We study the properties of Modified Riemann extensions evolving under dissipative hyperbolic geometric flow with examples

Harish, D.

Nagaraja, H.G.

ePrints@Bangalore University

Communications in Mathematics and ApplicationsVol. 6, No. 2, pp. 55–60, 2015ISSN 0975-8607 (online); 0976-5905 (print)Published by RGN Publications http://www.rgnpublications.comDissipative Hyperbolic Geometric Flowon Modified Riemann Extensions Research ArticleH.G. Nagaraja* and Harish D.Department of Mathematics, Bangalore University, Central College Campus, Bengaluru 560001, India*Corresponding author: hgnraj@yahoo.comAbstract. We study the properties of Modified Riemann extensions evolving under dissipativehyperbolic geometric flow with examples.Keywords. Riemann extension; Evolution equationsMSC. 53C20; 53C44Received: September 2, 2015 Accepted: November 27, 2015Copyright © 2015 H.G. Nagaraja and Harish D. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.1. IntroductionDissipative hyperbolic geometric flow was introduced by Dai-Kong-Liu. It is a new geometrictool which was motivated by the well developed theory of the dissipative hyperbolic equations.We define it on general class of pseudo Riemann metrics and study the necessary and sufficientconditions for a Riemann extension under dissipative geometric flow to stay as a Riemannextension. More precisely we study the conditions under which the solutions of Dissipativegeometric flow on a Riemann extension are also Riemann extensions.Patterson and Walker [6] defined Riemann extensions and showed how a Riemannianstructure can be given to the 2n dimensional tangent bundle of an n-dimensional manifoldwith given non-Riemannian structure. This shows Riemann extension provides a solution ofthe general problem of embedding a manifold M carrying a given structure in a manifold M′carrying another structure, the embedding being carried out in such a way that the structure onM′ induces in a natural way the given structure on M . The Riemann extension of Riemannianor non-Riemannian spaces can be constructed with the help of the Christoffel coefficients Γijkof corresponding Riemann space or with connection coefficients Πijk in the case of the space of56 Dissipative Hyperbolic Geometric Flow on Modified Riemann Extensions: H.G. Nagaraja and Harish D.affine connection [4]. The theory of Riemann extensions has been extensively studied by Afifi [8].Modified Riemann extensions introduced recently in [1] and their properties we list briefly inthe next section.2. PreliminariesLet ∇ be a torsion-free affine connection of M . The modified Riemann extension of (M,∇) is thecotangent bundle T∗M equipped with a metric g¯ whose local components given byg¯ i j =−2ωlΓli j+ ci j, g¯ i j∗ = δ ji and g¯ i∗ j∗ = 0.The contravariant components areg¯i j = 0, g¯i j∗ = δ ji , g¯i∗ j∗ = 0 and g¯i∗ j∗ = 2ωlΓli j− ci jfor i, j ranging from 1 to n and i∗ , j∗ ranging from n+1 to 2n, where ωl are extendedcoordinates.We note following results for the connection coefficients on extended space,Γ¯ki j =Γki j ,Γ¯k∗i j =ωlR lk ji+12(∇i c jk+∇ j cik−∇hci j) ,Γ¯ki∗ j = 0, Γ¯k∗i∗ j =−Γijk, Γ¯ki∗ j∗ = 0, andΓ¯k∗i∗ j∗ = 0 .The components of the Riemann curvature tensor of the extended space are given byR¯ ijkl =R ijkl ,R¯ i∗jkl =12[∇ j(∇l cki−∇i ckl)−∇k(∇l c ji−∇i c jl)−Rmjkl cmi−Rmjki clm]+ωa(∇ jRailk−∇kRail j) ,R¯ i∗j∗kl =R jilk ,R¯ i∗jk∗l =−Rkil j, andR¯ i∗jkl∗ =R lk ji .The others are zero. i∗ , j∗ , k∗ , l∗ ranges from n+1 to 2n. We lower the index in the middleposition, to getRi jkl = gmkRmi jl . (2.1)Further, R¯i j = Ri j+R ji , R¯i∗ j = 0 and R¯i∗ j∗ = 0. Here bar is used for components of extendedspace. However in last section we do not use bar though they are components of modifiedRiemann extension as the calculations involve only the extended space.Communications in Mathematics and Applications, Vol. 6, No. 2, pp. 55–60, 2015Dissipative Hyperbolic Geometric Flow on Modified Riemann Extensions: H.G. Nagaraja and Harish D. 573. Main Results and DiscussionDefinition 3.1. Let M be a pseudo Riemannian Manifold with pseudo Riemannian metric g i j .Then the dissipative hyperbolic geometric flow is defined by∂2∂t2g i j =−2Ri j+2gpq∂g ip∂t∂g jq∂t−(λ+2gpq ∂gpq∂t)∂g i j∂t+ 1n−1((gpq∂gpq∂t)2+ ∂gpq∂t∂gpq∂t)g i j (3.1)where λ is a positive constant.Theorem 3.2. Dissipative hyperbolic geometric flow on Riemann extensions is given by∂2∂t2g i j =−2Ri j−λ∂g i j∂t. (3.2)Proof. In equation (3.1) the expression gpq ∂g ip∂t∂g jq∂t ,∂g ip∂t and∂g jq∂t to be non zero, i, p, j, q mustbe lesser than n.But then, gpq = 0 and hence the term is equal to zero. On similar lines we can simplify theexpression to get the result.It may be noted that when λ= 0, the equation (3.1) reduces to hyperbolic geometric flow[4].Though the Ricci tensor in equation (3.1) can be arbitrary chosen, to obtain a solution which isa modified Riemann extension, the Ricci tensor has to satisfy the following condition.Theorem 3.3. A Riemann extension under dissipative geometric flow to remain as Riemannextension, the Ricci tensor must satisfy the equation,∂3Rµν∂t3+3λ∂2Rµν∂t2+2λ2∂Rµν∂t= 0. (3.3)Proof. From elementary calculations it can be shown that the Ricci tensor for a modifiedRiemann extension is given byRµν = 12∂∂xα(gαm∂gµν∂xm)− 12∂∂xν(gαm∂gµα∂xm)− 14gαm gβn∂gµν∂xm∂gαβ∂xn+ 14gαm gβn∂gµβ∂xm∂gνα∂xn. (3.4)Differentiating partially with respect to ‘t’ and simplifying we get∂Rµν∂t=−12gαm∂3 gµα∂t∂xν∂xm+ 12gαm∂3 gµν∂t∂xα∂xm− 14gαm gβn∂gµν∂xm∂2 gαβ∂t∂xn− 14gαm gβn∂gαβ∂xn∂2 gµν∂t∂xm+ 14gαm gβn∂gµβ∂xm∂2 gνα∂t∂xn+ 14gαm gβn∂gνα∂xn∂2 gνβ∂t∂xm, (3.5)∂2Rµν∂t2= λ2gαm[∂3 gµα∂xν∂xm∂t− ∂3 gµν∂xα∂xm∂t]− 12gαm gβn∂2 gµν∂t∂xm∂2 gαβ∂t∂xn+ 12gαm gβn∂2 gµβ∂t∂xm∂2 gνα∂t∂xn+ λ4gαm gβn∂∂t(∂gµν∂xm∂gαβ∂xn)− λ4gαm gβn∂∂t(∂gµβ∂xm∂gνα∂xn). (3.6)Communications in Mathematics and Applications, Vol. 6, No. 2, pp. 55–60, 201558 Dissipative Hyperbolic Geometric Flow on Modified Riemann Extensions: H.G. Nagaraja and Harish D.and∂3Rµν∂t3=−λ22gαm∂3 gµα∂xν∂xm∂t+ λ22gαm∂3 gµν∂xα∂xm∂t+ 3λ2gαm gβn∂2 gµν∂t∂xm∂2 gαβ∂t∂xn− λ24gαm gβn∂∂t(∂gαβ∂xn∂gµν∂xm)− 3λ2gαm gβn∂2 gµβ∂t∂xm∂2 gνα∂t∂xn+ λ24gαm gβn∂∂t(∂gνα∂xn∂gµβ∂xm). (3.7)Clearly,∂3Rµν∂t3−λ2∂Rµν∂t= 3λ2gαm gβn(∂2 gµν∂t∂xm∂2 gαβ∂t∂xn− ∂2 gµβ∂t∂xm∂2 gνα∂t∂xn)(3.8)and∂2Rµν∂t2+λ∂Rµν∂t= 12gαm gβn[∂2 gµβ∂t∂xm∂2 gνα∂t∂xn− ∂2 gµν∂t∂xm∂2 gαβ∂t∂xn]. (3.9)From equations (3.8) and (3.9) we get∂3Rµν∂t3−λ2∂Rµν∂t∂2Rµν∂t2+λ∂Rµν∂t=−3λ . (3.10)Hence∂3Rµν∂t3+3λ∂2Rµν∂t2+2λ2∂Rµν∂t= 0 . (3.11)Solving equation (3.11) we getRµν = Aµνe−2λt+Bµνe−λt+Cµν (3.12)where Aµν,Bµν,Cµν are arbitrary tensor components for µ,ν≤ n and zero otherwise. Thus wehave obtained the general form, the Ricci tensor must have for a modified Riemann extensionunder dissipative hyperbolic geometric flow to stay as modified Riemann extension.Using this we now obtain the solution to equation (3.2).Theorem 3.4. A modified Riemann extension under dissipative hyperbolic geometric flow stayingas modified