HYPERSURFACE OF A FINSLER SPACE WITH DEFORMED BERWALD-MATSUMOTO METRIC

Transcription

1 Bulle n of the Transilvania University of Braşov Vol 11(60), No Series III: Mathema cs, Informa cs, Physics, HYPERSURFACE OF A FINSLER SPACE WITH DEFORMED BERWALD-MATSUMOTO METRIC V. K. CHAUBEY 1 and Brijesh Kumar TRIPATHI,2 Abstract In the present paper we have studied the Finslerian hypersurfaces of a Finsler space with the special deformed Berwald-Matsumoto metric. We also examined the hypersurfaces of this special metric as a hyperplane of first, second and third kinds Mathema cs Subject Classifica on: 53B40, 53C60. Key words: Finslerian hypersurface, Matsumoto metric and Berwald metric. 1 Introduc on P. Finlser was the first who introduced the slope of a mountain with respect to me measurement but he thought that (as he men oned in his le er [9] to Matsumoto) it was a typical model of Finsler mertric. Further in 1989, Matsumoto [9] worked on the above problem and introduced the concept of slope metric which is defined as α 2 vα wβ where α 2 = a ij (x)y i y j is a Riemannian metric, β = b i (x)y i is a 1 form on n-dimensional manifold M n, v and w are the non-zero constants. In 1990 Aikou and coauthors [1] studied the above metric in detail and named it Matsumoto metric and obtained very interes ng results for this metric in stand point of Finsler metric. In 1929 Berwald [2] introduced a very famous Finsler metric which was defined on unit ball B n (1) with all the straight line segments. Its geodesics has constant flag curvature K = 0 in the form of L = { 1 x 2 y 2 + x, y 2 + x, y } 2 (1 x 2 ) 2 1 x 2 y 2 + x, y 2 (1) From a modern point of view the above Berwald s metric belongs to a special kind of Finsler metric called Berwald type metric and defined as (α+β)2 α [12] and the authors of 1 Department of Applied Sciences, Buddha Ins tute of Technology, Sector-7, GIDA, Gorakhpur (U.P.) India, vkcoct@gmail.com 2 Department of Mathema cs, L.E. College Morbi (Gujarat) India, brijeshkumartripathi4@gmail.com Corresponding Author

2 38 V. K. Chaubey and Brijesh Kumar Tripathi the papers introduced very interes ng results in the field of Finsler geometry. The concept of Finslerian hypersurface was first introduced by Matsumoto in 1985 and further he defined three types of hypersurfaces that were called a hyperplane of the first, second and third kinds. Further, many authors studied these hyperplanes in different changes of the Finsler metric [3, 4, 5, 6, 10, 11] and obtained different results. In the present paper we combine the Berwald and Matsumoto metric and obtain a new metric named deformed Berwald-Matsumoto metric. We also examine the hypersurfaces of this special metric as a hyperplane of first, second and third kinds. 2 Preliminaries In the present paper we consider an n-dimensional Finsler space F n = {M n, L(α, β)}, that is, a pair consis ng of an n- dimensional differen able manifold M n equipped with a Fundamental func on L as a special Finsler Space with the metric L(α, β) = (α + β)2 α + α2 (α β) i.e. the combina on of Berwald and Matsumoto metric named deformed Berwald-Matsumoto metric and the Finsler space F n equiped with this metric is known as Berwald- Matsumoto Finsler space. Differen a ng equa on (2) par ally with respect to α and β we get L α = (α+β)(α β)3 +α 3 (α 2β), L α 2 (α β) 2 β = 2(α+β)(α β)2 +α 3 α(α β) 2 L αα = 2{α3 +(α β) 3 }β 2 L α 3 (α β) 3 ββ = 2{α3 +(α β) 3 } α(α β) 3 L αβ = 2{α3 +(α β) 3 }β α 2 (α β) 3 (2) where L α = L α, L β = L β, L αα = Lα α, L ββ = Lβ β, L αβ = Lα β. In Finsler space F n = {M n, L(α, β)} the normalized element of support l i = i L and angular metric tensor h ij are given by [7]: l i = α 1 L α Y i + L β b i h ij = pa ij + q 0 b i b j + q 1 (b i Y j + b j Y i ) + q 2 Y i Y j where Y i = a ij y j. For the fundamental metric func on (2) the above constants are p = 4α7 + β 7 6α 6 β 6α 5 β 2 6α 4 β 3 + 4α 3 β 4 3α 2 β 5 αβ 6 α 4 (α β) 3 (3)

