Arif Salimov

Kursat Akbulut

Sibel Turanli

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 351 (2013) 837–839Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comDifferential geometryOn an isotropic property of anti-Kähler–Codazzi manifoldsSur une propriété isotrope des variétés anti-Kähler–CodazziArif Salimov a, Kursat Akbulut a, Sibel Turanli ba Ataturk University, Faculty of Science, Dep. of Mathematics, 25240, Turkeyb Erzurum Technical University, Faculty of Science, Dep. of Mathematics, Erzurum, Turkeya r t i c l e i n f o a b s t r a c tArticle history:Received 15 June 2013Accepted after revision 27 September 2013Available online 7 November 2013Presented by the Editorial BoardWe give a proof of the fact that an anti-Kähler–Codazzi manifold reduces to an isotropicanti-Kähler manifold if and only if the Ricci tensor field coincides with the Ricci* tensorfield.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous donnons une preuve du fait qu’une variété de type anti-Kähler–Codazzi se réduità une variété isotrope du même type si et seulement si le champ de tenseurs de Riccicoïncide avec le champ de tenseurs de Ricci*.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionIn [3] we introduced the notion of an anti-Kähler–Codazzi manifold, for which a twin anti-Hermitian (Norden) metricsatisfies the Codazzi equation. Such a structure gives rise to a new class of integrable anti-Hermitian structures, and weemphasize the importance of the Ricci and associated Ricci* tensor fields in the study of these manifolds. In this paper, weextend this study to other property of anti-Hermitian geometry, such as the isotropicity of anti-Hermitian structures.We begin by collecting some basic materials that we need later. Let (M, J ) be a 2n-dimensional almost complex manifold,where J denotes its almost complex structure. We denote by rs(M) the module of all tensor fields of type (r, s) on M .A semi-Riemannian metric g of neutral signature (n,n) is an anti-Hermitian (Norden) metric if:g( J X, Y ) = g(X, J Y )for any X, Y ∈ 10(M). An almost complex manifold (M, J ) with an anti-Hermitian metric is referred to as an almost anti-Hermitian manifold. An anti-Kähler (Kähler–Norden) manifold can be defined as a triple (M, g, J ), which consists of asmooth manifold M endowed with an almost complex structure J and an anti-Hermitian metric g such that ∇ J = 0, where∇ is the Levi-Civita connection of g . It is well known that the condition ∇ J = 0 is equivalent to the C-holomorphicity(analyticity) of the anti-Hermitian metric g [2] (see p. 76), i.e. Φ J g = 0, where Φ J is the Tachibana operator [4]:(Φ J g)(X, Y , Z) = (L J X g − L X G)(Y , Z), and G(Y , Z) = (g ◦ J )(Y , Z) = g( J Y , Z) is the twin anti-Hermitian metric.E-mail addresses: asalimov@atauni.edu.tr (A. Salimov), kakbulut@atauni.edu.tr (K. Akbulut), sibelturanli@hotmail.com (S. Turanli).1631-073X/$ – see front matter © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2013.09.020838 A. Salimov et al. / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 837–8392. Statement of the resultLet now (M, g, J ) be an almost anti-Hermitian manifold. Then the pair ( J , g) defines, as usual, the twin anti-Hermitianmetric G(Y , Z) = (g ◦ J )(Y , Z) = g( J Y , Z). If the twin metric G satisfies the Codazzi equation:(∇X G)(Y , Z) − (∇Y G)(X, Z) = 0 (1)or equivalently if the almost complex structure J satisfies:(∇X J )Y − (∇Y J )X = 0for any X, Y ∈ 10(M), then the triple (M, J , g) is called an anti-Kähler–Codazzi manifold (or AKC-space) [3]. Anti-Kähler–Codazzi manifolds are integrable almost anti-Hermitian manifolds (see [3]).It is well known that the inner product in the vector space can be extended to an inner