Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds

The main purpose of the present paper is to study conditions for the eight-dimensional Walker manifolds which admit a field of parallel null 4-planes to be Einsteinian, Osserman, or locally conformally flat.

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Datum:	2012
Hauptverfasser:	Iscan, M., Gezer, A., Salimov, A.
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Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2012
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spelling	irk-123456789-1067062016-10-04T03:02:15Z Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds Iscan, M. Gezer, A. Salimov, A. The main purpose of the present paper is to study conditions for the eight-dimensional Walker manifolds which admit a field of parallel null 4-planes to be Einsteinian, Osserman, or locally conformally flat. Основной целью данной статьи является изучение условий, при которых восьмимерные многообразия Уолкера, допускающие поле параллельных нулевых четырехмерных плоскостей, являются эйнштейновыми, оссермановыми или локально конформно плоскими. 2012 Article Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds / M. Iscan, A. Gezer, A. Salimov // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 1. — С. 21-37. — Бібліогр.: 15 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106706 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The main purpose of the present paper is to study conditions for the eight-dimensional Walker manifolds which admit a field of parallel null 4-planes to be Einsteinian, Osserman, or locally conformally flat.
format	Article
author	Iscan, M. Gezer, A. Salimov, A.
spellingShingle	Iscan, M. Gezer, A. Salimov, A. Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds Журнал математической физики, анализа, геометрии
author_facet	Iscan, M. Gezer, A. Salimov, A.
author_sort	Iscan, M.
title	Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds
title_short	Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds
title_full	Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds
title_fullStr	Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds
title_full_unstemmed	Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds
title_sort	some properties concerning curvature tensors of eight-dimensional walker manifolds
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2012
url	http://dspace.nbuv.gov.ua/handle/123456789/106706
citation_txt	Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds / M. Iscan, A. Gezer, A. Salimov // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 1. — С. 21-37. — Бібліогр.: 15 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT iscanm somepropertiesconcerningcurvaturetensorsofeightdimensionalwalkermanifolds AT gezera somepropertiesconcerningcurvaturetensorsofeightdimensionalwalkermanifolds AT salimova somepropertiesconcerningcurvaturetensorsofeightdimensionalwalkermanifolds
first_indexed	2025-07-07T18:52:47Z
last_indexed	2025-07-07T18:52:47Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2012, vol. 8, No. 1, pp. 21–37 Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds M. Iscan, A. Gezer, and A. Salimov Ataturk University, Faculty of Science, Department of Mathematics 25240, Erzurum-Turkey E-mail: miscan@atauni.edu.tr agezer@atauni.edu.tr asalimov@atauni.edu.tr Received June 19, 2009 The main purpose of the present paper is to study conditions for the eight-dimensional Walker manifolds which admit a field of parallel null 4-planes to be Einsteinian, Osserman, or locally conformally flat. Key words: Walker manifolds, Osserman manifolds, Riemannian exten- sion, Einstein metric. Mathematics Subject Classification 2000: 53C50 (primary); 53B05, 32Q20 (secondary). 1. Introduction Let M2n be a Riemannian manifold with a neutral metric, i.e., with a semi- Riemannian metric g of signature (n, n).We denote by =p q(M2n) the set of all tensor fields of type (p, q) on M2n. Manifolds, tensor fields and connections are always assumed to be differentiable of class C∞. By a Walker n-manifold, we mean a semi-Riemannian manifold which admits a field of parallel null r-planes with r ≤ n 2 . The canonical forms of the metrics were studied by Walker in [1]. Of special interest are even dimensional Walker manifolds (n = 2m) admitting a field of null planes of maximum dimensionality (r = m). It is known that the Walker metrics have