K 4 + K 2 + K 4 2 (31)4 B = 2 (31)4 (21)3 (21) K 2 + 2K 2 K 4 + K 4 K (10)3 B = (10)3 + (21)3 + (20) (30)1 B = (30)1 (21)1 K 2 (32)1 + (40)2 + (31)2 (20)4 B = (20)4 + (31)4 (30)3 (30)1 (21)3 (21)1 K 2 K 4 K 2 K 4 (40)2 B = (40)2 (31)2 112 (30)3 (10)3 (21)3 K 2 K 4 K 2 K 4 K 2 (32)3 110 B = (21)1 112 B = (21)3 222 B = (32)3 332 B = (1 + K 2 ) (21)3 (31)2 + K 4 + 2K 4 + 2K B = (32)3 (32)1 2K 2 2K 4

9 Einstein s connection in -dimensional E-manifold B = (21)3 (21)1 2K 2 2K B = (1 + K 2 ) (31)4 (21)3 + K 4 + 2K 4 2K 2 K 4 2K B = (32)1 330 B = (1 + K 2 ) (31)0 (21)1 + K 4 + 2K 4 + 2K 4 2 (31)0 B = 2 (31) K 4 K (21)1 2K 2 (21)1 Proof. This assertion follows from (2.1) using (1.31) and (1.40)-(1.43). References [1] I.H. Hwang, A stud on the recurrence relations of -dimensional E-manifold, Korean J. ath. 24 (3) (2016), [2] D.k. Datta, ome theorems on smmetric recurrent tensors of the second order, Tensor (N..) 1 (1964), [3] A. Einstein, The meaning of relativit, Princeton Universit Press, 190. [4] R.. ishra, n-dimensional considerations of unified field theor of relativit, Tensor 9 (199), [] K.T. Chung, Einstein s connection in terms of g λν, Nuovo cimento oc. Ital. Fis. B 27 (1963), (X), [6] V. Hlavatý, Geometr of Einstein s unified field theor, Noordhoop Ltd., 197 In Ho Hwang Department of athematics Incheon National Universit Incheon 22012, Korea ho818@inu.ac.kr