In this paper we consider a two dimensional Wagner space of
Douglas type with zero curvature scalar, and we give the main scalar
function of this space

Szilágyi, Brigitta

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODICA POLYTECHNICA SER. MECH. ENG. VOL. 47, NO. 1, PP. 75–80 (2003)SOME PROBLEMS IN WAGNER SPACES WITH VANISHINGDOUGLAS TENSORBrigitta SZILÁGYIDepartment of GeometryInstitute of MathematicsBudapest University Technology and EconomicsH–1521 Budapest, HungaryReceived: November 5, 2002AbstractIn this paper we consider a two dimensional Wagner space of Douglas type with zero curvature scalar,and we give the main scalar function of this space.Keywords: two-dimensional Finsler spaces, Wagner space, Douglas tensor, projective change.1. IntroductionIn 1943 WAGNER [1] gave a generalization of the concept of BERWALD space by in-troducing a new connection with the surviving (h)h-torsion. Recently HASHIGUCHI[2] established an exact formulation of such a concept based on a theory of Finslerconnections developed by M. MATSUMOTO. Throughout the present paper, weshall use the terminology and definitions described in MATSUMOTO’s monograph[3].Definition 1 Let Fn be a Finsler space with a fundamental function L(x, y), yi :=ẋ i and let gij (x, y) be the fundamental tensor of Fn . There exists a unique Finslerconnection W(s) = (Fijk, Nij , Cijk) satisfying the following four conditions. ThisW(s) is called the Wagner connection with respect to si .(1) It is h- and v-metrical: gij ||k = 0 and gij∥∥k = 0, where the symbols ‖ and∥∥∥mean the covariant derivatives in Fn.(2) The (h)h-torsion tensor Tijk(= Fijk − Fikj ) is defined by T ijk = δij sk − δiks j ,where si is a given covariant vector field the components of which are func-tions of position alone.(3) The (v)v-torsion tensor Sijk(= Cijk − Cikj ) vanishes.(4) The deflection tensor Dij (= yγ Fiγ j − Nij ) vanishes.76 B. SZILÁGYIIf we denote by C = (∗ij k, ∗i0 j , Cijk), the Cartan connection of Fn , the aboveCijk of W(s) are nothing, but those of C and the difference Dijk = Fijk −∗ij k aregiven byDijk = L2(Sijkr +CijsCskr )sr +(yiC jkr −y jCikr −ykCijr)+Cijks0+g jksi −δiks j , (1)where si = gij s j and Sijkr is the v-curvature tensor of C. Throughout this workthe subscript 0 stands for contraction by yi .Definition 2 If a Wagner connection W(s) of a Finsler space Fn is linear, namelythe connection coefficients Fijk of W(s) are functions of position xi alone, Fn iscalled a Wagner space with respect to the vector field si(x).We have many interesting results concerned with Wagner spaces.Theorem 1 (HASHIGUCHI [2]) A Finsler space is a Wagner space with respect toa vector field si (x), if and only if the C-tensor Chi j satisfies Chi j‖k = 0 (Covariantderivative with respect to W(s)).M. MATSUMOTO [4] found all the Wagner spaces of dimension two.In the present paper we shall restrict our consideration to two-dimensionalWagner spaces so we drop some words about Berwald frame.The special and useful Berwald frame was introduced and developed byBERWALD [5], [8]. We study two dimensional Finsler space and define a localfield of orthonormal frame (l, m) called the Berwald frame. We give a normalizedsupporting element li = yiL and another further on let be given the fundamentaltensor with the