served as a powerful tool of con- structing interesting indefinite metrics which exhibit various aspects of geometric properties not given by any positive definite metrics. Recently, it was shown [2, 3] that the Walker 4-manifolds of neutral signature admit a pair comprising an almost complex structure and an opposite almost complex structure, and that c© M. Iscan, A. Gezer, and A. Salimov, 2012 M. Iscan, A. Gezer, and A. Salimov Petean’s nonflat indefinite Kähler-Einstein metric on a torus was obtained as an example of a Walker 4-manifold [4]. Moreover, an indefinite Ricci flat strictly almost Kähler metric on eight-dimensional torus was reported in [5]. Thus the Walker 4- and 8-manifolds display a large variety of indefinite geometry in four and eight-dimensions ([6–10]). Our purpose is to systematically study the Walker metrics by focusing on their curvature properties. The main results of the present paper, the Walker metrics which are Einstein or locally conformally flat, are determined (Theorem 1, 2). To this end, the present paper is organized as follows. Section 2 develops some basic facts about Walker metrics. Their Levi–Civita connection and the curvature tensor are explicitly written for the purpose of the present analysis. In Section 3, the Walker metrics which are Einstein are studied, obtaining a family for the scalar curvature being zero (Theorem 1). In Section 4, the locally conformally flat Walker metrics are studied, and the examples of indefinite metrics of harmonic curvatures are exhibited. Finally, the Osserman property of Walker metrics in dimension eight are studied. 2. The canonical form of Walker metrics in dimension eight A neutral g on an 8-manifold M8 is said to be a Walker metric if there exists a 4-dimensional null distribution D on M8 which is parallel with respect to g. From Walker theorem [1], there is a system of the coordinates (x1, . . . , x8) with respect to which g takes the local canonical form g = (gij) = ( 0 I4 I4 B ) , (2.1) where I4 is the unit 4×4 matrix and B is a 4×4 symmetric matrix whose entries are functions of the coordinates (x1, . . . , x8). Note that g is of neutral signature (+ + + + − − −−), and that the parallel null 4-plane D is spanned locally by {∂1, ∂2, ∂3, ∂4}, where ∂i = ∂ ∂xi , (i = 1, . . . , 8). In this paper, we consider the specific Walker metrics on M8 with B of the form B =   a 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0   , (2.2) where a, b are smooth functions of the coordinates (x1, . . . , x8). The Walker met- rics with conditions (2.1) and (2.2) are studied by Matsushita et al. in [5]. There is a famous Goldberg conjecture, which states that the almost complex struc- ture of a compact almost Kahler–Einstein Riemannian manifold is integrable. In [5], by considering the 8-dimensional Walker manifolds with conditions (2.1) and 22 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 Eight-dimensional Walker manifolds (2.2), the authors have shown that the neutral-signature version of Goldberg’s conjecture fails. It is well known that if the Walker manifolds M4 and M8 are Einstein (or ∗-Einstein), then the Walker metrics on M4 and M8 can be viewed as Rieman- nian extensions from manifolds (M2,∇) and (M4, ∇̃) to their cotangent bundles, respectively (see [11], [9] and Section 5 in the present paper). Moreover, let there be given on both manifolds M2 and M4 holomorphic (analytic) connections with respect to complex structures (for example, these structures naturally appeare in anti-Kähler geometry). Then (M2,∇) is locally flat [12, p. 113], but (M4, ∇̃) is not locally flat. Hence the case M8 with Walker metrics is of special interest and can be viewed as a Riemannian extension of non-flat holomorphic connection ∇̃. Indeed, if (M2,∇) is locally flat, then the Riemannian extension R∇ to the cotangent bundle ∗T (M2) has the components of the form R∇ = ( 0 δj i δi j 0 ) , i.e. the metric R∇ is trivial, and hence it is not localy isometric to the Walker manifold (M4, w g) with conditions a 6= 0, b 6= 0, c 6= 0 (even if a = b 6= 0, c = 0), where wg =   0 0 1 0 0 0 0 1 1 0 a c 0 1 c b   . A straightforward calculation using the fact that the inverse of the metric tensor, g−1 = (gαβ), gives g−1 = ( −B I4 I4 0 ) . The Levi–Civita connection of a Walker metric (2.1), with B given as in (2.2), is given by ∇∂1∂5 = 1 2 a1∂1, ∇∂1∂7 = 1 2 b1∂3, ∇∂2∂5 = 1 2 a2∂1, ∇∂2∂7 = 1 2 b2∂3, ∇∂3∂5 = 1 2 a3∂1, ∇∂3∂7 = 1 2 b3∂3, ∇∂4∂5 = 1 2 a4∂1, ∇∂4∂7 = 1 2 b4∂3, ∇∂5∂5 = 1 2 a5∂1, ∇∂5∂6 = 1 2 a6∂1, ∇∂5∂7 = 1 2 a7∂1 + 1 2 b5∂3, ∇∂5∂8 = 1 2 a8∂1, ∇∂6∂7 = 1 2 b6∂3, ∇∂7∂7 = 1 2 b7∂3, ∇∂7∂8 = 1 2 b8∂3, (2.3) where ai means a partial derivative ∂ ∂xi a(x1, . . . , x8). Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 23 M. Iscan, A. Gezer, and A. Salimov Let R denote the Riemannian curvature tensor, taken with the sign convention R(X, Y ) = [∇X ,∇Y ] − ∇[X,Y ]. From (2.3), after a long but straightforward calculation we get that the nonzero components of the (0,4)-curvature tensor R(X, Y, Z, V ) = 〈R(X,Y )Z, V 〉 of any Walker metric (2.1) are determined by R1515 = 1 2 a11, R1525 = 1 2 a12, R1535 = 1 2 a13, R1545 = 1 2 a14, R1565 = 1 2 a16, R1575 = 1 2 a17 − 1 4 a3b1, R1585 = 1 2 a18, R1717 = 1 2 b11, R1727 = 1 2 b12, R1737 = 1 2 b13, R1747 = 1 2 b14, R1757 = 1 2 b15 − 1 4 a1b1, R1767 = 1 2 b16, R1787 = 1 2 b18, R2525 = 1 2 a22, R2535 = 1 2 a23, R2545 = 1 2 a24, R2565 = 1 2 a26, R2575 = 1 2 a27 − 1 4 a3b2, R2585 = 1 2 a28, R2727 = 1 2 b22, R2737 = 1 2 b23, R2747 = 1 2 b24, R2757 = 1 2 b25 − 1 4 a2b1, R2767 = 1 2 b26, R2787 = 1 2 b28, R3535 = 1 2 a33, R3545 = 1 2 a34, R3565 = 1 2 a36, R3575 = 1 2 a37 − 1 4 a3b3, R3585 = 1 2 a38, R3737 = 1 2 b33, R3747 = 1 2 b34, R3757 = 1 2 b35 − 1 4 a3b1, R3767 = 1 2 b36, R3787 = 1 2 b38, R4545 = 1 2 a44, R4565 = 1 2 a46, R4575 = 1 2 a47 − 1 4 a3b4, R4585 = 1 2 a48, R4747 = 1 2 b44, R4757 = 1 2 b45 − 1 4 a4b1, R4767 = 1 2 b46, R4787 = 1 2 b48, R6565 = 1 2 a66, R6575 = 1 2 a67 − 1 4 a3b6, R6585 = 1 2 a68, R6767 = 1 2 b66, R6787 = 1 2 b68, R7575 = 1 2 a77 − 1 4 a3b7, R7585 = 1 2 a78 − 1 4 a3b8, R8585 = 1 2 a88, R8787 = 1 2 b88. (2.4) 3. Einstein-Walker metrics We now turn our attention to the Einstein conditions for the Walker metric (2.1) with B given as in (2.2). As a matter of notation, let Rij and Sc denote the Ricci tensor and the scalar curvature of a Walker metric (2.1). From equations (2.4), the nonzero components of the Ricci tensor Rij are characterized by 24 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 Eight-dimensional Walker manifolds R15 = 1 2 a11, R25 = 1 2 a12, R35 = 1 2 a13, R45 = 1 2 a14, R17 = 1 2 b13, R27 = 1 2 b23, R37 = 1 2 b33, R47 = 1 2 b34, R56 = 1 2 a16, R58 = 1 2 a18, R67 = 1 2 b36, R78 = 1 2 b38, R57 = 1 2 b35 − 1 2 a3b1 + 1 2 a17, R55 = −a26 − a37 − a48 + 1 2 (aa11 + ba33 + a3b3), R77 = −b15 − b26 − b48 + 1 2 (ab11 + a1b1 + bb33). (3.1) From (2.1) and (3.1), the scalar curvature of a Walker metric (2.1) is given by Sc = 8∑ i,j=1 gijRij = a11 + b33. (3.2) R e m a r k 1. There exists a Walker metric with the prescribed scalar curvature, because from (3.2) we can always choose two suitable functions a and b. Next we prove the main result in this section. Theorem 1. A Walker metric (2.1) is Einstein if and only if the defining functions a(x1, . . . , x8) and b(x1 , . . . , x8) are as follows: { a(x1, . . . , x8) = a(x1, x3, x5, x7) = x1R(x5, x7) + x3S(x5, x7) + ξ(x5, x7) b(x1, . . . , x8) = b(x1, x3, x5, x7) = x3P (x5, x7) + x1Q(x5, x7) + η(x5, x7), (3.3) where ξ(x5, x7) and η(x5, x7) are arbitrary smooth functions, while P (x5, x7), R(x5, x7), S(x5, x7)and Q(x5, x7) are smooth functions satisfying S7 = 1 2 SP,Q5 = 1 2 RQ, R7 + P5 = SQ. (3.4) P r o o f. The Einstein equations defined by Gij = Rij − 1 8Scgij = 0 for a Walker metric (2.1) are as follows: (i)    G25 = 1 2a12 = 0, G35 = 1 2a13 = 0, G45 = 1 2a14 = 0, G56 = 1 2a16 = 0, G58 = 1 2a18 = 0, G17 = 1 2b13 = 0, G27 = 1 2b23 = 0, G47 = 1 2b34 = 0, G67 = 1 2b36 = 0, G78 = 1 2b38 = 0, Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 25 M. Iscan, A. Gezer, and A. Salimov (ii)    G26 = G48 = −1 8(a11 + b33) = 0, G15 = 1 8(3a11 − b33) = 0, G37 = 1 8(3b33 − a11) = 0, (iii) G57 = 1 2(a17 − a3b1 + b35) = 0, (iv) G55 = −a26 − a37 − a48 + 1 8 a(3a11 − b33) + 1 2 (ba33 + a3b3) = 0, (v) G77 = −b15 − b26 − b48 + 1 2 (ab11 + a1b1) + 1 8 b(3b33 − a11) = 0. (3.5) We divide the proof of this theorem into two steps. Step 1. The (i) PDEs in the Einstein conditions ( 3.5) imply that a and b take the following forms: { a(x1, . . . , x8) = ā(x1, x5, x7) + â(x2, x3, x4, x5, x6, x7, x8), b(x1, . . . , x8) = b̄(x3, x5, x7) + b̂(x1, x2, x4, x5, x6, x7, x8). (3.6) From the (ii) PDEs in the Einstein conditions (3.5), we get { a11 = 0, b33 = 0. (3.7) Substituting these functions a and b from (3.6) into (3.7), we get ā(x1, x5, x7) = x1R(x5, x7) + ξ(x5, x7) b̄(x3, x5, x7) = x3P (x5, x7) + η(x5, x7). Therefore we have { a(x1, . . . , x8) = x1R(x5, x7) + â(x2, x3, x4, x5, x6, x7, x8) b(x1, . . . , x8) = x3P (x5, x7) + b̂(x1, x2, x4, x5, x6, x7, x8) (3.8) for some functions R(x5, x7) and P (x5, x7). Step 2. The functions a and b in (3.8) satisfy the (i) and (ii) PDEs in the Einstein conditions (3.5). We must consider further conditions for a and b to satisfy the (iii)–(v) PDEs in (3.5). Inserting the functions a and b from (3.8) into the (iii) PDE in (3.5), we have that â3(x2, x3, x4, x5, x6, x7, x8)b̂1(x1, x2, x4, x5, x6, x7, x8) = R7(x5, x7) + P5(x5, x7). From this equation, we see that â3 does not depend on x2, x3, x4, x6 and x8, and, similarly, b̂1 does not depend on x1, x2, x4, x6 and x8. That is, 26 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 Eight-dimensional Walker manifolds â(x2, x3, x4, x5, x6, x7, x8) = x3S(x5, x7) +ξ(x5, x7), and b̂(x1, x2, x4, x5, x6, x7, x8) = x1Q(x5, x7) + η(x5, x7). Hence, we can put { a(x1, . . . , x8) = x1R(x5, x7) + x3S(x5, x7) + ξ(x5, x7) b(x1, . . . , x8) = x3P (x5, x7) + x1Q(x5, x7) + η(x5, x7). Finally, placing these expressions into the (iii)–(v) PDEs in (3.5), we get G57 = 1 2 (R7 − SQ + P5) = 0, G55 = −S7 + 1 2 S.P = 0, G77 = −Q5 + 1 2 RQ = 0, which hold if and only if R7 + P5 = SQ, S7 = 1 2SP and Q5 = 1 2RQ. This completes the proof. Corollary 1. Let Sc be the scalar curvature of the Einstein–Walker metric. Then the following equation holds: Sc = 0. Now, we will analyze the Ricci flat property of the Einstein–Walker metric. For some special cases of the Einstein–Walker metric there can be written the following remarks. R e m a r k 2. Let g be an Einstein–Walker metric. That is, the functions a and b of the metric g in (2.1) are given by the solution (3.5). g is necessarily of Ricci flat. Case 1: R = S = P = Q = 0. Since there is no restriction on ξ and η, it follows that any Einstein–Walker metric of this case, a = ξ(x5, x7), b = η(x5, x7), is Ricci flat. Case 2: S = Q = 0. In this case, the PDEs in (3.4) reduce to R7 + P5 = 0, and thus the Einstein–Walker metric of the form a = x1R(x5, x7) + ξ(x5, x7), b = x3P (x5, x7) + η(x5, x7) is Ricci flat if and only if R7 + P5 = 0. Case 3: R = P = 0. In this case, the PDEs in (3.4) reduce to S7 = 0, Q5 = 0, SQ = 0. It immediately follows that S(x5, x7) = S(x5), Q(x5, x7) = Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 27 M. Iscan, A. Gezer, and A. Salimov Q(x7). The condition S(x5, x7)Q(x5, x7) = S(x5)Q(x7) holds if and only if either S(x5) = 0 or Q(x7) = 0. Consequently, the Ricci flat Einstein–Walker metric must be either{ a = ξ(x5, x7) b = x1Q(x7) + η(x5, x7) or { a = x3S(x5) + ξ(x5, x7) b = η(x5, x7). R e m a r k 3. Let us assume that S and Q are nonzero. From the solutions of the first two PDEs in (3.4), we can respectively write S(x5, x7) = γ(x5)e 1 2 ∫ P (x5,x7)dx7 , Q = (x5, x7) = δ(x7)e 1 2 ∫ R(x5,x7)dx5 for some functions γ(x5), δ(x7). If we choose α(x5, x7) = ∫ P (x5, x7)dx7 and β(x5, x7) = ∫ R(x5, x7)dx5, then the a and b functions in (3.3) transform into { a(x1, . . . , x8) = x1β5(x5, x7) + x3γ(x5)e 1 2 α(x5,x7) + ξ(x5, x7) b(x1, . . . , x8) = x3α7(x5, x7) + x1γ(x7)e 1 2 β(x5,x7) + η(x5, x7). Here, the third PDE in (3.4) reduces to β57 + α57 = γδe 1 2 (α+β) for any two functions α, β. Let R denote the (0,4)-curvature tensor and consider its covariant derivative ∇R, whose vanishing is the condition for a metric to be locally symmetric. We complete this section by giving some results concerning the locally symmetric Ricci flat Einstein Walker metrics. R e m a r k 4. Locally symmetric Ricci flat Walker metrics can be constructed from Remark 2. Such metrics are necessarily of Einstein as given in (3.3). Assume R = S = P = Q = 0. Then the Ricci flat Einstein walker metric of the form a = ξ(x5, x7), b = η(x5, x7) is locally symmetric if and only if η555 + ξ577 = 0 and η755 + ξ777 = 0. Assume S = Q = 0. Then it follows from equations (3.4) that R7 + P5 = 0. A locally symmetric Ricci flat Walker metric is of the form a = x1R(x5) + ξ(x5, x7), b = x3P (x7) + η(x5, x7), provided that the following two conditions hold: η555 + ξ577 = 0 and η755 + ξ777 = 0. 28 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 Eight-dimensional Walker manifolds Assume R = P = 0. There are two cases. In one case, we have that a = ξ(x5, x7), b = x1Q(x7) + η(x5, x7) give a locally symmetric metric if and only if 2(ξ577 + η555) = Qξ55 and 2(η755 + ξ777) = Qξ75 − ξ5Q7. In the other case, a = x3S(x5) + ξ(x5, x7), b = η(x5, x7) is locally symmetric if and only if 2(η555+ξ577) = S5η7+Sη57 and 2(η755+ξ777) = Sη77. 