A THEORY OF THREE DIMENSIONAL FINSLER SPACES IN TERMS OF SCALARS AND ITS APPLICATIONS

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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLV, s.i a, Matematică, 1999, f.1. A THEORY OF THREE DIMENSIONAL FINSLER SPACES IN TERMS OF SCALARS AND ITS APPLICATIONS BY MAKOTO MATSUMOTO In 197 it has been proved by the present author [5] that a C reducible and Landsberg space F n, n 3, is reduced to a Berwald space ([8], Theorem 30.4). Next year he with HASHIGUCHI and HŌJŌ [1] found two dimensional Landsberg spaces with (α, β) metric, but their second paper [] has showed that those spaces are really Berwald spaces. On the other hand, he with SHIBATA [11] has proved that a semi C reducible and Landsberg space F n, n 4, satisfying nλ n + 1 (cf. 9) is a Berwald space. This is quite remarkable fact, because all of spaces with (α, β) metric and C i 0 are semi C reducible [9]. As a consequence we have no information yet on three dimensional semi C reducible and Landsberg spaces. For the three dimensional case we have an interesting and useful tool, the Moór frame [1],[6],([8], 9). The paper [6] was to have cultivated deep and long friendship between Professor Dr. Radu Miron and the present author ([7], p.vi). Since the (α, β) metric is not always positive definite, we have to generalize the theory based on the Moór frame to three dimensional Finsler spaces with any signature for our present purpose. The first eight sections are devoted to such subject and the scalar forms of the Bianchi identities are necessary for the detailed discussions. It gives the present author great pleasure to dedicate Prof. Miron the new version of the memorable paper [6]. Dedicated to Professor Dr. Radu Miron on his 70th Birthday

2 116 MAKOTO MATSUMOTO 1. Scalar components. Let F 3 = {M 3, L (x, y)} be a three dimensional Finsler space equipped with fundamental function L (x, y). We have the basic tensor fields of F 3 ; the normalized supporting element l i, its covariant components l i, the fundamental tensor g ij, the angular metric tensor h ij, the C tensor C ijk and the C vector C i, defined by l i = yi L, l i = i L, g ij = i j ( L / ), h ij = L i j L, C ijk = i j k ( L /4 ), C i = g jk C ijk, where i = / y i and (g jk ) is the inverse matrix of (g jk ). Throughout the present paper, suppose that the metric of F 3 be not restricted to be positive definite and F 3 be strongly non Riemannian [10], that is, g ij C i C j = ε (C), ε = ±1, C > 0. Thus a unit vector m i = C i /C can be introduced and is obviously independent of l i from m i l i = 0. Further it is observed from h ij = g ij l i l j that (h ij ε m i m j ) l j = (h ij ε m i m j ) m j = 0, which show that the matrix (h ij ε m i m j ) is of rank one and so we have ε 3 and n i such that h ij ε m i m j = ε 3 n i n j, ε 3 = ±1. Therefore (l i, m i, n i ) as above contitute an orthonormal frame field {e i α) } = = {l i, m i, n i }, which is called the Moór frame [6], ([8], 9). Thus we have g ij = l i l j + ε m i m j + ε 3 n i n j. The essential relation among e i α), α = 1,, 3, is written in the form (1.1) g ij e i α) ej β) = g αβ, (g αβ ) = ε ε ε 3 The first sign ( ε ) 1 is, of course, equal to +1 from l i l j = 1. Let and ( g αβ) be the inverse matrices of e α) i ( e i α) ) and (g αβ ) respectively and put e α)i = g ij e j α), eα)i = g ij e α) j. Then the following relations are easily derived: e α) i = g αβ e β)i, e α)i = g αβ e i β),

3 3 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 117 and we have (1.) {e α) i } = {l i, ε m i, ε 3 n i }, {e α)i } = {l i, ε m i, ε 3 n i }. Now we consider, for instance, a tensor T i jk of (1,) type and define Then we have T α βγ = T i jke α) i e j β) ek γ). T i jk = T α βγe i α) eβ) j eγ) k. Thus any tensor can be written in terms of scalars Tβγ α with respect to the Moór frame. Tβγ α are called the scalar components of T jk i. Example 1. The generalized Kronecker s deltas δ λµν αβγ, α,..., ν =1,, 3, give the symbols δ αβγ = δαβγ 13 and δλµν = δ λµν 13, which define the so called ε tensors: ε ijk = δ αβγ e α) i e β) j eγ) k, εijk = δ λµν e i λ) ej µ) ek ν). Thus δ αβγ are scalar components of ε ijk. Put E = det(e i α) ) and g = det (g ij). Then (1.1) gives ge = ε 1 ε ε 3 and and similarly ε ijk = δ ijk /E. e i 1) e j 1) e k 1) ε ijk = e i ) e j ) e k ) = δ ijk E, e i 3) e j 3) e k 3) Lemma 1. 1) Let T i jk be components of (1,)-type tensor and T α βγ its scalar components. The scalar components of the transvection T i j0=t i jky k are equal to LT α β1. ) T ijk = g ir T r jk has the scalar components g α T βγ.. Connection vectors. In the present paper the space F 3 is mainly to be equipped with the Carten connection CΓ = {Γ j i k, G i i j, C j k }. Hence we have g ij k = g ij = 0, where and denote the h and v covariant k differentiations in CΓ ([8], 17).