product in the tensor space. Infact, if T and L are tensors of type (r, s) with components T i1...irj1... js and Lk1...krl1...ls, then:g(T , L) = gi1k1 . . . girkr g j1l1 . . . g jsls T i1...irj1... js Lk1...krl1...ls.If T = L = ∇ J ∈ 12(M), then the square norm ‖∇ J‖2 of ∇ J is defined by:‖∇ J‖2 = gij gkl gms(∇ J )mik(∇ J )sjl.An almost anti-Hermitian structure (M, g, J ) is said to be isotropic anti-Kähler if ‖∇ J‖2 = 0. The notion of isotropic Kählerstructure is originally introduced in [1]. Some examples of isotropic anti-Kähler structures were given in [2]. From definitionof isotropic anti-Kähler we have ∇ J = 0, in general. Conversely, from property ∇ J = 0, we immediately see that ‖∇ J‖2 = 0,i.e. the anti-Kähler manifold is isotropic anti-Kähler.Let now R(X, Y ) = [∇X ,∇Y ]−∇[X,Y ] be the curvature operator of the Levi-Civita connection ∇ on an anti-Kähler–Codazzimanifold. Then the Ricci tensor S is defined as S(X, Y ) = trace{Z → R(Z , X)Y }. The Ricci* tensor field ∗S of the anti-Kähler–Codazzi manifold is locally defined by:∗Sij = −RhjtlGlh J ti ,where Glh = J ls gsh(GisGsj = −δ ji ) and Rhjtl are the covariant components of curvature tensor R . The fact that if an anti-Kähler–Codazzi manifold is anti-Kähler (∇ J = 0), then S = ∗S , is proved in [3]. The main result of this paper is the following.Theorem 1. An anti-Kähler–Codazzi manifold is isotropic anti-Kähler (‖∇ J‖2 = 0) if and only if S = ∗S, where S and ∗S are the Ricciand Ricci* tensor fields, respectively.3. Proof of the theoremEq. (1) locally is equivalent to:∇kGij − ∇i Gkj = 0.From here, using contraction with Gij , we find:(∇i Gkj)Gij = 0 (2)by virtue of (∇kGij)Gij = 0. In fact, since Gij Gij = −δii = −2n, we have:(∇kGij)Gij + Gij∇kGij = 0,(∇kGij)Gij + J si gsj∇k(J it gt j) = (∇kGij)Gij + J sj gis∇k(J it gt j)= (∇kGij)Gij + J sj gt j∇k(J it gis) = (∇kGij)Gij + Gst∇kGst= 2(∇kGij)Gij = 0.Applying ∇l to Eq. (2), we obtain:(∇l∇i Gkj)Gij + (∇kGij)∇lGi j = 0 (3)by virtue of (1). Using the contraction with gkl , from (3) we have:A. Salimov et al. / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 837–839 839gkl(∇l∇i Gkj)Gij + gkl(∇kGij)∇lGi j =(∇k∇kGij)Gij + gkl(∇k(gis Jsj))∇l(git J jt)= (∇k∇kGij)Gij + gkl gis(∇k J sj)(∇l(gt j J it))= (∇k∇kGij)Gij + gkl gis gt j(∇k J sj)(∇l J it)= (∇k∇kGij)Gij + ‖∇ J‖2 = 0, (4)where ‖∇ J‖2 is the square norm of ∇ J . In an anti-Kähler–Codazzi manifold, the complex structure J satisfies (see [3]):∇h∇ j J hi =(S jk −∗S jk)J ki ,which is equivalent to the following equation:∇k∇kGij =(S jk −∗S jk)J ki (5)by virtue of:∇h∇ j J hi = δsh∇s∇ j(Gik gkh) = gks∇s∇ j Gik = ∇k∇kG ji .Substituting (5) into (4), we find:(S jk −∗S jk)g jk = ‖∇ J‖2.This means that a necessary and sufficient condition for an anti-Kähler–Codazzi manifold to reduce to an isotropic anti-Kähler manifold is that S = ∗S . Thus the proof is complete.AcknowledgementThe paper was supported by TUBITAK project TBAG-112T111.References[1] E. García-Río, Y. Matsushita, Isotropic Kähler structures on Engel 4-manifolds, J. Geom. Phys. 33 (2000) 288–294.[2] A. Salimov, Tensor Operators and Their Applications, Nova Science Publishers, New York, 2012.[3] A. Salimov, S. Turanli, Curvature properties of anti-Kähler–Codazzi manifolds, C. R. Acad. Sci. Paris, Ser. I 351 (5–6) (2013) 225–227.[4] S. Tachibana, Analytic tensor and its generalization, Tohoku Math. J. 12 (2) (1960) 208–221.

On an isotropic property of anti-Kähler–Codazzi manifolds

Salimov, Arif

Akbulut, Kursat

Turanli, Sibel

Numérisation de Documents Anciens Mathématiques

http://www.numdam.org/item/10.1016/j.crma.2013.09.020.pdf

On an isotropic property of anti-Kähler–Codazzi manifolds

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