following equation:gij = li l j + mim j .In the present paper we give an example for Wagner–Douglas space in the twodimensional case, and we determine its main scalar. We will use the followingnotions:l i = yiL,li = ∂̇i L , where ∂̇i = ∂∂yi,hij = L ∂̇i ∂̇ j L ,gij = li l j + hij .Since the angular metric tensor hij has the matrix (hij ) of rank n −1, we can definethe vector m = (m1, m2) by hij = εmi m j , ε = ±1. (The sign ε is called thesignature of F2.)SOME PROBLEMS IN WAGNER SPACES 77Then we getgij = li l j + εmi m j ,det(gij ) = ε(l1m2 − l2m1)2.The C-tensor (Cijk = ∂̇i ∂̇ j ∂̇k( L24 )) has no components in the direction li (Cijk yi =0). The tensor Cijk is written using the frame (l, m) in the following formula:LCijk = I mim j mk .The scalar field I is called the main scalar of F2.Now we define the covariant differentiations in F2.Denote by ; , . and |,∣∣∣ the covariant differentiations with respect to the Berwald con-nection B(Gijk , Gij , 0) and with respect to the Cartan connection C(∗ij k , Gij , Cijk)respectively. Then for scalar field S(x, y) we get the following derivations:S;i = S|i = ∂i S − (∂̇r S)Gri ,S.i = S∣∣i = ∂̇i S.We write S|i and L S∣∣i in the frame (l, m) as follows:S|i = S,1li + S,2mi ,L S∣∣i = S;1li + S;2mi .(S,1, S,2) and (S;1, S;2) are called the h- and the v-scalar derivatives of S.The commutation formulæ for scalar derivatives are written in the form{(1) S,1,2 − S,2,1 = −RS;2,(2) S,1;2 − S;2,1 = S,2,(3) S,2;2 − S;2,2 = −ε(S,1 + I S.2 + I,1S;2),(2)where R is called curvature scalar of F2.Finally the Bianchi identities for an F2 reduce to the following identity:I,1,1 + RI + εR;2 = 0. (3)As it is well known, the Berwald connection coefficients Gij , Gijk can be derivedfrom the function Gi , namely Gij = Gi. j and Gijk = Gi. j.k , whereGi = 14gis(yr(∂L2.s∂xr)− ∂L2∂xs).Let us consider two Finsler spaces: Fn(Mn, L) and F̂n(Mn, L̂) on a commonunderlying manifold Mn .78 B. SZILÁGYIDefinition 3 The change L(x, y) → L̂(x, y) of metrics is called projective and Fnis projective to F̂n if any geodesic of Fn is a geodesic of F̂n as a point set and viceversa.Definition 4 A Finsler space is called projectively flat, if it has a covering bycoordinate neighborhoods in which it is projective to a locally Minkowski space.From Ghjk = Gi.h. j.k we get a projective invariant Dih jk called the Douglastensor [5], [8], [7]:Dih jk = Gihjk =1n + 1(yi Ghj.k + δihG jk + δij Gkh + δik Ghj ),where Ghj = Grhjr and Ghj.k = ∂̇k Ghj . In particular the Dih jk of a two dimensionalFinsler space F2 can be written in the form [10]:3LDih jk = −(6I,1 + εI2;2 + 2I I2)mhlim j mk, (4)whereI2 = I,1;2 + I,2.Thus there arises an interesting question: Can we give the main scalar of a Wagnerspace of Douglas type in an exact formula?2. The Main Scalar of a Special Wagner Space of Douglas TypeFurther on we use the following results:Theorem [8] If Fn is an n-dimensional, projectively flat Finsler space, thenthe v(h)-torsion tensor Whi j and the projective v(h)-curvature tensor Dhi jk are zeroidentically.It is a well known result that the Douglas tensor and Weyl tensors vanishidentically in a projectively flat Finsler space [6].MATSUMOTO proved [7]: Whi j = 0 and Dhi jk = 0 imply:(1) Hijk = 0 and Kij = 0, if the dimension is higher than two.