On a five dimensional Finsler space with vanishing v-connection vectors

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1 South Asian Journal of Mathematics 2017, Vol. 7 ( 2): ISSN RESEARCH ARTICLE On a five dimensional Finsler space with vanishing v-connection vectors Anamika Rai 1, S. K. Tiwari 1 2 Department of Mathematics, K. S. Saket Post Graduate College, Ayodhya, Faizabad India. anamikarai2538@gmail.com Received: Feb ; Accepted: Mar *Corresponding author Abstract In 1977, M. Matsumoto and R. Miron [1] constructed an orthonormal frame for an n- dimensional Finsler space, called Miron frame. Gauree Shanker, G. C. Chaubey and Vinay Pandey [3] discussed five-dimensional Finsler space on the basis of Miron frame and find some interesting results. The aim of the present paper is to find sufficient condition for a five-dimensional Finsler space to be S-3 like and to obtain some fruitful results for a five-dimensional Finsler space with vanishing v-connection vectors. Key Words Finsler space, Miron frame, S-3 like space, v-connection vectors MSC B40, 53C05 1 Orthonormal Frame and Connection Vectors Let M 5 be a five-dimensional smooth manifold and F 5 = (M 5, L) be a five-dimensional Finsler space equipped with a metric function L(x, y) on M 5. The normalized supporting element, the metric tensor, the angular metric tensor and Cartan tensor are defined as l i = i L, g ij = 1 2 i j L 2, h ij = L i j L and C ijk = 1 2 k g ij respectively. The torsion vector C i is defined by C i = Cjk i gjk. Throughout this paper, we use the symbol i and i for y i and x i respectively. The Cartan connection in the Finsler space is given as CΓ = (F jk i, Gi j, Ci jk ). The h- and v-covariant derivatives of a covariant vector X i (x, y) with respect to the Cartan connection are given by X i j = j X i ( h X i )G h j F r ijx r, (1.1) and X i j = j X i C r ijx r. (1.2) The Miron frame for a five-dimensional Finsler space is constructed by the unit vectors (e i 1), ei 2), ei 3), ei 4), e i 5) ). The first vector ei 1) is the normalized supporting element li and the second e i 2) is the normalized Citation: Anamika Rai, S. K. Tiwari, On a five dimensional Finsler space with vanishing v-connection vectors, South Asian J Math, 2017, 7(2),

2 Anamika Rai, et al: On a five dimensional Finsler space with vanishing v-connection vectors torsion vector m i = C i /C, where C is the length of the vector C i. The third vector e i 3) = ni, the fourth e i 4) = pi and the fifth vector e i 5) = qi are constructed by the Method of Matsumoto and Miron [1]. With respect to this frame, the scalar components of an arbitrary tensor Tj i are defined by T αβ = Tj i e α)i e j β), (1.3) then we get T i j = T αβe i α) e β)j, (1.4) where summation convention is also applied to Greek indices. The scalar components of the metric tensor g ij