In this paper, we discuss some examples of two dimensional spaces of constant curvature and their Riemannian extensions.Publisher's Versio

Harish, Dammu

Nagaraja, Halammanavar Gangadharappa

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.6, N.1, 2016, pp. 95-101.EXTENSIONS OF CONSTANT CURVATUREH. G. NAGARAJA1, D. HARISH1, §Abstract. In this paper, we discuss some examples of two dimensional spaces of con-stant curvature and their Riemannian extensions.Keywords: Riemannian curvature, Riemann extension, flat metric, geodesic equations,constant positive curature.AMS Subject Classification: 53C20, 53C21.1. IntroductionPatterson and Walker[2] have defined Riemann extensions and showed how a Riemann-ian structure can be given to the 2n dimensional tangent bundle of an n− dimensionalmanifold with given non-Riemannian structure.A Riemann extension provides a solution of the general problem of embedding a man-ifold M carrying a given structure in a manifold M ′ carrying another structure, the em-bedding being carried out in such a way that the structure on M ′ induces in a naturalway the given structure on M.The Riemann extension of Riemannian or non-Riemannian spaces can be constructedwith the help of the Christoffel coefficients Γijk of corresponding Riemann space or withconnection coefficients Πijk in the case of the space of affine connection. Here we discussexamples on flat spaces, spaces of constant positive and negative curvatures.In each case it may be noted that the Christoffel symbols and hence the geodesic equa-tions are extended. We give examples where extension of flat metric is flat and that ofextension of a metric of non zero constant curvature is not a metric of constant curvature.The theory of Riemann extensions has been extensively studied by Afifi [1], Dryuma,[3],[4], [5], [6].PreliminariesLet (M, g) be a Riemannian manifold and let gij be components of the metric tensor g.The metric g is of constant curvature ifRijkl = λ(gikgjl − gilgjk), (1)1 Department of Mathematics, Bangalore University, Bengaluru-560001, India.e-mail: hgnraj@yahoo.com, itsme.harishd@gmail.com;§ Manuscript received: February 12, 2015; Accepted: May 6, 2016.TWMS Journal of Applied and Engineering Mathematics, Vol.6, No.1; c© Işık University, Departmentof Mathematics 2016; all rights reserved.2 The second author is fully supported by CSIR 09/039(0106)2012-EMR-I.9596 TWMS J. APP. ENG. MATH. V.6, N.1, 2016where Rijkl are components of the Riemannian curvature tensor on M and λ is a constant.For a metricds2 = gijdxidxj , (2)the system of geodesic equations is given byd2xkds2+ Γkijdxidsdxjds= 0, (3)where Γkij are the Christophel symbols constructed from g.Γkij =12gkm(gmi,j + gmj,i − gij,m). (4)The Riemann extension of the metric (2) is given byds2 = −2Γkij(xl)ψkdxidxj + 2dψkdxk, (5)where ψk are the coordinates of additional space. The extended metric(5) has the follow-ing property. The geodesic equations of (5) consists of two parts.d2xkds2+ Γkijdxidsdxjds= 0 (6)andδ2ψkds2+Rlkjidxjdsdxidsψl, (7)whereδ2ψkds2=dψkds− Γljkψldxjds. (8)The system of equations (6) is the system of geodesic equations for geodesics of basic spacewith local coordinates xi and it does not contain the coordinates ψk.The system of equations (7) is a n × n linear matrix system of second order of the formgiven byd2ψds2+A(s)dψds+B(s)ψ = 0, (9)where A(s) and B(s) are matrices. The geometric properties of the basic space are pre-served by Riemann extension. For example, Riemann extension of flat space is flat. Wegive an example for this below. This gives us the possibility of the basic space to use thelinear system of equations for