4 118 MAKOTO MATSUMOTO 4 The h covariant derivatives e i α) j of ei α) of the Moór frame are denoted by (.1) e i α) j = H α β γe i β) eγ) j. β It is remarked that α of H α γ shows only the number α of e i α). Then we have e α)i j = e k α) j g δ ik = (H α γ e k δ) eγ) j )(g βσe β) i e σ) k ) = H α σ γg σβ e β) i e γ) j, which shows that scalar components of e α)i j are H αβγ = H α σ γ g σβ. It is observed that g αβ k =0=(g ij e i α) ej β) ) k gives rise to g β H α γ + g α H β γ =0, which shows H αβγ = H βαγ. Next e i 1) j = li j = 0 gives H 1 β γ = 0. Consequently the matrices H γ = (H γαβ ) may be written as Then it is observed that H γ = (H αβγ ) = m i j = e i ) j = H β γe i β) eγ) j = h γ ε 3 n i e γ) j h γ 0 h γ 0 = H δγ g δβ e i β) eγ) j. = H 3γ g 3β e i β) eγ) j Similarly we get n i j = h γ ε m i e γ) j. Hence we shall introduce the vector field h j = h γ e γ) j, with the scalar components h γ. Then we have m i j = ε 3 n i h j, n i j = ε m i h j. h j as thus defined is called the h connection vector. Next we are concerned with the v covariant derivatives e i α) j of e i α) in CΓ. Let us denote it in the form (.) Le i β α) j = V α γ e i β) eγ) Remark 1. In the following all the scalar components we treat are restricted to be positively homogeneous of degree zero in y i. Since e i α) and j.

5 5 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 119 e i α) j are clearly positively homogeneous of degree zero 0 and -1 in y i β, V α γ, the scalar components of Le i α) j are of degree zero in y i. If the components T i j of a tensor T is positively hmogeneous of degree r in y i, then we put L r T i j = T α βe i α) eβ) j throughout the following. Similarly to the case of β H α γ, if we put V αβγ = V α γ g β, g ijk = 0 leads to V αβγ = V βαγ. Next it is observed that Transvecting by e i β) ej γ), we get Le 1)i j = Ll i j = g ij l i l j = V 1σ e ) i eσ) j. (.3) V 1βγ = g βγ g β1 g γ1 = h βγ, which are scalar components of the angular metric tensor h ij. Therefore the matrix V γ = (V γαβ ) may be written as V γ = (V αβγ ) = Then it is observed that Lm i j = Le i β ) j = V γ e i β) eγ) j 0 g γ g 3γ g γ 0 v γ g 3γ v γ 0. = V γ g β e i β) eγ) j = V 1γ l i e γ) j + V 3γ ε 3 n i e γ) j = g γ l i e γ) j + v γ ε 3 n i e γ) j, Ln i j = g 3γ l i e γ) j v γ ε m i e γ) j. Hence we shall introduce the vector field v j = v γ e γ) j, with the scalar components v γ, called the v connection vector. have from (1.) Then we (.4) Lm i j = l i m j + ε 3 n i v j, Ln i j = l i n j ε m i v j. Lemma. The first scalar components v 1 of the v connection vector v i vanishes identically, so that v i is orthogonal to l i.

6 10 MAKOTO MATSUMOTO 6 Proof. Since m i is positively homogeneous of degree zero in y i, we have m i j y j = 0 and (.4) shows Lm i j y j = ( ε 3 n i) v j y j. We consider the covariant derivatives of e α) i : e α) i j = (g αβ e β)i ) j = g αβ H βγδ e γ) Le α) i j = V γ α δ e γ) i i eδ) α j = H γ δ e γ) i eδ) j. eδ) j, Next we are concerned with the covariant differentiation of tensor fields. For instance, we consider T i j of (1,1) type and observe that = T α β ke i α) eβ) j T i j k = (T α βe i α) eβ) j ) k + T α β(h α γ e i ) eγ) k )eβ) j T α βe i α) (H β γe ) j eγ) k ) = (T α β re r γ) + T βh α γ T α H β γ )e i α) eβ) j eγ) k. Hence the scalar components of T i j k are equal to T α β,γ = T α β re r γ) + T βh α γ T α H β γ, which are called the h scalar derivatives of T α β. It is noted that T α β r are the h covariant derivative of scalar fields T α β: T α β r = δ r T α β, δ r = r G i r i, where r = / x r. Similarly the scalar components T α β;γ of the v covariant derivative k multiplied by L are given by LT i j T α β;γ = LT α β r e r γ) + T βv α γ T α V β γ, where T α β r are the v covariant derivative of scalar fields T α β : T α β r = r T α β. T α β;γ are called the v scalar derivatives of T α β. Lemma 3. 1) If T i jk is positively homogeneous of degree 0, then we have T α βγ;1 = 0.

7 7 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 11 ) If the scalar components T α βγ of T i jk satisfy T α β1 = 0, then we have T α β1,γ = 0, and T α β1;γ = T α βγ. Proof. T α β1 = 0 means T i j0 = 0 from Lemma 1. Hence T i j0 k = 0, which shows T α β1,γ = 0. Or we have T α β1,γ = (δ i T α β1) e i γ) + T β1h α γ T α 1H β γ T α βh 1 γ, where the last term also vanished from H 1 γ = g β H 1βγ = 0. On the other hand, we have ( ) T α β1;γ = k T α β1 e k γ) + T α β1v γ T α 1V β γ T α βv 1 γ = T α βv 1σγ g σ = T α β (g σ g σ1 g 1 ) g σ = T α β ( δ γ δ 1 g γ1) = T α βγ. Lemma 4. Let T i j be a tensor field, positively homogeneous of degree r in y i. If we denote by T α β the scalar components of L r T i j, then the scalar components of L r+1 T i j k are equal to T α β;γ + rt α βg γ1. Proof. From L r T i j = T α βe i α) eβ) j we have L ( ) L r T i k j = T α β;γe i α) eβ) j eγ) k, L r+1 T i j k rl r T i je 1)k = T α β;γe i α) eβ) L r+1 T i j k = T α β;γe i α) eβ) j eγ) k j eγ) k, + rt α βe i α) eβ) j e 1)k = (T α β;γ + rt α βg γ1 )e i α) eβ) j eγ) k. Remark. In opposition to Lemma 3, T α β = 0, for instance, does not imply T α β,γ = 0. If T α βδ = 0, then we have T α βδ,γ = 0 obviously. But T α β is one of T α βγ and T α β,γ is one of T α βδ,γ. In fact we observe that T α β,γ = (δ i T α β)e i γ) + T βh α γ T α H β γ T α βh γ = T α βh γ = T α β3ε 3 h γ. 3. The Ricci identities. We have the Ricci identities which show the commutation formulae of h and v covariant differentiations. For a vector field T i (x, y), for instance, they are written in the form T i j k T i k j = T h i R h jk T i h R h jk, k T i k j = T h i P h jk T i h hc j k T i h P h jk, T i j T i j k T i k j = T h S h i jk.