are δ αβ. Therefore, we get g ij = l i l j + m i m j + n i n j + p i p j + q i q j. (1.5) Let H α)βγ and V α)βγ /L be scalar components of the h and v covariant derivatives e i α))j and ei α) j respectively of the vectors e i α), then e i α) j = H α)βγ e i β) e γ)j, (1.6) and L e i α) j = V α)βγ e i β) e γ)j, (1.7) H α)βγ and V α)βγ are called h and v connection scalars respectively and are positively homogeneous of degree zero in y. Orthogonality of the Miron frame yields [2] H α)βγ = H β)αγ and V α)βγ = V β)αγ. Also, we have H 1)βγ = 0 and V 1)βγ = δ βγ δ 1β δ 1γ. Now, we define Finsler vector fields: h i = H 2)3γ e γ)i, J i = H 2)4γ e γ)i, k i = H 2)5γ e γ)i, h i = H 3)4γ e γ)i, J i = H 3)5γ e γ)i, k i = H 4)5γ e γ)i, and u i = V 2)3γ e γ)i, v i = V 2)4γ e γ)i, w i = V 2)5γ e γ)i, u i = V 3)4γ e γ)i, v i = V 3)5γ e γ)i, w i = V 4)5γ e γ)i. The vector fields h i, J i, k i, h i, J i, k i are called h connection vectors and the vector fields u i, v i, w i, u i, v i, w i are called v connection vectors. The scalars H 2)3γ, H 2)4γ, H 2)5γ, H 3)4γ, H 3)5γ, H 4)5γ and V 2)3γ, V 2)4γ, V 2)5γ, V 3)4γ, V 3)5γ, V 4)5γ are considered as the scalar components h γ, J γ, k γ, h γ, J γ, k γ and u γ, v γ, w γ, u γ, v γ, w γ of the h and v connection vectors respectively. From (1.7), we get (a) Le i 1) j = Ll i j = m i m j + n i n j + p i p j + q i q j = h i j (b) Le i 2) j = Lm i j = l i m j + n i u j + p i v j + q i w j, (c) Le i 3) j = Ln i j = l i n j m i u j + p i u j + q i v j, (1.8) (d) Le i 4) j = Lp i j = l i p j m i v j n i u j + qi w j, (e) Le i 5) j = l i q j m i w j n i v j p i w j. 74

3 South Asian J. Math. Vol. 7 No. 2 Since m i, n i, p i, q i are homogeneous function of degree zero in y i, we have Lm i j l j = Ln i j l j = Lp i j l j = Lq i j l j = 0. These imply u 1 = v 1 = w 1 = u 1 = v 1 = w 1 = 0. Consequently, we have Proposition 1.1. The first scalar components u 1, v 1, w 1, u 1, v 1, w 1 of v connection vectors u i, v i, w i, u i, v i, w i vanish identically. 2 Scalar Derivatives Taking h covariant differentiation of both sides of (1.4), we get Tj k i = (δ kt αβ )e i α) e β)j + T αβ e i α) k e β)j + T αβ e i α) e β)j k. (2.1) If T αβ,γ are the scalar components of Tj k i, i.e., Tj k i = T αβ,γ e i α) e β)j e γ)k, (2.2) then, we obtain Similarly, if we put T αβ,γ = (δ k.t αβ )e k γ) + T µβh µ)αγ + T αµ H µ)βγ. (2.3) LT i j k = T αβ;γ e i α) e β)j e γ)k, (2.4) then the scalar components T αβ;γ of LTj i k are given by T αβ;γ = L( k T αβ )e k γ) + T µβv µ)αγ + T αµ V µ)βγ. (2.5) The scalar components T αβ,γ and T αβ;γ respectively are called h and v scalar derivatives of the scalar components T αβ of Tj i. From (1.7), it follows that L 2 e i α) j k + Le i α) j l k = L(Le i α) j) k = V α)βγ;δ e i β) e γ)j e δ)k, which implies L 2 e i α) j k = (V α)βγ;δ V α)βγ δ 1δ )e i β) e γ)j e