studying the properties of basic space.In section 1 we give an example of a flat space where geodesic parallel to an axis is trans-formed in to a curve with 4-components in the extended space. In section 2 we studythe metric similar to the metric of the 2-sphere embedded in three dimensional Euclideanspace, with constant radius. We study the properties of geodesics in the extended space.It will be shown that stationary points are transformed into geodesics with extendedcomponents changing at constant rate. It can further be noted that stationary extendedcomponents are not solutions to this system of equations except for the components beingzero. In section 3 we see that a line parallel to an axis can be transformed in to a curvewith the extended components related parabolically. Throughout this paper we use thesame notation for Christoffel symbols of base and extended spaces.H. G. NAGARAJA, HARISH D.:EXTENSIONS OF CONSTANT CURVATURE 972. Two dimensional flat metricConsider the metric of the form,ds2 = dx2 + x2dy2. (10)For this metric we have Γ122 = x , Γ212 =1x rest being zero.The geodesic equations ared2xds2− x(dyds)2= 0 (11)andd2yds2+1xdxdsdyds= 0. (12)Simplifying we get, (dxds)2= klogx (13)anddyds=cx. (14)On applying (5) we obtain the extended metric given byds2 = 4xPdy2 − 4Qxdxdy + 2dPdx+ 2dQdy, (15)where P and Q are the extended coordinates.For this extended metric, the Christoffel symbols are given by,Γ122 = x, Γ212 =1x, Γ312 =2Qx2, Γ322 = −P, Γ324 = −1x,Γ411 =2Qx2, Γ412 = −P, Γ414 = −1x, Γ422 = −2Q, Γ423 = x.The geodesic equations of the extended metric are(dxds)2= klogx, (16)dyds=cx, (17)d2Pds2+4Qx2dxdsdyds− P(dyds)2− 2xdydsdQds= 0 (18)andd2Qds2+2Qx2(dxds)2− 2pdxdsdyds− 2xdxdsdQds− 2Q(dyds)2+ 2xdydsdpds= 0.(19)Making k = 0, the system of equations reduces tox = m, (20)dyds=cm, (21)d2Pds2− c2Pm2− 2cmdQds= 0, (22)d2Qds2− 2Q( c2m2) + 2cdpds= 0, (23)98 TWMS J. APP. ENG. MATH. V.6, N.1, 2016where m and c are constants. This is a second order linear system and can be solved.The solution is lengthy though. The point of discussing such a direction is that straightline motions parallel to y-axis with constant velocity are transformed into motion withexponential characteristics in the extended components. Clearly, this is geodesical exten-sion and equation (15) is a flat metric which can be easily verified. If a metric is flat itsextension is also flat. But for a non zero constant metric this is not so. In fact there canbe no Riemann extension of constant curvature.3. two dimensional space of constant positive curvatureConsider two dimensional metricds2 =11 − kx2dx2 + x2dy2. (24)For this metric, we haveΓ111 =kx1 − kx2, Γ122 = −x(1 − kx2), Γ212 =1x.It is easy to check that metric (24) is a metric of constant curvature. On applying (3),the geodesic equations of this metric ared2xds2+kx1 − kx2(dxds)2− x(1 − kx2)(dyds)2= 0 (25)andd2yds2+2xdxdsdyds= 0. (26)On applying (5) to the metric (24), we get the extended metric asds2 =−4kxp1 − kx2dx2 + 4px(1 − kx2)dy2 − 4Qxdxdy + 2dxdp+ 2dQdy, (27)Γ111 =kx1 − kx2, Γ122 = −x(1 − kx2), Γ212 =1x, Γ311 = −kP(1 − kx2)2,Γ312 =2Qx2, Γ313 = −kx1 − kx2, Γ322 = 2kx2P − P, Γ324 = −1x,Γ411 =2Qx2(1 − kx2), Γ412 = −2kx2P, Γ414 = −1x, Γ422 = −2(1 − kx2)Q,Γ423 = x(1 − kx2).It may be noted that it is not a metric of constant curvature. For instance R1113 =2k2x2(1−kx2)2as obtained from usual calculations. But the right side of equation(1) is zero. The systemof geodesic equations ared2xds2+kx1 − kx2(dxds)2− x(1 − kx2)(dyds)2= 0, (28)d2yds2+2xdxdsdyds= 0, (29)d2Pds2− kP(1 − kx2)2(dxds)2+4Qx2dxdsdyds− 2kx1 − kx2dxdsdPds+ (5kx2P − 4P )(dyds)2− 2xdydsdQds= 0(30)H. G. NAGARAJA, HARISH D.:EXTENSIONS OF CONSTANT CURVATURE 99andd2Qds2+2Qx2(1 − kx2)(dxds)2− (P + 4kx2P )dxdsdyds− 2xdxdsdyds− 2Q(1 − kx2)(dyds)2+ 2x(1 − kx2)dydsdPds= 0.