8 1 MAKOTO MATSUMOTO 8 Here three torsion tensors and three curvature tensors take place: R h jk = y i R i h jk... (v)h torsion, C j h k = g hi C jik... (h)hv torsion, P h jk = y i P i h jk... (v)hv torsion. The h curvature tensor R h i jk, hv curvature tensor P h i jk and v curvature tensor S h i jk are positively homogeneous of degree 1,-1 and - respectively. The scalar components of these tensors are denoted by the same letters: L 1 R h jk... R α βγ, i β R h jk... R α γδ, P h jk... P α βγ, i β LP h jk... P α γδ, h α LC j k... C β γ, L i β S h jk... S α γδ. According to our notations we have T i = T α e i α), T i j = T α,βe i α) eβ) j, T i j k = T α,β,γe i α) eβ) j eγ) k. Thus the first Ricci identity can be written as T α,β,γ T α,γ,β = T R α βγ T α ;R βγ. Lemma 1 implies R α βγ = R 1 α βγ. Similarly the second Ricci identity can be written as T α,β;γ T α ;γ,β = T P α βγ T α,c β γ T α ;P βγ. We have also P α βγ = P 1 α βγ. Thirdly we have L(LT i j ) k = T α ;β;γe i α) eβ) j eγ) k = Ll kt i j + L T i j k, L T i j k = (T α ;β;γ T α ;βg γ1 )e i α) eβ) j eγ) k. Therefore the third of the Ricci indentities is written as T α ;β;γ T α ;γ;β = T S α βγ + T α ;βg γ1 T α ;γg β1. Next we deal with (.1) and (.): e i α) j k = H α β γ,δe i β) eγ) j eδ) k, Le i β α) j k = V α γ,δ e i β) eγ) j eδ) k, L(Le i α) j ) k = Ll k e i α) j + L e i α) Lei α) j β k = H α γ;δ e i β) eγ) j eδ) j k = V α β γ;δ e i β) eγ) j eδ) k, k.

9 9 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 13 Therefore the Ricci identities yield β β β β H α γ,δ H α δ,γ = R α γδ V α R γδ, (3.1) β β β β β H α γ;δ V α δ,γ = P α γδ H α C γ δ V α P γδ, β β β β β V α γ;δ V α δ;γ V α γ g δ1 + V α δ g γ1 = S α γδ. β First we are concerned with (3.1) of α = 1. Since H α γ are scalar components of e i α) j, we have H α β γ,δ = (δ i H α β γ )e i δ) + H α γh β δ H α β H γ δ, H α β γ;δ = ( i H α β γ )e i δ) + H α γv β δ H α β V γ δ, β β where the terms H γ H α δ and H γ V α δ do not take place. Then β β β β H 1 γ = 0 implies H 1 γ,δ = H 1 γ;δ = 0. Next V 1 δ = h β β δ leads to V 1 δ,γ = 0 β and V 1 δ;γ = h β γg δ1 h δγ δ β 1 from Lhi j k = h i kl j h jk l i ([8], (17.31)). Consequently (3.1) of α = 1 are all trivial. Tranvecting (3.1) by g βσ and putting (α, σ) = (, 3) we are led to the following equations: h γ,δ h δ,γ = R 3γδ v R γδ, (3.) h γ;δ v δ,γ = P 3γδ h C γ δ v P γδ, v γ;δ v δ;γ g γ g δ3 + g γ3 g δ v γ g δ1 + v δ g γ1 = S 3γδ. In the last equation we have to pay attention to the fact that V 3γ;δ = ( i v γ )e i δ) V γv 3 γ V 3 V γ δ = 1 (v γ;δ + v V γ δ ) V 1γ V 3 δ v V γ δ = v γ;δ g γ g 3δ. Proposition 1. The h and v connection vectors satisfy the differential equations (3.). 4. The main scalars. We are concerned with the scalar components C αβγ of LC ijk. C αβγ are clearly symmetric and C 1βγ = 0 from y i C ijk = 0. Next LC i = LCe )i = C αβγ g βγ e α) i leads to LCg = C ε + C 33 ε 3. If we put C = ε H, C 33 = ε I and C 333 = ε 3 J, then the above shows LC = ε H + ε 3 I and 0 = C 3 ε + J. Consequently, putting ε = ε ε 3, we obtain C = H, C 3 = J, C 3 3 = I, 3 C = εi, 3 C 3 = εi, 3 C 3 3 = J.

10 14 MAKOTO MATSUMOTO 10 Three scalars H, I and J are called the the main scalars of F 3. follows from LC ijk = C αβγ g αλ g βµ g γν e λ)i e µ)j e ν)k that Then it (4.1) LC ijk = Hm i m j m k εj(m i m j n k + m j m k n i + m k m i n j ) + I(m i n j n k + m j n k n i + m k n i n j ) + Jn i n j n k. Proposition. The scalar components C αβγ of the C tensor LC ijk multiplied by L are written in terms of the main scalars H, I, J as (1) (C 1βγ, C, C 3, C 33, C 333 ) = = (0, ε H, ε J, ε I, ε 3 J), and LC ijk can be written as (4.1). We have () LC = ε (H + εi), ε = ε ε 3. Example. We shall find the main scalars of a C reductible Finsler space F 3 ([8], 30). The C tensor of such an F 3 is written in the form that is, 4C ijk = h ij C k + h jk C i + h ki C j, 4LC ijk = LC[3ε Cm i m j m k + ε 3 (m i n j n k + m j n k n i + m k n i n j )]. Thus (4.1) leads to H = 3LC 4 ε, I = LC 4 ε 3, J = 0, and so H 3εI = 0. Next we treat the T tensor ([8], (8.0)): Since we have T hijk = LC hij k + l h C ijk + l i C hjk + l j C hik + l k C hij. L C hij k = (C αβγ;δ C αβγ g δ1 )e α) h eβ) i e γ) j eδ) k,