δ)k. (2.6) According to the formula (2.5), V α)βγ;δ are given by V α)βγ;δ = L( k V α)βγ )e k δ) + V α)µγv µ)βδ + V α)βµ V µ)γδ. For α = 2, β = 3, we get V 2)3γ;δ = L( k V 2)3γ )e k δ) + V 2)µγV µ)3δ + V 2)3µ V µ)γδ = L( k V 2)3γ )e k δ) + V 2)1γV 1)3δ + V 2)4γ V 4)3δ + V 2)5γ V 5)3δ + V 2)3µ V µ)γδ V 2)3γ;δ = L( k u γ )e k δ) V 1)2γV 1)3δ v γ u δ w γ v δ + u µ V µ)γδ V 2)3γ;δ = L( k u γ )e k δ) δ 2γδ 3δ v γ u δ w γv δ + u µv µ)γδ V 2)3γ;δ = u γ;δ δ 2γ δ 3δ v γ u δ w γ v δ; 75

4 Anamika Rai, et al: On a five dimensional Finsler space with vanishing v-connection vectors where u γ;δ = L( k u γ )e k δ) + u µv µ)γδ. Similarly, we get V 2)4γ;δ = V γ;δ δ 2γ δ 4δ w γ w δ + u γ u δ V 2)5γ;δ = w γ;δ δ 2γ δ 5δ + u γ v δ + v γw δ V 3)4γ;δ = u γ;δ δ 3γ δ 4δ u γ v δ v γw δ V 3)5γ;δ = V γ;δ δ 3γδ 5δ u γ w δ + u γ w δ V 4)5γ;δ = w γ;δ δ 4γ δ 5δ v γ w δ u γv δ. 3 v-curvature Tensor The v curvature tensor is defined by S hijk = C r hkc ijr C r hjc ikr. The scalar components S αβγδ of L 2 S hijk are written as L 2 S hijk = S αβγδ e α)h e β)i e γ)j e δ)k. (3.1) Since S hijk is skew-symmetric in h and i as well as j and k and S 0ijk = S hi0k = 0, the surving independent components of S αβγδ are twenty two and these are S 2323, S 2324, S 2325, S 2334, S 2335, S 2424, S 2434, S 2435, S 3434, S 3435, S 4545, S 2425, S 3525, S 3535, S 3545, S 3425, S 3445, S 4523, S 4524, S 4525, S 4535, S A Finsler space F n (n 4) is called S-3 like, if there exists a scalar S such that the curvature tensor S hijk of F n is written in the form L 2 S hijk = S(h hj h ik h hk h ij ), (3.2) where h ij = g ij l i l j is the angular metric tensor. In terms of scalar components, the Ricci identity e i α) j k e i α) k j = e r α) Si rjk, may be written as (V α)βγ;δ V α)βγ δ 1δ ) (V α)βδ;γ V α)βδ δ 1γ ) = S αβγδ. 76

5 South Asian J. Math. Vol. 7 No. 2 For different values of α and β, we get (u γ;δ δ 2γ δ 3δ v γ u δ w γv δ u γδ 1δ ) (u δ;γ δ 2δ δ 3γ v δ u γ w δ v γ u δ δ 1γ ) = S 23γδ (v γ;δ δ 2γ δ 4δ w γ w δ + u γu δ v γδ 1δ ) (v δ;γ δ 2δ δ 4γ w δ w γ + u δ u γ v δ δ 1γ ) = S 24γδ (w γ;δ δ 2γ δ 5δ + u γ v δ + v γw δ w γδ 1δ ) (w δ;γ δ 2δ δ 5γ + u δ v γ + v δ w γ w δ δ 1γ ) = S 25γδ (u γ;δ δ 3γ δ 4δ u γ v δ v γw δ u γδ 1δ ) (u δ;γ δ 3δ δ 4γ u δ v γ (3.3) which gives us v δ w γ u δ δ 1γ) = S 34γδ (V γ;δ δ 3γ δ 5δ u γ w δ + u γw δ v γδ 1δ ) (V δ;γ δ 3δ δ 5γ u δ w γ + u δ w γ v δ δ 1γ) = S 35γδ (w γ;δ δ 4γ δ 5δ v γ w δ u γv δ w γδ 1δ ) (w δ;γ δ 4δ δ 5γ v δ w γ u δ v γ w δ δ 1γ) = S 45γδ S 2323 = (u 2;3 u 3;2 v 2 u 3 w 2v 3 + v 3u 2 + w 3v 2 ) 1 S 2324 = (u 2;4 u 4;2 v 2 u 4 w 2v 4 + v 4u 2 + w 4v 2 ) S 2325 = (u 2;5 u 5;2 v 2 u 5 w 2 v 5 + v 5 u 2 + w 5 v 2) S 2334 = (u 3;4 u 4;3 v 3 u 4 w 3v 4 + v 4u 3 + w 4v 3 ) S 2335 = (u 3;5 u 5;3 v 3 u 5 w 3 v 5 + v 5 u 3 + w 5 v 3) S 2424 = (V 2;4 V 4;2 w 2 w 4 + u 2u 4 + w 4w 2 u 4u 2 ) 1 S 2434 = (V 3;4 V 