www.ajms.com 46 ISSN 2581-3463 RESEARCH ARTICLE On Certain Special Vector Fields in a Finsler Space III P. K. Dwivedi, S. C. Rastogi, A. K. Dwivedi 1Department of Applied Science, Ambalika Institute of Management and Technology, Lucknow, Uttar Pradesh, India, 2 Professor (Rtd), Shagun Vatika, Lucknow, Uttar Pradesh, India, 3Department of Applied Science, Central Institute of Plastics Engineering and Technology, Lucknow, Uttar Pradesh, India Received: 15-07-2019; Revised: 15-09-2019; Accepted: 19-10-2019 ABSTRACT In an earlier paper in 2017, Rastogi and Bajpai [1] defined and studied a special vector field of the first kind in a Finsler space as follows: Definition 1: A vector field X i (x), in a Finsler space, is said to be a special vector field of the first kind, if (i) X i /j = - δ i j and (ii) X i h ij = Ɵ j , where Ɵ j is a non-zero vector field in the given Finsler space. In 2019, some more special vector fields in a Finsler space of two and three dimensions have been defined and studied by the authors Dwivedi et al .[2] and Dwivedi et al .[3] In Dwivedi et al .[3] , the authors defined and studied six kinds of special vector fields in a Finsler space of three dimensions and, respectively, called them special vector fields of the second, third, fourth, fifth, sixth, and seventh kind. In the present paper, we shall study some curvature properties of special vector fields of the first and seventh kind in a Finsler space of three dimensions. Key words: Curvature tensors, Finsler space,vector fields INTRODUCTION Let F 3 , be a three-dimensional Finsler space, with metric function L(x,y), metric tensor g ij = l i l j + m i m j + n i n j and angular metric tensor h ij = m i m j + n i n j , where l i = Δ i L and Δ i = ∂/∂y i , while m i and n i are vectors orthogonal to each other Matsumoto. [4] The torsion tensor A ijk = L C ijk = (L/2) Δ k g ij . The h- and v-covariant derivatives of a tensor field T i j (x,y) are defined as Matsumoto: [4] T i j/k = ∂ k T i j – N m k Δ m T i j + T m j F i mk – T i m F m jk (1.1) and T i j//k = Δ k T i j + T m j C i mk – T i m C m jk (1.2) where, ∂ k = ∂/∂x k and other terms have their usual meanings Matsumoto. [4] Corresponding to h- and v-covariant derivatives, in F 3 , we have: l i/j = 0, m i/j = n i h j , ni/j = - m i h j , (1.3) and l i//j = L -1 h ij , m i//j = L -1 (-l i m j + n i vj ), n i//j = - L -1 (l i nj + m i vj ) (1.4) where, h j and v j are respectively h- and v-connection vectors in F 3 . Furthermore, C ijk = C (1) m i m j m k – ∑ (I,j,k) {C (2) m i m j n k – C (3) m i nj n k } + C (2) ni nj n k (1.5) Corresponding to these covariant derivatives, we have following: T i j/k//h – T i j//h/k = T r j P i rkh – T i r P r jkh – T i j/r Cr kh – T i j//r P r kh (1.6) and T i j//k//h – T i j//h//k = T r j S i rkh – T i r S r jkh – T i j//r S r kh (1.7) where, P r jkh and Sr jkh are, respectively, the second and third curvature tensors of F 3 , Rund. [5] Address for correspondence: P. K. Dwivedi, drpkdwivedi@yahoo.co.in. Dwivedi, et al .: Special vector fields in a Finsler space AJMS/Oct-Dec-2019/Vol 3/Issue 4 47 PROPERTIES OF SPECIAL VECTOR FIELD OF THE FIRST KIND RELATED WITH THE SECOND CURVATURE TENSOR In F 3 , we assume X i (x) = A l i + B m i + D n i , (2.1) where, A, B, and D are scalars satisfying X i l i = A, X i m i = B, and X i n i = D such that for X i /j = - δ i j , we get A /j = - l j , B /j = D h j – m j , D j = -(B h j + n j ) (2.2) and A //j = L -1 (B m j + D n j ), B //j = (C (1) B – C (2) D - L -1 A) m j + (C (3) D- C (2) B) nj + L -1 D v j D //j = (C (3) D – C (2) B) m j + (C (3) B + C (2) D – L -1 A) n j – L -1 B v j (2.3) From definition 1., we can obtain Ɵ j = B m j + D n j , Ɵ j/k = - h jk . Furthermore, we get Ɵ j//r = (C (1) B – C (2) D) m j m r + (C (3) B + C (2) D) n j nr + (C (3) D – C (2) B) (m j nr + m r nj ) – L -1 (A h jr + l j Ɵ r ). (2.4) These equations help us to give Ɵ j/k//r = L -1 (l j hkr + l k h jr ) (2.5) and Ɵ j//r/k = {(C (1)/k + 3 C (3) hk ) B – (C (2)/k – (C(1) – 2 C (3) ) h k ) – C (1) m k + C (2) n k } m j m r + {(C (3)/k – 3 C (2) hk )B + (C (2)/k + 3 C (3) hk ) -C (3) m k – C (2) n k } n j nr + {(C (3)/k – 3 C(2) hk )D – (C (2)/k + (2 C (3) – C (1) )h k )B + C (2) m k – C (3) n k }} (m j nr + m r nj ) + L -1 (l j hkr + l k h jr ) (2.6) Using Equations (1.6), (2.5), and (2.6) on simplification, we obtain Ɵ t P t jkr + Ɵ j//t P t kr - {(C (1)/k + 3 C (3) hk ) B – (C (2)/k – (C(1) – 2 C (3) ) h k ) D} m j m r - {(C (3)/k – 3 C (2) hk )B + (C (2)/k + 3 C (3) hk ) D} n j nr - {(C (3)/k – 3 C (2) hk )D – (C (2)/k + (2 C (3) – C (1) )h k )B} (m j nr + m r nj ) = 0. (2.7) Hence: Theorem 2.1 In a three-dimensional Finsler space F 3 , for a special vector field of the first kind, the second curvature tensor Pt jkr satisfies Equation (2.7). PROPERTIES OF SPECIAL VECTOR FIELDS OF THE FIRST KIND RELATED WITH THE THIRD CURVATURE TENSOR From Equation (2.4), we can get on simplification Ɵ j//r//k = B{C (1)//k m j m r + C (3)//k nj nr – C (2)//k (m j nr + m r nj )} + D{C (2)//k nj nr – C (2)//k m j m r + C (3)//k (m j nr + m r nj )} + B //k {C (1) m j m r + C (3) nj nr – C (2) (m j nr + m r nj )} + D //k {C (2) nj nr – C (2) m j m r + C (3) (m j nr + m r nj )} + m j//k {(C (1) B – C (2) D - L -1 A) m r + (C (3) D – C (2) B) nr } + n j//k {(C (3) B + C (2) D – L -1 A) n r + (C (3) D – C (2) B) m r } + m r//k {(C (1) B – C (2) D – L -1 A) m j + (C (3) D – C (2) B) n j } + n r//k {(C (3) B + C (2) D – L -1 A) n j + (C (3) D – C (2) B) m j } + L -2 {l k(A h jr + l j Ɵ r ) – L A //k h jr – hjk Ɵ r – L l j Ɵ r//k } (3.1) Using Equations (1.7) and (3.1), after some lengthy calculation, we can obtain Ϛ (k,r) [B{(C (1)//r + 2 L -1 C (2) vr ) m j m k – (C(2)//r + 2 L -1 C (3) vr ) m j n k + L -1 C(1) m j (l r m k + v r n k ) – L -1 C (2) m j (l r n k + m r v k ) + (C (3)//r – 3 L -1 C (2) vr ) n j n k – (C (2)//r + (C (1) C (3) – 2 C (2) 2 – C (3) 2 )n r ) n j m k + L -1 C (3) (l r n k + m r v k ) n j – L -1 ((C (3) – C (1) ) v r + C (2) l r )n j m k Dwivedi, et al .: Special vector fields in a Finsler space AJMS/Oct-Dec-2019/Vol 3/Issue 4 48 – L -2 nj n k m r +L -1 l j l r m k } + D{(C (3)//r – 3 L -1 C (2) vr + L -1 C (3) l r )m j n k - (C (2)//r + (C (1) C (3) + C (3) 2)+ L -2 )n r – L -1 ((2 C (3) - C(1) ) v r + C (2) l r ) m j m k + (C (2)//r – 2 L -1 C (2) vr + L -1 C (3) l r ) n j m k + (C (3)//r + L -1 (3C(3) vr + C (2) l r ))n j n k + L -1 (C(2) nj m r v k – L -1 l j l k nr )} – L -2 A{(l r n k + m r v k ) m j – (l r n k – vr m k ) n j }] + Ɵ p S p jkr + Ɵ j//p S p kr = 0. (3.2) Hence: Theorem 3.1 In a three-dimensional Finsler space F 3 , for a special vector field of the first kind, the third curvature tensor St jkr satisfies Equation (3.2). PROPERTIES OF SPECIAL VECTOR FIELD OF THE SEVENTH KIND The special vector field of the seventh kind is defined as follows: [3] Definition 2 A vector field X i (x), satisfying i) X i /j = - δ i j and X i Y ij = Y i , where Y i is a non-zero vector field in F 3 and Y ij = m i nj – m j n i is a tensor field, is called special vector field of the seventh kind. From this definition, we can observe that Y j = B n j – D mj , Y j/k = Y jk (4.1) and Y j//k = (C (1) B – C (2) D) m k nj – (C(3) B + C (2) D) m j n k + L -1 {A (m j n k – m k nj ) - l j Y k } + (C (3) D – C (2) B)(n j n k – m j m k ) (4.2) From Equation (4.1), we can obtain Y j/k//r = L -1 (l j Y kr + l k Y jr ) (4.3) while from Equation (4.2), we get Y j//r/k = {C (1)/k B – C (2)/k D + C (1) (D h k – m k ) + C (2) (B hk + n k )}m r nj – ((C (1) B - C (2) D)(n j nr – m j m r ) h k + {C (3)/k B + C (2)/k D + C (3) (D h k – m k ) – C(20) (B hk + n k )}m j nr + (C (3) B + C (2) D). (n j nr – m j m r ) h k + L -1 {l k (m j nr – m r nj ) + l j Y rk } – {C (3)/k D - C (2)/k B – C (3) (B hk + n k ) – C(2) (D h k – m k )}(n j nr – m j m r ) + 2(C (3) D – C (2) B)(m j nr + m r nj ) h k (4.4) Equations (4.3) and (4.4) with the help of Equation (1.6) lead to B[(C (3)/k – 3 C (2) hk ) m j nr – (C (1)/k + 3 C (2) hk ) m r nj + {C (2)/k + (2 C (3) – C (1) ) h k }(n j nr – m j m r )] + D[{C (2)/k + (2 C (3) – C (1) ) h k } m r nj + (C (2)/k + 3 C (3) hk ) m j nr -(C (3)/k – 3 C (2) hk )(n j nr – m j mr)] – m k {C (2) (n j nr – m j m r ) – C (1) m r nj + C (3) m j nr } + n k {C (3) (n j nr – m j m r ) – C (2) (m r nj + m j nr )} + Y t P t jkr + Y j//t P t kr = 0. (4.5) Hence: Theorem 4.1 In a three-dimensional Finsler space F 3 , for a special vector field of the seventh kind, the second curvature tensor satisfies Equation (4.5). From Equation (4.2), we can get Y j//k//r = m j [{C (2)//r B – C (3)//r D + C (2) B //r – C (3) D //r )m k + (C (2) B – C (3) D). L -1 (-l k m r + n k vr )} – {C (3)//r B + C (2)//r D – L -1 (B m r + D n r ) + L -2 A l r + C (3) B //r + C (2) D //r )} n k + (C (3) B + C (2) D- L -1 A) L -1 (l k nr + m k vr )] Dwivedi, et al .: Special vector fields in a Finsler space AJMS/Oct-Dec-2019/Vol 3/Issue 4 49 + n j [{(C (1)//r B -C (2)//r D + C (1) B //r – C (2) D //r + L -2 A l r – L -2 (B m r + D n r )} m k + L -1 (-l k m r +n k vr )(C (1) B- C (2) D – L -1 A) +(C (3)//r D – C (2)//r B+ C (3) D //r – C (2) B //r )n k + L -1 (C(3) D – C (2) B). (l k nr + m k vr )] + L -1 (-l j m r + n j vr ){(C (2) B – C (3) D) m k – (C(3) B + C (2) D -L -1 A) n k } – L -1 (l j nr + m j vr ){(C (1) B – C (2) D – L -1 A)m k + (C (3) D – C (2) B) n k } + L -2 Y k (l j l r – h jr ) – L -1 l j Y kr , (4.6) which by virtue of Equation (1.7) after some lengthy calculation leads to Y t S t jkr + Ϛ (k,r) [m j m k {B(C (2)//r – (C(2)2 – C (3) 2 ) n r – L -1 (C(1) + C (2) ) v r ) - D(C (3)//r - L -1 C (2) vr ) – L -1 A C (3) nr } – n j n k {B(C (2)//r + C (2) (C(1) + C (3) )m r + 2 L -1 C (3) vr ) + D(C (3)//r – (C(2)2 + C (3) 2) m r – L -1 C (2) vr ) + L -1 A C (2) m r } + m j n k {B(C (2) 2 -C(1) C (3) )m r – C (3)//r + L -1 m r + L -1 (C(1) +3 C (2) ) v r ) + D(C (2)//r + 2 C (2) C (3) m r – L -1 m r - L -1 (C(2) – 3 C (3) ) v r ) + L -1 A C (3) m r } - m k nj {B(C (1)//r + C (2) (C(1) – C (3) ) n r ) + D(C (2)//r – (C(2)2 + C (1) C (3) ) n r – L -1 C (1) vr ) – L -1 A C (2) nr } – L -1 l k {C (3) h jr + C (2) (m j nr + m r nj )}] = 0. (4.7) Hence: Theorem 4.2 In a three-dimensional Finsler space, for a special vector field of the seventh kind, the third curvature tensor St jkr satisfies Equation (4.7). REFERENCES 1. Rastogi SC, Bajpai P. On certain special vector fields in a finsler space. Acta Cienc Indic 2017;43:149-52. 2. Dwivedi PK, Rastogi SC, Dwivedi AK. On certain special vector fields in a finsler space-1. Int J Adv Innov Res 2019;8:1-6. 3. Dwivedi PK, Rastogi SC, Dwivedi AK. On certain special vector fields in a finsler space-2 (Under Publication). Int J Sci Res Math Stat Sci 2019;6:108-12. 4. Matsumoto M. Foundations of Finsler Geometry and Special Finsler Spaces. Otsu, Japan: Kaiseisha Press; 1986. 5. Rund H. The Differential Geometry of Finsler Spaces. Berlin: Springer-Verlag; 1959.