(2) The two dimensional case: Hijk = 0, where Hijk = Hi. j.k + G jk;i , Hi =L(3Rli + R.2mi), andG jk = I2m j mk/L . (5)Let us consider a two dimensional Wagner space with vanishing Douglas tensorwith the following assumption R = 0.From R = 0 we get Hi = 0.In the case of n = 2, from the theorems above we have Gjk;i = 0.SOME PROBLEMS IN WAGNER SPACES 79Now, using the following formulas:mi, j = l i; j mi + l imi; j = 0,l imi = 0,l ii j = 0,we finally obtain by the help of (5):G jk;i = I2;i m j mkL , that is I2;2 = 0.We obtain from (4):3I,1 + I I2 = 0. (6)If we use MATSUMOTO’s conditions [4] for the Wagner spaces,I,1 = I;2s2,I,2 = −I;2(s1 + I s2),(s1);2 − s2 = 0,(s2);2 + s1 + I s2 = 0,(s1);1 = 0,(s2);1 = 0,then we get the following equations:I;2,1 = I;2;2s2, (7)I2 = I;2;2s2 − 2I;2(s1 + I s2). (8)Using by the Bianchi identity the Eqs. (7) and (8) we have:I;2;2s2 = − I;2s2,1s2, s2 = 0. (9)(If s2 = 0, then a two dimensional Wagner space is a Landsberg space. We knowthat an F2 Finsler space is a Landsberg if and only if I,1 = 0 [4].)Substituting (9) into (8), then from (8) followsI2 = − I;2s2,1s2 − 2I;2(s1 + I s2). (10)Now (6) leads toI;2(3s2 − 2I (s1 + I s2) − I s2,1s2 ) = 0. (11)If I;2 = 0, then a Wagner space is a Berwald space [4], [9].80 B. SZILÁGYIIf we put I;2 = 0, then (11) implies3s2 − 2I (s1 + I s2) − I s2,1s2= 0. (12)This is a quadratic equation. Consequently the main scalar is written asI =−(2s1 + s2,1s2 ) ±√4s21 + 4s1s2,1s2 +s22,1s22+ 24s224s2. (13)In general for the main scalar in (13), I;2 = 0. Thus we have the following:Theorem 2 The main scalar of a two dimensional Wagner space of Douglas typewith the assumption R = 0 can be given in the formula (13).References[1] WAGNER, V., On Generalized Berwald Spaces, C.R. (Doklady) Acad. Sci. URSS, 39 (1943),pp. 3–5.[2] HASHIGUCHI, M., On Wagner’s Generalized Berwald Spaces, J. Korean Math. Soc., 12 (1975),pp. 51–61.[3] MATSUMOTO, M., Foundations of Finsler Geometry and Special Finsler Spaces, KaiseishaPress, Saikawa 3-32-4, Otsu-Shi, Shiga-Ken 520, Japan.[4] MATSUMOTO, M., On Wagner’s Generalized Berwald Spaces of Dimension Two, Tensor, N.S., 36 (1982), pp. 303–311.[5] BERWALD, L., On Cartan and Finsler Geometries, III, Two Dimensional Finsler Spaces withRectilinear Extremal, Ann. of Math., 42 No. 2 (1941), pp. 84–122.[6] ANTONELLI, P. L. – INGARDEN, R. S. – MATSUMOTO, M., The Theory of Sprays andFinsler Spaces with Applications in Physics and Biology, FTP 58, Kluver Academic Publishers,Dordrecht-Boston-London, 1993.[7] MATSUMOTO, M., Projective Changes of Finsler Metrics and Projectively Flat Finsler Spaces,Tensor, N. S., 34 (1980), pp. 303–315.[8] BERWALD, L., Über Finslersche und Cartansche Geometrie IV., Projektiv-krümmung allge-meiner Räume skalarer Krümmung, Ann. of Math., 48 (1947), pp. 755–784.[9] MATSUMOTO, M., On Three-Dimensional Finsler Spaces Satisfying the T - and B P-Conditions, Tensor, N. S., 29 (1975), pp. 13–20.[10] BÁCSÓ, S. – MATSUMOTO, M., Reduction Theorems of Certain Landsberg Spaces to BerwaldSpaces, Publicationes Mathematicæ, 48 No. 3–4, (1996), pp. 357–366.

Some problems in Wagner spaces with vanishing Douglas tensor

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Some problems in Wagner spaces with vanishing Douglas tensor

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