11 11 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 15 the scalar components T αβγδ of LT hijk are given by T αβγδ = C αβγ;δ + g α1 C βγδ + g β1 C αγδ + g δ1 C αβγ. T hijk are completely symmetric because of C hij k being so. Thus we must have C αβγ;δ C αβδ;γ C αβγ g δ1 + C αβδ g γ1 = 0. In the case (γ, δ) = (1, ) and (1, 3) the above is trivial because of Lemma 3. The case (γ, δ) = (, 3) yields C αβ;3 C αβ3; = 0. The case α = 1 is also trivial from Lemma 3. Consequently we put (α, β) = (, ), (, 3) and (3, 3). For instance, we have C ;3 C 3; = ( i C )e i 3) 3C V 3 ( i C 3 )e i ) + C 3V + C V 3 = ε H ;3 3C 3 V 33 ε 3 + ε J ; + C 33 V 3 ε 3 C V 3 ε. It is noted that H ;3 and J ; are v scalar derivatives of single scalars H and J respectively. Thus we obtain (4.) ε (H ;3 + J ; ) = (H εi)v 3εJv 3, ε (I ; + J ;3 ) = 3Jv + (H εi)v 3, ε I ;3 ε 3 J ; = 3(Jv 3 + Iv ). Consequently the scalar components T αβγδ of LT hijk are written as (4.3) T 1βγδ = 0, T = ε H ; + 3εJv, T 3 = ε H ;3 + 3εJv 3 = (H εi)v ε J ;, T 33 = ε J ;3 + (H εi)v 3 = ε I ; 3Jv, T 333 = ε I ;3 3Jv 3 = ε 3 J ; + 3Iv, T 3333 = ε 3 J ;3 + 3Iv 3. Theorem 1. The T tensor of F 3 with C i 0 vanishes, if and only if the v connection vector v i vanishes and all the main scalars H, I, J are functions of position alone.

12 16 MAKOTO MATSUMOTO 1 Proof. T 33 = T 333 = T 333 = 0 give I ; = 3ε Jv, I ;3 = 3ε Jv 3, J ; = 3ε 3 Iv, J ;3 = 3ε 3 Iv 3, and T 3 = T 33 = 0 give (H + εi)v = (H + εi)v 3 = 0. Since H + εi = 0 contradicts to C > 0 from Proposition, we have v = v 3 = 0, i.e., v i = 0 from Lemma. Then (4.3) yields H ; = H ;3 = I ;3 = = I ;3 = J ; = J ;3 = 0. H ;1 = I ;1 = J ;1 = 0 from Lemma The v curvature tensor. First we consider a tensor T ijk, for instance, of (0, 3) type which is skew symmetric in (i, j). Using the ε tensor ε ijk given in Example 1, if we define a tensor T h k = 1 εhij T ijk, then we get T ijk of the form T ijk = ε ijh T h k. This operation on T ijk is called the shortening of T ijk. Let us denote by T αβγ and T α β the scalar components of T ijk and T h k. Then we have T α β = 1 δασ T σβ, T αβγ = δ αβ T γ. Consequently we have ( T 1 β, T β, T 3 β) = (T 3β, T 31β, T 1β ). Now we are concerned with the v curvature tensor S hijk, which is written in terms of the C tensor as (5.1) S hijk = C hkr C i r j C hjr C i r k. This is skew symmetric in (h, i) and (j, k). Thus, by the double shortening, we can put S hi = 1 4 εhjk ε ilm S jklm, S jklm = ε jkh ε lmi S hi.

13 13 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 17 Let S αβ be the scalar components of L S hi. Then it follows from 3 that S αβ = 1 4 δαγδ δ βσ S γδσ, S γδσ = δ γδα δ σβ S αβ, and further S αβ = 1 4 δαγδ δ βσ (C γσπ C δ π C γπ C δ π σ ). This shows that S αβ vanishes except S 11, because of C l α β = 0; Proposition leads to S 11 = C 3π C 3 π C π C 3 π 3 = ε 3 I ε (HI J ), which is denoted by εs. S is called the v scalar curvature. Then As a consequence we obtain and (1.) gives S αβγδ = δ αβ1 δ γδ1 εs. L S hijk = εsδ 1αβ δ 1γδ e α) h eβ) i e γ) j eδ) k = εs(e ) h e3) i e 3) h e) i )(e) j e3) k e3) j e) k ), = εs(m h n i m i n h )(m j n k m k n j ) = S(h hj h ik h hk h ij ). Next we remember the third of (3.): v ;3 v 3; ε = S 33, which is equal to S 11 = εs. Summarizing up all above, we have Theorem. The v curvature tensor S hijk of F 3 with C 0 can be written in the form (1) L S hijk = S(h hj h ik h hk h ij ). The v scalar curvature S is written as () S = ε I ε 3 (HI J ) = ε(v ;3 v 3; ε).