4. Locally conformally flat Walker metrics A semi-Riemannian manifold is locally conformally flat if and only if its Weyl tensor vanishes, where the Weyl tensor is given by W (X, Y, Z, V ) = R(X,Y, Z, V ) + Sc (n−1)(n−2) {g(X, Z)g(Y, V )− g(X, V )g(Y, Z)} − 1 (n−2) {Ric(X, Z)g(Y, V )−Ric(Y,Z)g(X,V ) +Ric(Y, V )g(X,Z)−Ric(X,V )g(Y,Z)} . The nonzero components of Weyl tensor of a special Walker metric are given by W1525 = 7 12a12,W1535 = 7 12a13,W1545 = 7 12a14,W1565 = 7 12a16, W1585 = 7 12a18, W2525 = 1 2a22,W2535 = 1 2a23,W2545 = 1 2a24,W2585 = 1 2a28,W3535 = 1 2a33, W3545 = 1 2a34,W3565 = 1 2a36,W3585 = 1 2a38,W4545 = 1 2a44,W4565 = 1 2a46, W4585 = 1 2a48,W6565 = 1 2a66,W6585 = 1 2a68,W8585 = 1 2a88,W1717 = 1 2b11, W1727 = 1 2b12,W1737 = 7 12b13,W1747 = 1 2b14,W1767 = 1 2b16,W1787 = 1 2b18, W2727 = 1 2b22,W2737 = 1 2b23,W2747 = 1 2b24,W2767 = 1 2b26,W2787 = 1 2b28, W3747 = 7 12b34,W3767 = 7 12b36,W3787 = 7 12b38,W4747 = 1 2b44, W4767 = 1 2b46, W6767 = 1 2b66,W6787 = 1 2b68,W8787 = 1 2b88,W1515 = 1 42(27a11 − b33), W3737 = 1 42(27b33 − a11),W4575 = 1 2a47 − 1 4a3b4 − 1 12ab34, W2575 = 1 2a27 − 1 4a3b2 − 1 12ab23,W2757 = 1 2b25 − 1 4a2b1 − 1 12ba12, W4757 = 1 2b45 − 1 4a4b1 − 1 12ba14,W5767 = 1 2b56 − 1 4a6b1 − 1 12ba16, W5787 = 1 2b58 − 1 4a8b1 − 1 12ba18,W6575 = 1 2a67 − 1 4a3b6 − 1 12ab36, W7585 = 1 2a78 − 1 4a3b8 − 1 12ab38,W1575 = 7 12a17 − 1 3a3b1 − 1 12ab13 + 1 12b35, W3757 = 7 12b35 − 1 3a3b1 − 1 12ba13 + 1 12a17, W1757 = 2 3b15 + 1 6b26 + 1 6b48 − 1 12ab11 − 1 3a1b1 − 5 84b(a11 − b33), Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 29 M. Iscan, A. Gezer, and A. Salimov W2565 = 2 3a26 + 1 6a37 + 1 6a48 − 1 12ba33 − 1 12a3b3 − 1 84a(5a11 − 2b33), W3575 = 2 3a37 + 1 6a26 + 1 6a48 − 1 12ba33 − 1 3a3b3 − 5 84a(a11 + b33), W4787 = 2 3b48 + 1 6b15 + 1 6b26 − 1 12ab11 − 1 12a1b1 − 1 84b(5b33 − 2a11), W5757 = 1 2b15 − 1 4a5b1 + 1 6b(a26 + a37 + a48) + 1 6a(b15 + b26 + b48) − 5 84ab(a11 + b33)− 1 12b(ba33 + a3b3)− 1 12a(ab11 + a1b1), W7575 = 1 2a77 − 1 4a3b7 + 1 6b(a26 + a37 + a48) + 1 6a(b15 + b26 + b48) − 5 84ab(a11 + b33)− 1 12b(ba33 + a3b3)− 1 12a(ab11 + a1b1). (4.1) Now it is possible to obtain the form of a locally conformally flat Walker metric as follows. Theorem 2. A Walker metric (2.1) is locally conformally flat if and only if the defining functions a = a(x1, . . . , x8) and b = b(x1, . . . , x8) satisfy    a = x1K(x5, x7) + x2T (x5, x7) + x3L(x5) + x4R(x5, x7) + x6F (x5, x7) + x8D(x5, x7) + ξ(x5, x7) b = x1S(x7) + x2Q(x5, x7) + x3P (x5, x7) + x4M(x5, x7) + x6N(x5, x7) + x8V (x5, x7) + η(x5, x7) (4.2) for any functions K(x5, x7), T (x5, x7), L(x5), R(x5, x7), F (x5, x7), D(x5, x7), ξ(x5, x7), S(x7), Q(x5, x7), P (x5, x7), M(x5, x7), N(x5, x7), V (x5, x7), η(x5, x7), satisfying K7 = 1 2LS, T7 = 1 2LQ,R7 = 1 2LM, F7 = 1 2LN,D7 = 1 2LV, ξ7 = 1 2Lη, Q5 = 1 2ST, P5 = 1 2SL, M5 = 1 2SR, N5 = 1 2SF, V5 = 1 2SD, η5 = 1 2Sξ, KS = 0, LP = 0. (4.3) P r o o f. Since an eight-dimensional manifold is locally conformally flat if and only if the Weyl tensor vanishes, we consider (4.1) as a system of PDEs. We will prove this theorem in three steps. Step 1. Considering the following components of the Weyl tensor of (4.1), we have    W1525 = 7 12a12 = 0 W1535 = 7 12a13 = 0 W1545 = 7 12a14 = 0 W1565 = 7 12a16 = 0 W1585 = 7 12a18 = 0    W1727 = 1 2b12 = 0 W1737 = 7 12b13 = 0 W1747 = 1 2b14 = 0 W1767 = 1 2b16 = 0 W1787 = 1 2b18 = 0. (4.4) First, the PDEs (4.4) imply that a and b take the form { a = a(x1, . . . , x8) = ā(x1, x5, x7) + â(x2, x3, x4, x5, x6, x7, x8) b = b(x1, . . . , x8) = b̄(x1, x5, x7) + b̂(x2, x3, x4, x5, x6, x7, x8). (4.5) 30 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 Eight-dimensional Walker manifolds From the two PDEs, W1515 = 1 42(27a11−b33) = 0 and W3737 = 1 42(27b33−a11) = 0, we obtain { a11 = 0 b33 = 0. Then we have a11 = 0 so ā11 = 0, then ā(x1, x5, x7) = x1K(x5, x7) + ξ(x5, x7), moreover, b11 = 0 so b̄11 = 0, then b̄(x1, x5, x7) = x1S(x5, x7) + η(x5, x7). There- fore equations (4.5) transform into { a = x1K(x5, x7) + â(x2, x3, x4, x5, x6, x7, x8) b = x1S(x5, x7) + b̂(x2, x3, x4, x5, x6, x7, x8). (4.6) Substituting the functions a and b from (4.6) into the following PDEs :    W2535 = 1 2a23 = 0 W2545 = 1 2a24 = 0 W2585 = 1 2a28 = 0,    W2737 = 1 2b23 = 0 W2747 = 1 2b24 = 0 W2767 = 1 2b26 = 0 W2787 = 1 2b28 = 0, the functions a and b become of the forms { â(x2, x3, x4, x5, x6, x7, x8) = ¯̂a(x2, x5, x6, x7) + ˆ̂a(x3, x4, x5, x6, x7, x8) b̂(x2, x3, x4, x5, x6, x7, x8) = ¯̂ b(x2, x5, x7) + ˆ̂ b(x3, x4, x5, x6, x7, x8). Since W2525 = 1 2a22 = 0 and W2727 = 1 2b22 = 0, we have that ¯̂a(x2, x5, x6, x7) = x2T (x5, x6, x7) +ξ(x5, x6, x7) and ¯̂ b(x2, x5, x7) = x2Q(x5, x7) + η(x5, x7). Hence we can put { a = x1K(x5, x7) + x2T (x5, x6, x7) + ˆ̂a(x3, x4, x5, x6, x7, x8) b = x1S(x5, x7) + x2Q(x5, x7) + ˆ̂ b(x3, x4, x5, x6, x7, x8). (4.7) On inserting the functions a and b from (4.7) into the following PDEs :    W3545 = 1 2a34 = 0 W3565 = 1 2a36 = 0 W3585 = 1 2a38 = 0,    W3747 = 7 12b34 = 0 W3767 = 7 12b36 = 0 W3787 = 7 12b38 = 0, we get    ˆ̂a(x3, x4, x5, x6, x7, x8) = ¯̂̂ a(x3, x5, x7) + ˆ̂̂ a(x4, x5, x6, x7, x8) ˆ̂ b(x3, x4, x5, x6, x7, x8) = ¯̂̂ b(x3, x5, x7) + ˆ̂̂ b(x4, x5, x6, x7, x8). Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 31 M. Iscan, A. Gezer, and A. Salimov From W3535 = 1 2a33 = 0 and b33 = 0, it can be written that ¯̂̂ a(x3, x5, x7) = x3L(x5, x7)+ ξ(x5, x7) and ¯̂̂ b(x3, x5, x7) = x3P (x5, x7)+η(x5, x7). Therefore, the functions a and b take the form    a = x1K(x5, x7) + x2T (x5, x6, x7) + x3L(x5, x7) + ˆ̂̂ a(x4, x5, x6, x7, x8) b = x1S(x5, x7) + x2Q(x5, x7) + x3P (x5, x7) + ˆ̂̂ b(x4, x5, x6, x7, x8). (4.8) Analogously, the functions a and b, satisfying the following PDEs :    W4545 = 1 2a44 = 0 W4565 = 1 2a46 = 0 W4585 = 1 2a48 = 0, { W4747 = 1 2b44 = 0 W4767 = 1 2b46 = 0,    W6565 = 1 2a66 = 0 W6585 = 1 2a68 = 0 W8585 = 1 2a88 = 0,    W6767 = 1 2b66 = 0 W6787 = 1 2b68 = 0 W8787 = 1 2b88 = 0, take the form    a = x1K(x5, x7) + x2T (x5, x6, x7) + x3L(x5, x7) + x4R(x5, x7) + x6F (x5, x7) + x8D(x5, x7) + ξ(x5, x7) b = x1S(x5, x7) + x2Q(x5, x7) + x3P (x5, x7) + x4M(x5, x7, x8) + x6N(x5, x7) + x8V (x5, x7) + η(x5, x7) for any functions K(x5, x7), T (x5, x7), L(x5), R(x5, x7), F (x5, x7), D(x5, x7), ξ(x5, x7), S(x7), Q(x5, x7), P (x5, x7), M(x5, x7), N(x5, x7), V (x5, x7), η(x5, x7). Step 2. Considering the result of step one, placing this result into the fol- lowing PDEs: W2575 = 1 2(a27 − 1 2a3b2 − 1 6ab23) = 0,W4757 = 1 2(b45 − 1 2a4b1 − 1 6ba14) = 0, there can be obtained the following conditions: T7 = 1 2LQ,M5 = 1 2SR. (4.9) From (4.9), we can see that T does not depend on x6, and M does not depend on x8. Therefore we get    a = x1K(x5, x7) + x2T (x5, x6, x7) + x3L(x5, x7) + x4R(x5, x7) + x6F (x5, x7) + x8D(x5, x7) + ξ(x5, x7). b = x1S(x5, x7) + x2Q(x5, x7) + x3P (x5, x7) + x4M(x5, x7, x8) + x6N(x5, x7) + x8V (x5, x7) + η(x5, x7). (4.10) 32 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 Eight-dimensional Walker manifolds Inserting the functions a and b from (4.10) into to the following PDEs : W4575 = 1 2(a47 − 1 2a3b4 − 1 6ab34) = 0,W2757 = 1 2(b25 − 1 2a2b1 − 1 6ba12) = 0, W5767 = 1 2(b56 − 1 2a6b1 − 1 6ba16) = 0, W5787 = 1 2(b58 − 1 2a8b1 − 1 6ba18) = 0, W6575 = 1 2(a67 − 1 2a3b6 − 1 6ab36) = 0,W7585 = 1 2(a78 − 1 2a3b8 − 1 6ab38) = 0, W1575 = 1 12(7a17 − 4a3b1 − ab13 + b35) = 0, W3757 = 1 12(7b35 − 4a3b1 − ba13 + a17) = 0, W1757 = 2 3b15 + 1 6b26 + 1 6b48 − 1 12ab11 − 1 3a1b1 − 5 84b(a11 − b33) = 0, W4787 = 2 3b48 + 1 6b15 + 1 6b26 − 1 12ab11 − 1 12a1b1 − 1 84b(5b33 − 2a11) = 0, W2565 = 2 3a26 + 1 6a37 + 1 6a48 − 1 12ba33 − 1 12a3b3 − 1 84a(5a11 − 2b33) = 0, W3575 = 2 3a37 + 1 6a26 + 1 6a48 − 1 12ba33 − 1 3a3b3 − 5 84a(a11 + b33) = 0, we obtain, respectively, the following conditions: R7 = 1 2LM,Q5 = 1 2ST, N5 = 1 2SF, V5 = 1 2SD, F7 = 1 2LN,D7 = 1 2LV, K7 = P5 = 1 2LS, S5 = 1 2KS, L7 = 1 2LP. Step 3. It follows from the previous steps that the last two equations (4.1) reduce to the equations W5757 = 1 2b15 − 1 4a5b1 + 1 6b(a26 + a37 + a48) + 1 6a(b15 + b26 + b48) − 5 84ab(a11 + b33)− 1 12b(ba33 + a3b3)− 1 12a(ab11 + a1b1) = 0, W7575 = 1 2a77 − 1 4a3b7 + 1 6b(a26 + a37 + a48) + 1 6a(b15 + b26 + b48) − 5 84ab(a11 + b33)− 1 12b(ba33 + a3b3)− 1 12a(ab11 + a1b1) = 0. (4.11) From the first equation in (4.11), we conclude that S = S(x7), η5 = 1 2Sξ. (4.12) From the first equation in (4.12) and S5 = 1 2KS, we find S5 = 0, (4.13) that is, KS = 0. Similarly, from the second equation in (4.11) we conclude that L = L(x5), ξ7 = 1 2Lη. (4.14) From the first equation in (4.14) and L7 = 1 2LP , we obtain L7 = 0, (4.15) Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 