3 Hypersurface of Finsler space with q 0 = 8α6 8αβ 5 + 2α 2 β 4 + 8α 3 β 3 2α 4 β 2 6α 5 β + 2β 6 α 2 (α β) 4, q 1 = 8α6 β + 8αβ 6 2α 2 β 5 8α 3 β 4 + 2α 4 β 3 + 6α 5 β 2 2β 7 α 4 (α β) 4, q 2 = = 4α8 4α 6 β α 3 β 5 26α 5 β α 7 β + 10α 4 β 4 4α 2 β 6 6αβ 7 + 3β 8 α 6 (α β) 4 Fundamental metric tensor g ij = 1 2 i j L 2 and its reciprocal tensor g ij for L = L(α, β) are given by [7] where g ij = pa ij + p 0 b i b j + p 1 (b i Y j + b j Y i ) + p 2 Y i Y j (5) p 0 = q 0 + L 2 β, (6) p 1 = q 1 + L 1 pl β p 2 = q 2 + p 2 L 2 The reciprocal tensor g ij of g ij is given by g ij = p 1 a ij s 0 b i b j s 1 (b i y j + b j y i ) s 2 y i y j (7) where b i = a ij b j and = a ij b i b j s 0 = 1 τp {pp 0 + (p 0 p 2 p 2 1)α 2 }, (8) s 1 = 1 τp {pp 1 + (p 0 p 2 p 2 1)β}, s 2 = 1 τp {pp 2 + (p 0 p 2 p 2 1) }, τ = p(p + p 0 + p 1 β) + (p 0 p 2 p 2 1)(α 2 β 2 ) The hv-torsion tensor C ijk = 1 2 k g ij is given by [11] where, 2pC ijk = p 1 (h ij m k + h jk m i + h ki m j ) + γ 1 m i m j m k (9) γ 1 = p p 0 β 3p 1q 0, m i = b i α 2 βy i (10) Here m i is a non-vanishing covariant vector orthogonal to the element of support y i. Let { i jk } be the component of christoffel symbols of the associated Riemannian space R n and k be the covariant deriva ve with respect to x k rela ve to this christoffel symbol. Now we define, 2E ij = b ij + b ji, 2F ij = b ij b ji (11)

4 40 V. K. Chaubey and Brijesh Kumar Tripathi where b ij = j b i. Let CΓ = (Γ i jk, Γ i 0k, Γi jk ) be the Cartan connec on of F n. The difference tensor Djk i = Γ i jk {i jk } of the special Finsler space F n is given by where D i jk = Bi E jk + F i k B j + F i j B k + B i jb 0k + B i k b 0j b 0m g im B jk (12) C i jma m k Ci km Am j + C jkm A m s g is + λ s (C i jmc m sk + C i km Cm sj C m jk Ci ms) B k = p 0 b k + p 1 Y k, B i = g ij B j, F k i = g kj F ji (13) B ij = {p 1(a ij α 2 Y i Y j ) + p 0 β m im j }, Bi k = g kj B ji 2 A m k = Bm k E 00 + B m E k0 + B k F0 m + B 0 Fk m λ m = B m E B 0 F0 m, B 0 = B i y i where 0 denotes contrac on with y i except for the quan es p 0, q 0 and s o. 3 Induced Cartan Connec on Let F n 1 be a hypersurface of F n given by the equa on x i = x i (u α ) where {α = 1, 2, 3...(n 1)}. The element of support y i of F n is to be taken tangen al to F n 1, that is [8], y i = Bα(u)v i α (14) the metric tensor g αβ and hv-tensor C αβγ of F n 1 are given by g αβ = g ij B i αb j β, C αβγ = C ijk B i αb j β Bk γ and at each point (u α ) of F n 1, a unit normal vector N i (u, v) is defined by g ij {x(u, v), y(u, v)}bαn i j = 0, g ij {x(u, v), y(u, v)}n i N j = 1 Angular metric tensor h αβ of the hypersurface are given by h αβ = h ij BαB i j β, h ijbαn i j = 0, h ij N i N j = 1 (15) (Bi α, N i) inverse of (Bα, i N i ) is given by Bi α = g αβ g ij B j β, Bi αb β i = δα, β Bi αn i = 0, BαN i i = 0 N i = g ij N j, B k i = g kj B ji, B i αb α j + N i N j = δ i j The induced connec on ICΓ = (Γ α βγ, Gα β, Cα βγ ) of F n 1 from the Cartan s connec on CΓ = (Γ i jk, Γ i 0k, C i jk ) is given by [8].

5 Hypersurface of Finsler space with Γ α βγ = Bα i (Bi βγ + Γ i jk Bj β Bk γ) + M α β H γ G α β = Bα i (Bi 0β + Γ i 0j Bj β ), Cα βγ = Bα i Ci jk Bj β Bk γ where M βγ = N i C i jk Bj β Bk γ, M α β = gαγ M βγ, H β = N i (B i 0β + Γ i oj Bj β ) and B i βγ = Bi β u γ, B i 0β = Bi αβ vα The quan es M βγ and H β are called the second fundamental v-tensor and normal curvature vector respec vely [8]. The second fundamental h-tensor H βγ is defined as [8] H βγ = N i (B i βγ + Γ i jk Bj β Bk γ) + M β H γ (16) where M β = N i C i jk Bj β N k (17) The rela ve h and v-covariant deriva ves of projec on factor B i α with respect to ICΓ are given by B i α β = H αβn i, B i α β = M αβ N i It is obvious form equa on (15) that H βγ is generally not symmetric and The above equa on yields H βγ H γβ = M β H γ M γ H β (18) H 0γ = H γ, H γ0 = H γ + M γ H 0 (19) We shall use the following lemmas which are due to Matsumoto [8] in the coming sec on Lemma 1. The normal curvature H 0 = H β v β vanishes if and only if the normal curvature vector H β vanishes. Lemma 2. A hypersurface F (n 1) is a hyperplane of the first kind with respect to connec on CΓ if and only if H α = 0. Lemma 3. A hypersurface F (n 1) is a hyperplane of the second kind with respect to connec on CΓ if and only if H α = 0 and H αβ = 0. Lemma 4. A hypersurface F (n 1) is a hyperplane of the third kind with respect to connec on CΓ if and only if H α = 0 and H αβ = M αβ = 0.