14 18 MAKOTO MATSUMOTO 14 The scalar components S αβγδ of L S hijk is of the form (3) S αβγδ = εδ 1αβ δ 1γδ S. Remark 3. The special form (1) of S hijk of F 3 was first shown by the present author [4] and this is the origin of the concept S3 like, which has given rise to many discussions [13], [3]. 6. The hv curvature tensor. The hv curvature tensor P hijk is written in terms of the C tensor as (6.1) P hijk = C ijk h + C h r j C rik 0 (h/i), where (h/i) denotes the interchange of indice (h, i) in all the previous terms. Thus P hijk is skew symmetric in (h, i), so that the shortening gives P i jk = 1 εihm P hmjk, P hmjk = ε hmi P i jk. It should be remarked that the (v)hv torsion tensor P i jk differs from P i jk. From P i jk = y h P h i jk it follows that P ijk = g ir P r jk is written in the form P ijk = y h ε hir P r jk. In terms of the scalar components P αβγδ, P α βγ and P αβγ of LP hijk, L P i jk and P ijk respectively we have P α βγ = 1 δασ P σβγ, P σβγ = δ σα P α βγ, P αβγ = δ 1α P βγ. From P ijk = C ijk 0 and P ij0 = 0 it follows that P βγ is symmetric in (β, γ) and P β1 = 0. We consider the equation (6.) P αβγδ = δ αβ P γδ = C βγδ,α + C α γ C βδ,1 (α/β). We first deal with C βγδ,α. Lemma 3 gives C 1γδ,α = 0. Next, for instance, C 3,α = {δ i ( ε J)}e i α) C 3H α C H 3 α = = ε J,α C 33 H 3α ε 3 + C H 3α ε = ε J,α εih α + Hh α.

15 15 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 19 Remark 4. We have C 3 = ε J, but C 3,α = ε J,α is a mistake! The symbol C 3,α is one of the h scalar derivatives C βγδ,α of C βγδ, while the symbol J,α shows the h scalar derivatives of a single scalar J. Similarly we obtain (6.3) C,α = ε H,α + 3εJh α, C 3,α = ε J,α + (H εi)h α,, C 33,α = ε I,α 3Jh α, C 333,α = ε 3 J,α + 3Ih α. Putting (α, β)=(1,), (,3) (3,1) in (6.), we have (6.4) P 1 γδ = C 3γδ, C γδ,3 + C γ C 3δ,1 C 3 γ C δ,1, P γδ = C 3γδ,1, P 3 γδ = C γδ,1. We have interesting two classes of special Finsler spaces, that is, Landsberg spaces and Berwald spaces which are characterized respectively by C hij 0 = 0 and C hij k = 0. From C αβγ,δ = 0 and (6.3) we have H,α = 3ε 3 Jh α, I,α = 3ε Jh α, J,α = ε (H εi)h α = 3ε 3 Ih α. The latter implies (H + εi)h α = 0 and () of Proposition gives h α = 0. Therefore we obtain Theorem 3. 1) F 3 with C 0 is a Landsberg space, if and only if the h curvature vector h i is orthogonal to the supporting element, i.e., h 1 of h α vanishes, and the main scalars satisfy H,1 = I,1 = J,1 = 0. ) F 3 with C i 0 is a Berwald space, if and only if h i vanishes and all the main scalars are h covariant constant. 7. The h curvature tensor. The h curvature tensor R hijk is skew symmetric in (h, i) and (j, k) by means of the h metrical property g ij k = 0 of CΓ. Then, by the double shortening we get R ij = 1 4 εihk ε jlm R hklm, R hklm = ε hki ε lmj R ij. If we denote by R αβ the scalar components of R ij, then we have R αβ = 1 4 δαγδ δ βσ R γδσ, R γδσ = δ γδα δ σβ R αβ.

16 130 MAKOTO MATSUMOTO 16 Thuus the scalar components R αβγ of the (v)h torsion tensor R ijk /L is written as R αβγ = δ 1αβ δ βγσ R σ. Let us introduce from R hijk the three tensor fields as follows: R ij = g hk R hikj, R = g ij R ij, L ij = R ij R 4 g ij. R ij and R are called the h-ricci tensor and the h-scalar curvature respectively. The scalar components R αβ of the h Ricci tensor are given by (7.3) R αβ = g λµ δ λα δ µβσ R σ = ε λ δ λα δ λβσ R σ, where we put g 11 (of g αβ ) = +1 = ε 1. Then we get which is written in the form R = ε λ δ λα δ λβσ R σ g αβ = ε λ ε µ δ λµ δ λµσ R σ = (ε 1 ε R 33 + ε ε 3 R 11 + ε 3 ε 1 R ), (7.4) R = εε R, ε = ε ε 3. Then it is observed from (7.3) and (7.4) that, for instance, Consequently we get R 11 = ε λ δ λ1 δ λ1σ R σ = ε R 33 + ε 3 R ( ) R = ε ε 3 R 11 ε 1 = R ε 1 ε R 11, R 3 = ε λ δ λ δ λ3σ R σ = ε 1 R 3 = ε(ε ε 3 R 3 ). (7.5) R αβ = R g αβ εε α ε β R βα. Next we deal with L ij. Its scalar components are L αβ = R αβ (R/4)g αβ, which is equal to L αβ = (R/4)g αβ ε α ε β Rβα from (7.5). If we put L hijk = g hj L ik + g ik L hj g hk L ij g ij L hk,

17 17 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 131 then its scalar components are, for instance, L 13 = g L 13 = ε ( εε 1 ε 3 R 31 ) = R 13, L 11 = g 11 L + g L 11 = ε 1 ( R 4 ε ε R ) + ε ( R 4 ε 1 ε R 11 ) = ε R ε 3(ε 1 R 11 + ε R ) = ε R ε 3 = R 33 = R 11. ( ) R ε ε 3 R 33 Consequently we have L hijk = R hijk. Thus R hijk is written in the form (7.6) R hijk = g hj L ik + g ik L hj g hk L ij g ij L hk. Theorem 4. The h curvature tensor of F 3 with C 0 can be written in the form (7.6) where L ij = R ij (R/4)g ij. For the latter use we shall write the scalar components R α βγ of R i jk/l. From (7.) we have which gives R α βγ = δ 1λ δ βγσ R σ g αλ = (δ 1 g α + δ 13 g α3 ) R σ δ βγσ, (7.7) R α βγ = (g α R 3σ g α3 R σ )δ βγσ. 8. The Bianchi identities. In the general theory of Finsler connections we have eleven Bianchi identities ([8], 11.), which are divided into four groups. For the Cartan connection one of every group can be directly derived from the other because of R i jk = y h R h i jk, P i jk = y h P h i jk and S i jk = y h S h i jk = 0. Further two of the remaining seven identities are equivalent to (5.1) and (6.1). Consequently we have essentially five Bianchi identities in our case: (8.1) (LC hir )(L 1 R r jk) + R hijk + (i, j, k) = 0, (8.) (LP mhir )(L 1 R r jk) + R mhij k + (i, j, k) = 0,