33 M. Iscan, A. Gezer, and A. Salimov that is, LP = 0. Therefore, from (4.10), (4.13) and (4.15), we have that   a = x1K(x5, x7) + x2T (x5, x7) + x3L(x5) + x4R(x5, x7) + x6F (x5, x7) + x8D(x5, x7) + ξ(x5, x7) b = x1S(x7) + x2Q(x5, x7) + x3P (x5, x7) + x4M(x5, x7) + x6N(x5, x7) + x8V (x5, x7) + η(x5, x7), which finishes the proof. Let Rij and Sc denote the Ricci tensor and the scalar curvature of a Walker metric (2.1). It follows from the equation expression for the components of the curvature tensor that the Ricci operator 〈R̂ic(X), Y 〉 = Ric(X, Y ) satisfies R̂ic(∂1) = 1 2a11∂1 + 1 2b13∂3, R̂ic(∂2) = 1 2a12∂1 + 1 2b23∂3, R̂ic(∂3) = 1 2a13∂1 + 1 2b33∂3, R̂ic(∂4) = 1 2a14∂1 + 1 2b34∂3, R̂ic(∂6) = 1 2a16∂1 + 1 2b36∂3, R̂ic(∂8) = 1 2a18∂1 + 1 2b38∂3, R̂ic(∂5) = 1 2(−2a26−2a37−2a48+ba33 + a3b3)∂1 + 1 2a16∂2 +1 2(b35 − a3b1 + a17 − ba13)∂3 + 1 2a18∂4 + 1 2a11∂5 +1 2a12∂6 + 1 2a13∂7 + 1 2a14∂8, R̂ic(∂7) = 1 2(b35 − a3b1 + a17 − ab13)∂1 + 1 2b36∂2 + +1 2(−2b15 − 2b26 − 2b48 + ab11 + a1b1)∂3 +1 2b38∂4 + 1 2b13∂5 + 1 2b23∂6 + 1 2b33∂7 + 1 2b34∂8. (4.16) The result follows by a straightforward calculation. R e m a r k 5. Considering (4.16), we see that the locally conformally flat Walker metric satisfying (4.2) has a vanishing Ricci operator. R e m a r k 6. Considering (3.2) and (4.2), we see that the locally conformally flat Walker metric (4.2) has a vanishing scalar curvature. From Remark 6 we have R e m a r k 7. Locally conformally flat Walker metrics are semi-Riemannian manifolds with harmonic curvature [13]. 5. Osserman–Walker 8-manifolds Let Mn be an n-dimensional differentiable manifold of class C∞, CT (Mn) be its cotangent bundle, and π, the natural projection CT (Mn) → Mn. A system of local coordinates (U ;xi), i = 1, . . . , n in Mn induces on CT (Mn) a system of lo- cal coordinates (π−1(U);xi, xı̄ = pi), i = 1, . . . , n, ı̄ = n + i = n + 1, . . . , 2n, 34 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 Eight-dimensional Walker manifolds where xı̄ = pi is the cartesian coordinates of covectors p in each cotangent spaces CTx(Mn), x ∈ U with respect to the natural coframe { dxi } . We denote by =r s(Mn) (=r s( CT (Mn))) the module over F (Mn) (F (CT (Mn))) of C∞ tensor field of type (r, s), where F (Mn) (F (CT (Mn))) is the ring of the real-valued C∞functions of Mn (CT (Mn)). Let X = Xi ∂ ∂xi and ω = ωidxi be the local expression in U ⊂ Mn of a vector field X ∈ =1 0(Mn), and a 1-form ω ∈ =0 1(Mn), respectively. Then the horizontal lift HX ∈ =1 0( CT (Mn)) of X and the vertical lift V ω ∈ =1 0( CT (Mn)) of ω are given, respectively, by HX = Xi ∂ ∂xi + ∑ i phΓh ijX j ∂ ∂xı̄ (5.1) and V ω = ∑ i ωi ∂ ∂xı̄ (5.2) with respect to the natural frame { ∂ ∂xi , ∂ ∂xı̄ } , where Γh ij are the components of a symmetric (torsion-free) affine connection ∇ on Mn. We now consider a tensor field R∇ ∈ =0 2( CT (Mn)), whose components in π−1(U) are given by R∇ = (R∇JI) = ( −2phΓh ji δi j δj i 0 ) (5.3) with respect to the natural frame, where δi j denotes the Kronecker delta. The in- dices I, J,K, . . . = 1, . . . , 2n indicate the indices with respect to the natural frame{ ∂ ∂xi , ∂ ∂xı̄ } . This tensor field defines a pseudo-Rimannian metric in CT (Mn), and the line element of the pseudo-Riemannian metric R∇ is given by ds2 = 2dxiδpi, where δpi = dpi − phΓh jidxi. This metric is called the Riemannian extension of the symmetric affine connection ∇ [14, p. 268]. The complete lift of a vector field X ∈ =1 0(Mn) to the cotangent bundle CT (Mn) is defined by CX = Xi ∂ ∂xi − ∑ i ph∂iX h ∂ ∂xı̄ . (5.4) Using (5.3) and (5.4), we easily see that R∇(CX,C Y ) = −γ(∇XY +∇Y X), (5.5) where γ(∇XY +∇Y X) = ph(Xi∇iY h + Y i∇iX h). Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 35 M. Iscan, A. Gezer, and A. Salimov Since the tensor field R∇ ∈ =0 2( CT (Mn)) is completely determined by action on vector fields of type CX and CY (see Proposition 4.2 of [14, p. 237]), we have an alternative definition of R∇: the tensor field R∇ is completely determined by condition (5.5). From Theorem 1, we see that the metrics (3.3) can be viewed as the Rieman- nian extensions, i.e. they are locally isometric to the cotangent bundle CT (Mn) with metric R∇ + π∗G, where π∗G is the pull-back on CT (Mn) of a symmetric tensor field G ∈ =0 2(Mn). On the other hand, we can be able to state the PDEs in Theorem 1 in terms of the geometry of the corresponding torsion-free connec- tion on the 4-dimensional base manifold. The condition (3.4) in Theorem 1 is just the condition to be affine Osserman on the base 4-manifold [11]. In [15], it is proved that (CT (Mn), R∇) is a pseudo-Riemannian Osserman space if and only if (Mn,∇) is an affine Osserman space. Thus we have the following theorem. Theorem 3. A Walker metric (2.1) is Osserman if and only if it is Einstein with the metric given by (3.3) and (3.4). 