4;3 w 3 w 4 + u 3 u 4 + w 4 w 3 u 4 u 3) S 2435 = (V 3;5 V 5;3 w 3 w 5 + u 3 u 5 + w 5 w 3 u 5 u 3) S 3434 = (u 3;4 u 4;3 u 3v 4 v 3 w 4 + u 4v 3 + v 4 w 3 ) 1 S 3435 = (u 3;5 u 5;3 u 3 v 5 v 3w 5 + u 5 v 3 + v 5w 3) S 4545 = (w 4;5 w 5;4 v 4w 5 u 4 v 5 + v 5w 4 + u 5 v 4 ) 1 S 2425 = (V 2;5 w 2 w 5 + u 2 u 5 V 5;2 + w 5 w 2 u 5 u 2) S 3525 = (V 2;5 V 5;2 u 2w 5 + u 2 w 5 + u 5w 2 u 5 w 2 ) S 3535 = (V 3;5 V 5;3 u 3w 5 + u 3 w 5 + u 5w 3 u 5 w 3 ) 1 S 3545 = (V 4;5 V 5;4 u 4 w 5 + u 4w 5 + u 5 w 4 u 5w 4) S 3425 = (u 2;5 u 5;2 u 2v 5 v 2 w 5 + u 5v 2 + v 5 w 2 ) S 3445 = (u 4;5 u 5;4 u 4 v 5 v 4w 5 + u 5 v 4 + v 5w 4) S 4523 = (w 2;3 w 3;2 v 2w 3 u 2 v 3 + v 3w 2 + u 3 v 2 ) S 4524 = (w 2;4 w 4;2 v 2 w 4 u 2v 4 + v 4 w 2 + u 4v 2) (3.4) 77

6 Anamika Rai, et al: On a five dimensional Finsler space with vanishing v-connection vectors S 4525 = (w 2;5 w 5;2 v 2w 5 u 2 v 5 + v 5w 2 + u 5 v 2 ) S 4535 = (w 3;5 w 5;3 v 3 w 5 u 3v 5 + v 5 w 3 + u 5v 3) S 2545 = (w 4;5 w 5;4 + u 4 v 5 + v 4w 5 u 5v 4 v 5w 4 ). If the Finsler space F 5 is S 3 like then L 2 S hijk = S(h hj h ik h hk h ij ), i.e. L 2 S hijk = S{m i m k n h n j + m i m k p h p j + m i m k q h q j + m h m j n i n k + n i n j n k n h + n i n k p h p j + n i n k q h q j + p i p k m h m j + p i p k n h n j + p i p k p h p j + p i p k q h q j + q i q k m h m j + q i q k n h n j + q i q k p h p j + q i q k q h q j m i m j n h n k m i m j p h p k m i m j q h q k m h m k n i n j n i n j n k n h n i n j p h p k n i n j q h q k p i p j m h m k p i p j n h n k p i p j p h p k p i p j q h q k q i q j m h m k q i q j n h n k q i q j p h p k q i q j q h q k } which implies S 2323 = S, S 2324 = 0, S 2325 = 0, S 2334 = 0, S 2335 = 0, S 2424 = S, S 2434 = 0, S 2435 = 0, S 3434 = S, S 3435 = 0, S 4545 = S, S 2425 = 0 S 3525 = 0, S 3535 = S, S 3545 = 0, S 3425 = 0, S 3445 = 0, S 4523 = 0, S 4524 = 0, S 4525 = 0, S 4535 = 0, S 2525 = S. (3.5) From (3.4) and (3.5), we get u 2;3 u 3;2 v 2 u 3 w 2v 3 + v 3u 2 + w 3v 2 = S + 1 u 2;4 u 4;2 v 2 u 4 w 2 v 4 + v 4 u 2 + w 4 v 2 = 0 u 2;5 u 5;2 v 2 u 5 w 2 v 5 + v 5 u 2 + w 5 v 2 = 0 u 3;4 u 4;3 v 3 u 4 w 3v 4 + v 4u 3 + w 4v 3 = 0 u 3;5 u 5;3 v 3 u 5 w 3 v 5 + v 5 u 3 + w 5 v 3 = 0 v 2;4 v 4;2 w 2 w 4 + u 2u 4 + w 4w 2 u 4u 2 = S + 1 v 3;4 v 4;3 w 3 w 4 + u 3 u 4 + w 4 w 3 u 4 u 3 = 0 v 3;5 v 5;3 w 3 w 5 + u 3u 5 + w 5w 3 u 5u 3 = 0 u 3;4 u 4;3 u 3v 4 v 3 w 4 + u 4v 3 + v 4 w 3 = S + 1 u 3;5 u 5;3 u 3 v 5 v 3w 5 + u 5 v 3 + v 5w 3 = 0 w 4;5 w 5;4 v 4w 5 u 4 v 5 + v 5w 4 + u 5 v 4 = S + 1 v 2;5 w 2 w 5 + u 2 u 5 v 5;2 + w 5 w 2 u 5 u 2 = 0 v 2;5 v 5;2 u 2w 5 + u 2 w 5 + u 5w 2 u 5 w 2 = 0 v 3;5 v 5;3 u 3 w 5 + u 3w 5 + u 5 w 3 u 5w 3 = S + 1 (3.6) 78