18 13 MAKOTO MATSUMOTO 18 (8.3) (L S mhkr )(L 1 R r ij) + LR mhij k + [R mhir (LC j r k ) +(LP mhir )P r jk + LP mhjk i (i/j)] = 0, (8.4) L S mhij k + [(LP mhri )(LC k r j ) (L S mhir )P r kj L P mhkj i (i/j)] = 0, (8.5) L 3 S mhij k + (i, j, k) = 0, where (i, j, k) and (i/j) denote the cyclic permutation of the indices i, j, k and the interchange of indices i, j respectively. The purpose of the present section is to write these identities in terms of the scalar components. First (8.1) can be written as C αβ R γδ + R αβγδ + (β, γ, δ) = 0. From (7.7) and (7.1) it follows that this is written as (δ αβ R σ + C α β R 3σ C α 3 β R σ )δ σγδ + (β, γ, δ) = 0. In our three dimesnsional case, putting (β, γ, δ) = (1,, 3), we have the equation of the form A α σ 1 δ σ3 + A α σ δ σ31 + A α σ 3 δ σ1 = A α σ σ = 0. Consequently the indentity is equivalent to δ ασ R σ + C α σ R 3σ C α 3 σ R σ = 0. Putting α = 1,, 3, we obtain three identities as follows: (8.1 ) (1) R 3 R 3 = 0, () R 31 R 13 = (H εi) R 3 J( R 33 ε R ), (3) R 1 R 1 = I( R 33 ε R ) J R 3. Next (8.) is written in the scalar components as follows: δ λµα δ γδ A α β + (β, γ, δ) = 0, Aα β = R α,β + P α βσ(g σ R 3

19 19 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 133 g σ3 R ). It is obvious that this is equivalent to δ γδ A α β +(β, γ δ) = 0. Thus, similarly to the case (8.1), we can reduce this identity to A α = 0 only: (8. ) R α, + P α σ(g σ R 3 g σ3 R ) = 0. Next we deal with (8.3) which is wrriten in terms of scalar components as follows: δ γδσ ( R ασ ;β Sδ α 1 ε δ β R σ ) + [ P α δβ,γ + δ γσ R α C δ σ P δβ P α γ (γ/δ)] = 0. We put in the above (γ, δ) = (1, ), (, 3), (3, 1). σ From C 1 β = P α 1 = = P α 1 = 0 and Lemma 3 we obtain (8.3 ) 1) R α3 ;β Sδ1 α ε β R β3 + P α β,1 + R α3 C β R α 3 C β = 0, ) R α1 ;β Sδ1 α ε β R β1 + P α 3β, P α β,3 + R α1 C β + P β P α 3 P 3β P α = 0, 3) R α ;β Sδ1 α ε β R β P α 3β,1 R α3 C 3 β + R α 3 C 3 β = 0. Next we write (8.4) in terms of scalar components: Paying attention to the terms L P mhkj i and Lemma 4, we have εδ α 1 S,β δ γδ1 + [ P α σγc β σ δ εsδ γσ1 δ α 1 P σ βδ + P α βγ;δ In the case (γ, δ) = (, 3) we obtain P α βγg δ1 (γ/δ)] = 0. (8.4 ) εδ α 1 (S,β SP β) + P α σc β σ 3 P α σ3c β σ + P α β;3 P α β3; = 0. On the other hand the cases (γ, δ) = (1, ), (1, 3) are reduced to be trivial because of C β σ 1 = P σ β1 = P α σ1 = 0 and P α β1;δ = P α βδ, δ =, 3, from Lemma 3. Since a Landsberg space is characterized by C hij 0 = P hij = 0 which implies P hijk = 0 ([8], (19.7)), (8.4 ) leads to Theorem 5. For a Landsberg space F 3 with C 0, the v scalar curvature S is h covariant constant.

20 134 MAKOTO MATSUMOTO 0 Finally we consider (8.5). Paying attention to Lemma 4, it is written in the scalar components as δ αβ ( S σ ;γ S σ g γ1 ) + (α, β, γ) = 0. Since this is similar in form to the cases (8.1) and (8.), we have the equivalent identity S σ ; S σ g 1 = 0, which is reduced to be trivial, because S σ ; = ( i S σ )e i ) + S π V π σ + S σπ V π = ( i S σ1 )e i 1) + S 11 V 1 σ 1 + S σ1 V 1. The first term of the right hand side vanishes from the homogeneity of S σ1 ans e i 1) = yi /L. Thus (.3) leads to = S 11 h σ 1 + S σ1 h = S 1. Proposition 3. In F 3 with C 0 we have four Bianchi identities (8.1 ), (8. ), (8.3 ) and (8.4 ). 9. The semi C reducibility. Since the present author introduced the notion of the C-reducibility [5], this notion has been generalized to the semi-c-reducibility, quasi-c-reducibility and generalized C-reducibility [14]. If the C tensor is of the form C ijk = λ n + 1 [h ijc k + (i, j, k)] + 1 λ C i C j C k, C then the space F n is called semi C reducible. The case λ = 1 is C reducible and λ = 0 C like. In a previous paper [9] the following fact has been shown: For a Finsler space F n, n 3, with (α, β) metric, if C does not vanish, then F n is semi C reducible. A little generalization of the semi C reducibility is given by C ijk = A ij C k + A jk C i + A ki C j, for some tensor A ij satisfying A ij = A ji and A i0 = 0, and is called quasi C reducibility.