6. Conclusions A Walker n-manifold is a semi-Riemannian manifold which admits a field of parallel null r planes with r ≤ n 2 . In this article, we study the curvature proper- ties of a Walker 8-manifold (M, g) which admits a field of parallel null 4-palnes. The metric g is necessarily of neutral signature (++++−−−−). In [5], the au- thors consider Goldberg’s conjecture but for the metrics with neutral signature. They initially display examples of almost Kahler–Einstein neutral structures on R8 such that the almost complex structure is not integrable. Then, they obtain the structures of the same type on the torus T 8. Therefore, it is proved that the neutral version of Goldberg’s conjecture fails. For such restricted Walker 8-manifolds, we study mainly the curvature properties, e.g., the conditions for a Walker metric to be Einstein, Osserman, or locally conformally flat, etc. One of our main results is the exact solutions to the Einstein equations for a restricted Walker 8-manifold. The Walker metrics (3.3) can be alternatively described in terms of Riemannian extensions. As a particular case, when we consider that the base manifold is a four-dimensional Walker manifold, the condition (3.4) in The- orem 1 is just the condition to be affine Osserman on the base four-dimensional Walker manifolds [11]. Our another main result relates to the Osserman property of the Walker metrics (3.3). Acknowledgement The authors are grateful to Professor E. Garcia Rio for valuable comments. The first and third authors acknowledge the support of the project 108T590 (TUBITAK, TURKEY). 36 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 Eight-dimensional Walker manifolds References [1] A.G. Walker, Canonical Form for a Rimannian Space with a Paralel Field of Null Planes. — Quart. J. Math. Oxford 1 (1950), No. 2, 69–79. [2] Y. Matsushita, Four-Dimensional Walker Metrics and Symplectic Structure. — J. Geom. Phys. 52 (2004), 89–99. [3] Y. Matsushita, Walker 4-Manifolds with Proper Almost Complex Structure. — J. Geom. Phys. 55 (2005), 385–398. [4] Y. Petean, Indefinite Kähler–Einstein Metrics on Compact Complex Surfaces. — Commun. Math. Phys. 189 (1997), 227–235. [5] Y. Matsushita, S. Haze, and P.R. Law, Almost Kähler–Einstein Structure on 8-Dimensional Walker Manifolds. — Monatsh. Math. 150 (2007), 41–48. [6] A. Chudecki and M. Przanowski, From Hyperheavenly Spaces to Walker and Os- serman Spaces: I Class. — Class. Quantum. Grav 25 (2008), 145010 (18pp). [7] A. Chudecki and M. Przanowski, From Hyperheavenly Spaces to Walker and Os- serman Spaces: II Class. — Class. Quantum. Grav 25 (2008), 235019 (22pp). [8] J. Davidov, J.C. Dı́az-Ramos, E. Garćıa-Rı́o, Y. Matsushita, O. Muškarov, and R. Vázquez-Lorenzo, Almost Kähler Walker 4-Manifolds. — J. Geom. Phys. 57 (2007), 1075–1088. [9] J. Davidov, J.C. Dı́az-Ramos, E. Garćıa-Rı́o, Y. Matsushita, O. Muškarov, and R. Vázquez-Lorenzo, Hermitian–Walker 4-Manifolds. — J. Geom. Phys. 58 (2008), 307–323. [10] E. Garćıa-Rı́o, S. Haze, N. Katayama, Y. Matsushita, Symplectic, Hermitian and Kähler Structures on Walker 4-Manifolds. — J. Geom. 90 (2008), 56–65. [11] M. Chaichi, E. Garćıa-Rı́o, and Y. Matsushita, Curvature Properties of Four- Dimensional Walker Metrics. — Class. Quantum. Grav 22 (2005), No. 3, 559–577. [12] V.V. Vishnevskii, A.P. Shirokov, and V.V. Shurygin, Spaces over algebras. Kazan- skii Gosudarstvennyi Universitet, Kazan, 1985. (Russian) [13] A. Derdzinski, On Compact Riemannian Manifolds with Harmonic Curvature. — Math. Ann. 259 (1982), 145–152. [14] K. Yano and S. Ishihara, Tangent and Cotangent Bundle. Marcel Dekker, New York, 1973. [15] E. Garćıa-Rı́o, D.N. Kupeli, M.E. Vázquez-Abal, R. Vázquez-Lorenzo, Affine Osser- man Connections and Their Riemann Extensions. — Diff. Geom. Appl. 11 (1999), No. 2, 145–153. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 1 37

Some Properties Concerning Curvature Tensors of Eight-Dimensional Walker Manifolds

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