Riemann extension, the metric is given by,gµν =2Aµνλ2(e−λt+λte−λt−1)+ 2Bµνλ2(e−λt− 12e−2λt− 12)− (e−λt−1)(2Cµνλ+ 1λ∂∂tgµν(0))+ gµν(0) . (3.13)Communications in Mathematics and Applications, Vol. 6, No. 2, pp. 55–60, 2015Dissipative Hyperbolic Geometric Flow on Modified Riemann Extensions: H.G. Nagaraja and Harish D. 59Proof. Substituting equation (3.12) in (3.2) we get,∂2∂t2gµν+λ∂gµν∂t+2Aµνe−λt+2Bµνe−2λt+2Cµν = 0. (3.14)Integrating we get,∂gµν∂t+λgµν = 2Aµνe−λtλ+ Bµνte−2λtλ−2Cµνt−Fµν (3.15)which is a first order differential equation solving and substituting the initial conditions we getthe required result.4. ConclusionThus we have obtained the general solution for dissipative hyperbolic geometric flow on modifiedRiemann extension. It must be noted that for dissipative hyperbolic geometric flow on pseudoRiemann metric, the existence and uniqueness of solution must be proved first which is adifficult problem. But restricting the flow to modified Riemann extensions, we have solved theproblem.AcknowledgementThis work is supported by CSIR 09/039(0106)2012-EMR-I.Competing InterestsThe authors declare that they have no competing interests.Authors’ ContributionsAll the authors contributed significantly in writing this article. The authors read and approvedthe final manuscript.References[1] E. Calvin¯o-Louzao et al., The geometry of modified Riemannian extensions, Proc. R. Soc. Lond. Ser.A Math. Phys. Eng. Sci. 465 (2107) (2009), 2023–2040.[2] E. Calvin¯o-Louzao, E. García-Río and R. Vázquez-Lorenzo, Riemann extensions of torsion-freeconnections with degenerate Ricci tensor, Canad. J. Math. 62 (5) (2010), 1037–1057.[3] W.-R. Dai, D.-X. Kong and K. Liu, Dissipative hyperbolic geometric flow, Asian J. Math. 12 (3) (2008),345–364.[4] W.-R. Dai, D.-X. Kong and K. Liu, Hyperbolic geometric flow (I): short-time existence and nonlinearstability, Pure Appl. Math. Q. 6 (2) (2010), Special Issue: In honor of Michael Atiyah and IsadoreSinger, 331–359.[5] V.S. Dryuma, Teoret. Mat. Fiz. 146 (1) (2006), 42–54; translation in Theoret. and Math. Phys. 146 (1)(2006), 34–44.[6] L.P. Eisenhart, Fields of parallel vectors in Riemannian space, Ann. of Math. 39 (2) (1938), 316–321.Communications in Mathematics and Applications, Vol. 6, No. 2, pp. 55–60, 201560 Dissipative Hyperbolic Geometric Flow on Modified Riemann Extensions: H.G. Nagaraja and Harish D.[7] A. Gezer, L. Bilen and A. Cakmak, Properties of modified Riemannian extensions, arXiv:1305.4478v2 [math.DG] 26 May 2013.[8] D.-X. Kong and K. Liu, Wave character of metrics and hyperbolic geometric flow, J. Math. Phys. 48(10) (2007), 103508, 14 p.[9] O. Kowalski and M. Sekizawa, The Riemann extensions with cyclic parallel Ricci tensor, Math.Nachr. 287 (8-9) (2014), 955–961.[10] W. Lu, Evolution equations of curvature tensors along the hyperbolic geometric flow, Chin. Ann.Math. Ser. B 35 (6) (2014), 955–968.[11] E.M. Patterson and A.G. Walker, Riemann extensions, Quart. J. Math., Oxford Ser. 3 (2) (1952),19–28.[12] A.G. Walker, Canonical form for a Riemannian space with a parallel field of null planes, Quart. 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Dissipative Hyperbolic Geometric Flowon Modified Riemann Extensions

Dissipative Hyperbolic Geometric Flow on Modified Riemann Extensions

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Dissipative Hyperbolic Geometric Flowon Modified Riemann Extensions

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