21 1 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 135 Theorem 6. If F 3 with C 0 is quasi C reducible, then F 3 is semi C reducible and the main scalar are ( H = ε 1 λlc ) λlc, I = ε 3 4 4, J = 0. Proof. Let F 3 with C i 0 be quasi C reducible. Then (4.1) gives Hm i m j m k εj[m i m j n k + (i, j, k)] + I[m i n j n k + (i, j, k)] +Jn i n j n k = LC[A ij m k + (i, j, k)]. Transvecting successively by m k, m j and m i, we have J = 0, H = 3LCA, A i (= A ij m j ) = ε A m i and A ij = A m i m j + Consequently we have C ijk = I LC n in j = I LC ε 3h ij + H 3εI 3LC 3 C ic j. I LC ε 3[h ij C k + (i, j, k)] + H 3εI LC 3 C i C j C k, which is of the semi-c-reducible form with λ/4 = Iε 3 /LC and 1 λ = = (H 3εI)ε /LC. Cf. Example. From Theorems 1 and 6 we get Theorem 7. F 3 with C 0 is semi-c-reducible and has the vanishing T -tensor, if and only if the v-connection vector v i vanishes and the main scalars are functions of position alone, especially J = 0. The v-scalar curvature is S = ε 3 I(εI H). Remark 5. Compare Theorem 7 with Theorems 1 and 5 of [11]. If F n, n 4, is S3-like, that is, S hijk is of the form (1) of Theorem, then it is proved that S is a function of position alone ([8], Theorem 31.6). 10. Semi C reducible and Landsberg spaces F 3. In the paper [1], the following fact has been proved: All the semi-c-reducible and Landsberg spaces F n, n 4, are divided into two classes as follows: (1 ) nλ n + 1 : λ i = 0, C i j = 0, ( ) nλ = n + 1 : (C ) i = 0.

22 136 MAKOTO MATSUMOTO A space belonging to (1 ) is reduced to a Berwald space, while a space belonging to ( ) is S3-like. The purpose of the last section is to consider semi-c-reducible and Landsberg spaces F 3 with C 0. Owing to Theorems 3 and 6 such a space F 3 has h 1 = H,1 = I,1 = J = 0. Further S is h-covariant constant, which leads to (10.1) IH,α + KI,α = 0, K = H εi. From P hijk = C ijk 0 = 0 of F 3 and (6.3) if follows that C 3γδ, C γδ,3 = 0, which is written as (10.) H,3 = ε Kh, I, = ε Kh 3, I,3 = 3ε Ih. Then (10.1) is rewritten as (10.1 ) Ih K = 0, IH, + ε K h 3 = 0. Since I = K = 0 leads to a contradiction, (10.1 ) and (10.) show that the spaces under consideration are divided into three classes as follows: (I) I = 0 : h 3 = 0, H,3 = ε Hh, (II) K = 0 : H, = I, = H,3 = I,3 = 0, h = 0, (III) IK 0, h = 0 : IH, = ε K h 3, I, = ε Kh 3, H,3 = I,3 = 0. Further we should pay attention to identities (4.): ε H ;3 = Kv, ε 1 I ; = Kv 3, ε I ;3 = 3Iv. For an F 3 belonging to (I), we then have ε H ;3 = Hv and v 3 = 0. If an F 3 belongs to (II), then H ;3 = I ; = 0, hence H ; = I ;3 = 0 from K = 0, and consequently H and I satisfy H,α = H ;α = I,α = I ;α = 0. Therefore we have Theorem 8. A semi-c-reducible and Landsberg space F 3 with C 0 has J = H,1 = I,1 = h 1 = 0 and the h-covariant constant v-scalar curvature S. All of such spaces are divided into three classes as follows: (I) I = 0 : h 3 = v 3 = 0, H,3 = ε Hh, H ;3 = ε Hv, S = 0, (II) K(= H εi) = 0: H, I = const., h = 0, (III) IK 0, h = 0: IH, = ε K h 3, I, = ε Kh 3, H,3 = I,3 = 0, H ;3 = ε Kv, I ; = ε Kv 3, I ;3 = 3ε Iv.

23 3 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 137 If h 3 of an F 3 belonging to (II) vanishes, then h α = 0, α = 1,, 3, and hence Theorem 3 shows that F 3 is a Berwald space. Next, the similar fact is true for an F 3 belonging to (III), because we have h α = 0 and H,α = I,α = 0, α = 1,, 3. Therefore we have Corollary 1. A semi-c-reducible and Landsberg space F 3 with C 0 belonging to the classes (II) or (III) is reduced to a Berwald space, if and only if h 3 = 0, that is, h i n i = 0. The remainder of the present section is devoted to detailed discussions of spaces belonging to the class (I), because the hv- and v-curvature tensors of such a space vanish. Our discussions are specially based on the Bianchi identities (8.1 ) and (8.3 ). From I = J = 0 (8.1 ) is simply written in the form (10.3) R 3 = R 3, R 31 R 13 = H R 3, R 1 = R 1. From C α β γ = 0 except C = H (8.3 ) is also simply written in the form (10.4) (1) () (3) R α3 ;β + R α3 C β = 0 R α1 ;β + R α1 C β = 0 R α ;β = 0. The first term of each equation is one of the scalar components R αγ ;β. For instance, we have R α3 ;β = ( i R α3 )e i β) + R 3 ε α V αβ + R α ε 3 V 3β. In the following we shall regard R ab as a set of nine single scalars ( R 11, R 1,..., R 33 ) and denote as R ab. Similarly we denote v as v, because v 1 = v 3 = 0 of v α of the space belonging to (I). (C αβγ are already regarded as a set of single scalars (H, I, J) in Proposition ). Then we can write ( i R α3 )e i β = R α3;β and R α3 ;β = R α3;β + ε α (R 13 h αβ + R 3 V αβ + R 33 V 3αβ ) + ε 3 (R α1 h 3β + R α v β ). Consequently (10.4) is rewriten in the form (10.5) αβ R α3;β + R α1 ε 3 h 3β + R α ε 3 v β + R α3 C β + R 13ε α h αβ + R 3ε α V αβ + R 33ε α V 3αβ = 0,