(31)Clearly, dxds = 0 anddyds = 0 form a solution to the systems of equations (28) and (29).Then (30) and (31) reduce tod2Pds2= 0, (32)d2Qds2= 0. (33)Solving (32) and (33), we get P = As+B, Q = Cs+D.This shows that the extended velocity components are constants. Further we note thatstationary points are transformed in to either stationary points or points moving withconstant extended velocity components.4. Two dimensional space of constant negative curvatureConsider two dimensional metric asds2 =dx2 + dy2y2. (34)HereΓ112 = −1y, Γ211 =1y, Γ222 = −1y.It can be easily verified that this is a metric constant negative curvature -1.Its geodesic equations ared2xds2− 2y(dxds)2= 0 (35)andd2yds2+1y(dxds)2− 1y(dyds)2 = 0. (36)On applying (5) to (34), the four dimensional Riemann extension is given byds2 = −4Qydx2 +4Pydxdy +4Qydy2 + 2dxdp+ 2dydQ, (37)where p and q are coordinates in the extended space. We haveΓ112 = −1y, Γ211 =1y, Γ222 = −1y, Γ311 = −2Py2,Γ312 = −Qy2, Γ314 = −1y, Γ323 =1y, Γ411 = −3Qy2,Γ412 =2Py2, Γ413 =1y, Γ422 =Qy2, Γ424 =1y.100 TWMS J. APP. ENG. MATH. V.6, N.1, 2016The system of geodesic equations of (37) ared2xds2− 2y(dxds)2= 0, (38)d2yds2+1y(dxds)2− 1y(dyds)2= 0, (39)d2Pds2− 2Py2(dxds)2 − 2Qy2dxdsdyds− 2ydxdsdQds+2ydydsdPds= 0 (40)andd2Qds2− 3Qy2(dxds)2+4Py2dxdsdyds+2ydxdsdPds+Qy2(dyds)2+2ydydsdQds= 0. (41)Clearly, x = c, a constant is a solution of (38) and the system reduces to the formx = c, (42)dyds= cy, (43)d2Pds2+ 2cdPds= 0 (44)andd2Qds2+ 2cdQds+ c2Q = 0. (45)Solving above system of equations, we getx = c, (46)y = Aecs, (47)P =−B2ce−2cs +D (48)and Q = (E + Fs)e−cs. (49)Here again, lines parallel to y-axis are transformed into curves in 4-dimensions. ForD = F = 0, we have P = kQ2, where k = − B2cE2. Thus in all three cases we notethat the geodesics in the extended space generally give very drastic behaviour thoughpreserving the properties of the base space.References[1] Z. Afifi, Riemann extension of affine connected spaces., Quart. J. Math. Oxford (2), 5 (1954),pp.312-30.[2] E. M. Paterson and A.G. Walker., Riemann extensions., Quart,J. Math. Oxford, 3(1952), pp.19-28.[3] V. S. Dyruma., Toward a theory of spaces of constant curvature, Theor. Math. Phys., 146, (2006),pp.42-54.[4] V. S. Dryuma. Mat.Issled.,124(1992), pp.56-68.[5] V. S. Dryuma On Geodesical Extension of the Schwarzschild Space time. Proceedings of Institute ofMathematics of NAS of Ukraine, 50(3),(2004), pp.1294-1300.[6] V. S. Dryuma On Riemann extension of the Schwarzschild metric. Buletinul Academiei De Stiinte ARepublich Moldova, Mathematica, 3(43), (2003), pp.92-103.[7] R. Jackiw, A pure cotton kink in a funny place math-ph/0403044(2004).[8] A.M. Naveira, The Riemann Curvature through History RACSAM, Rev. R. Acad. Cien. Serie A. Mat.,99(2), (2005), pp. 195-210.H. G. NAGARAJA, HARISH D.:EXTENSIONS OF CONSTANT CURVATURE 101Dr. H. G. NAGARAJA is Professor of Mathematics at the BangaloreUniversity. He has over 20 years of teaching experience of PG/UG classes.Dr. H. G. Nagaraja has published many research articles in the areas ofRiemannian geometry in reputed mathematical journals.Harish D. did his M. Sc. from Bangalore University. He is currently doinghis Ph.D. He is a senior research fellow of CSIR(NET). His research interestsare Riemannian Geometry and its applications.

Extensions of constant curvature

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Extensions of constant curvature

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