7 South Asian J. Math. Vol. 7 No. 2 v 4;5 v 5;4 u 4w 5 + u 4 w 5 + u 5w 4 u 5 w 4 = 0 u 2;5 u 5;2 u 2 v 5 v 2w 5 + u 5 v 2 + v 5w 2 = 0 u 4;5 u 5;4 u 4v 5 v 4 w 5 + u 5v 4 + v 5 w 4 = 0 w 2;3 w 3;2 v 2 w 3 u 2v 3 + v 3 w 2 + u 3v 2 = 0 w 2;4 w 4;2 v 2w 4 u 2 v 4 + v 4w 2 + u 4 v 2 = 0 w 2;5 w 5;2 v 2w 5 u 2 v 5 + v 5w 2 + u 5 v 2 = 0 w 3;5 w 5;3 v 3 w 5 u 3v 5 + v 5 w 3 + u 5v 3 = 0 w 2;5 w 5;2 + u 2 v 5 + v 2w 5 u 5v 2 v 5w 2 = S + 1. Thus, we have: Theorem 3.1. A five-dimensional Finsler space is S 3 like if and only if the conditions (3.6) are satisfied. Corollary 3.1. A five-dimensional Finsler space is S 3 like if u γ;δ = v γ;δ = w γ;δ = u γ;δ = v γ;δ = w γ;δ = 0, v γ u δ + w γv δ v δ u γ + w δv γ = u γw δ + u δw γ u γ w δ + u δ w γ = u γu δ + w γ w δ w γ w δ + u δu γ = v γw δ + u γv δ v δ w γ + u δ v γ = v γu δ + v δ w γ u γ v δ + v γ w δ = u γv δ + v γw δ u δ v γ + v δ w γ. Corollary 3.2. A five-dimensional Finsler space with vanishing v-connection vectors is S 3 like v curvature 1. From Corollary 3.2, we have L 2 S hijk = 1(h hj h ik h hk h ij ). Taking h covariant differentiation on both sides and using L r = 0 and h ij r = 0, we get S hijk r = 0, and then the identity P hijk P hikj = S hijk r y r reduce to P hijk = P hikj, where P hijk is the hv curvature tensor defined by P hijk = C ijk h C hjk i + Chj r C rik s y s Cij r C rhk s y s. Thus, we have: Theorem 3.2. The hv curvature tensor P hijk of a five-dimensional Finsler space with vanishing v connection vectors is symmetric in j and k. A Finsler space is said to be a Landsberg space if hv curvature tensor P hijk vanishes. A P Finsler space is characterized by C ijk r y r = λc ijk, while a Finsler space is said to be a P 2 like Finsler space if P hijk = P h C ijk P i C hjk, where P i is a covariant vector field. M. Matsumoto [2] proved that if the hv curvature tensor P hijk is symmetric in j and k in a P Finsler space then either P hijk = 0 or S hijk = 0. Therefore, in view of theorem 3.2, P hijk = 0 or S hijk = 0 in a five-dimensional P Finsler space with vanishing v connection vectors. But for such 79

8 Anamika Rai, et al: On a five dimensional Finsler space with vanishing v-connection vectors space S hijk 0 in view of corollary 3.2. Therefore, P hijk = 0. Hence the space is a Landsberg space. Thus, we have: Theorem 3.3. A five-dimensional P Finsler space with vanishing v connection vectors is a Landsberg space. Since a P 2 like Finsler space is P Finsler space, we may conclude: Theorem 3.4. A five-dimensional P 2 like Finsler space with vanishing v connection vector is a Landsberg space. References 1 M. Matsumoto and R. Miron : On an invariant theory of Finsler spaces, Period. Math. Hunger, 8 (1977), M. Matsumoto : Foundations of Finsler geometry and special Finsler spaces, Kaiseisha press, Saikawa, Ostu, 520 (1986), Japan. 3 Gauree Shanker, G. C. Chaubey and Vinay Pandey : On the main scalars of a five-dimensional Finsler space, International Electronic J. Pure and Applied Math., 5 (2012),