24 138 MAKOTO MATSUMOTO 4 (10.6) αβ R α1;β + R α1 C β R α h β R α3 h 3β + R 11ε α h αβ + R 1ε α V αβ + R 31ε α V 3αβ = 0, (10.7) αβ R α;β + R α1 ε h β R α3 ε v β + R 1ε α h αβ + R ε α V αβ + R 3ε α V 3αβ = 0. We construct (10.5) (10.7) 3, (10.7) 1 (10.6), (10.6) 3 (10.5) 1. It then follows from (10.3) that we obtain R 3 ε Hh β + R 3 C β = 0, R 3 ε Hv β R 1 C β = 0, (R 3 H) ;β + R 3 HC β = 0. The first equations (β =, 3) are trivial. One (β = 3) of the second are also trivial and the other (β = ) yields (10.8) R 1 = R 3 ε v. One (β = 3) of the third is a consequence of (10.8) because of H ;3 = ε Hv from Theorem 8 and R 3;3 = R 1 from (10.5) 3. The other (β = ) is (10.9) (R 3 H) ; = R 3 H. Then, using (10.8), (10.9) and H ;3 = ε Hv, (10.5), (10.6) and (10.7) are reduced to independent twelve differential equations as follows : (10.10) R 3; = R 31 + (R 33 R ε)ε v, R 3;3 = R 1, R 1; = R 11 R 1 H + (R + R 31 v)ε, R 1;3 = R 3 ε 3, R 31; = R 3 ε R 31 H R 1 ε 3 v, R 31;3 = R 11 + R 33 ε 3, R 11; = R 1 ε R 11 H, R 11;3 = (R 31 R 3 H)ε 3, R ; = R ;3 = 0, R 33; = R 3 ε 3 v R 33 H, R 33;3 = R 31 + R 3 H. Here we remember () of Theorem. In the case we consider it gives v ;3 v 3; = ε. We have Thus we get v ;3 = ( i v )e i 3) v V 3 = v ;3, v 3; = ( i v 3 )e i ) v V 3 = v ε. (10.11) v ;3 = ε + v ε.

25 5 A THEORY OF THREE DIMENSIONAL FINSLER SPACES 139 Now we consider the integrability conditions R αβ;γ;δ R αβ;δ;γ = 0 of (10.10). Making use of H ;3 = ε Hv and (10.11), the conditions yield v (R 33 εr ) = 0, v(r 11 ε R + R 1 H R 31 ε v) = 0, (10.1) v(r 3 R 31 ε H R 1 εv) = 0, v(r 1 R 11 ε H) = 0, v(r 3 v + R 33 ε 3 H) = 0. Since v does not vanish from (10.11), we have six linear equations (10.8) and from (10.1). It is easy to show that these equations lead to In the case R 33 0, we get (3 + H ε + v ε 3 )R 33 = 0. HH ;3 ε + vv ;3 ε 3 = v(ε + H + v ε) = v = 0, contradiction. Therefore we have R 33 = 0, hence all R αβ = 0. Theorem 9. The three curvature tensors R, P, S of the Cartan connection of a semi C reducible and Landsberg space F 3 with C 0 vanish, if F 3 belongs to the class (I) of Theorem 8. Remark 6. It seems to the present author that R hijk = 0 of a Finsler space F n leads often to a knotty problem. For instance, H. Akber Zadeh showed ([8], Theorem.): If F n, n 3, is h isotropic, that is, R hijk = K(g hj g ik g hk g ij ), then K is a constant. If K does not vanish, then P hijk = P hikj and S hijk = 0. S. Numata showed ([8], Theorem 30.6): If F n, n 3, is a Landsberg space and of non zero scalar curvature K, that is, R i0k = KL h ik, then F n is a Riemannian space of constant curvature K. No one knows what space is an F n with R hijk = P hijk = 0. REFERENCES 1. HASHIGUCHI, M., HŌJŌ, S. and MATSUMOTO, M. On Landsberg spaces of two dimensions with (α,β) metric, J. Korean Math. Soc. 10 (1973), HASHIGUCHI, M., HŌJŌ, S. and MATSUMOTO, M. Landsberg spaces of dimension two with (α,β) metric, to appear in Tensor, N.S. 3. HŌJŌ, S. Structure of fundamental functions of S3 like Finsler spaces, J. Math. Kyoto Univ. 1 (1981),

26 140 MAKOTO MATSUMOTO 6 4. MATSUMOTO, M. On Finsler spaces with curvature tensors of some special forms, Tensor, N.S. (1971), MATSUMOTO, M. On C reducible Finsler spaces, Tensor, N.S. 4 (197), MATSUMOTO, M. A theory of three dimensional Finsler spaces in terms of scalars, Demonst. Math. 6 (1973), MATSUMOTO, M. Greetings, Anal. Şt. Univ. Al.I.Cuza, Iaşi, 30 (1984), Romanian Japanese Coll. on Finsler Geom., p.vi. 8. MATSUMOTO, M. Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press, Saikawa, Otsu, Japan, MATSUMOTO, M. Reduction theorems of Landsberg spaces with (α,β) metric, to appear in Tensor, N.S. 10. MATSUMOTO, M. and MIRON, R. On an invariant theory of the Finsler spaces, Period. Math. Hungar. 8 (1977), MATSUMOTO, M. and SHIBATA, C. On semi C reducibility, T tensor=0 and S4 likeness of Finsler spaces, J. Math. Kyoto Univ. 19 (1979), MOÓR, A. Über die Torsions und Krümmungsinvarianten der dreidimensionalen Finslerschen Räume, Math. Nachr. 16 (1957), OKUBO, K. Some theorems of S 3 (K) metric spaces, Rep.on Math. Phys. 16 (1979), OKUBO, T. and NUMATA, S. On generalized C reducible Finsler spaces, Tensor, N.S. 35 (1981), Received: 5.X , Zenbu cho, Shimogamo Sakyo ku, Kyoto JAPAN