6 42 V. K. Chaubey and Brijesh Kumar Tripathi 4 Hypersurface F (n 1) of a deformed Berwald-Matsumoto Finsler space Let us consider a Finsler space with the deformed Berwald-Matsumoto metric L(α, β) = (α+β) 2 α + α2 (α β), where, α2 = a ij (x)y i y j is a Riemannian metric and vector field b i (x) = b is a gradient of some scalar func on b(x). Now we consider a hypersurface F (n 1) x i given by equa on b(x) = c, a constant [11]. From the parametric equa on x i = x i (u α ) of F n 1, we get b(x) u α = 0 b(x) x i x i u α = 0 b i B i α = 0 Above it is shown that b i (x) are covarient components of a normal vector field of hypersurface F n 1. Further, we have b i B i α = 0 and b i y i = 0 i.e β = 0 (20) and induced matric L(u, v) of F n 1 is given by which is a Riemannian metric. L(u, v) = a αβ v α v β, a αβ = a ij B i αb j β (21) Wri ng β = 0 in equa ons (3), (4) and (6) we get s 0 = from (5) we get, g ij = 1 4 aij p = 4, q 0 = 8, q 1 = 0 q 2 = 4α 2 (22) p 0 = 17 p 1 = 6α 1 p 2 = 0 τ = 16(1 + 2 ), (1 + 2 s 1 = ) 8α(1 + 2 s 2 = ) 16α 2 (1 + 2 ) 1 2(1 + 2 ) bi b j thus along F n 1, (22) and (19) lead to 3 8α(1 + 2 ) (bi y j + b j y i ) + g ij b i b j = 4(1+2 ) 9 16α 2 (1 + ) yi y j (23) So we get b b i (x(u)) = 2 4(1 + 2 ) N i, = a ij b i b j (24)

7 Hypersurface of Finsler space with where b is the length of vector b i Again from (22) and (23), we get thus we have, b i = a ij b j = 4 (1 + 2 )N i + 3b2 y i 2α (25) Theorem 1. In a deformed Berwald-Matsumoto Finsler hypersurface F (n 1), the Induced Riemannian metric is given by (20) and the scalar func on b(x)is given by (23) and (24). Now the angular metric tensor h ij and metric tensor g ij of F n are given by h ij = 4a ij + 8b i b j 4 α 2 Y iy j and g ij = 4a ij + 17b i b j + 6 α (b iy j + b j Y i ) (26) From equa on (19), (25) and (14) it follows that if h (a) αβ denote the angular metric tensor of the Riemannian a ij (x), then we have along F n 1, h αβ = h (a) αβ. thus along F n 1, p 0 β = 26 α from equa on (9) we get r 1 = 10 α, m i = b i then hv-torsion tensor becomes C ijk = 3 4α (h ijb k + h jk b i + h ki b j ) 5 4α b ib j b k (27) in the deformed Berwald-Matsumoto Finsler hypersurface F (n 1). Due to facts from (14), (15), (17), (19) and (26) we have M αβ = 3 4α 4(1 + 2 ) h αβ and M α = 0 (28) Therefore, from equa on (18) it follows that H αβ is symmetric. Thus we have Theorem 2. The second fundamental v-tensor of the deformed Berwald-Matsumoto Finsler hypersurface F (n 1) is given by (27) and the second fundamental h-tensor H αβ is symmetric. Now from (19) we have b i B i α = 0. Then we have b i β B i α + b i B i α β = 0

8 44 V. K. Chaubey and Brijesh Kumar Tripathi Therefore, from (17) and using b i β = b i j B j β + b i j N j H β, we have since b i j = b h Cij h, we get bi jb Therefore from equa on (28) we have, b i j B i αb j β + b i jb i αn j H β + b i H αβ N i = 0 (29) i αn j = 0 4(1 + 2 ) H αβ + b i j B i αb j β = 0 (30) because b i j is symmetric. Now contrac ng (29) with v β and using (13) we get 4(1 + 2 ) H α + b i j B i αy j = 0 (31) Again contrac ng by v α equa on (4.12) and using (13), we have 4(1 + 2 ) H 0 + b i j y i y j = 0 (32) From lemma (1) and (2), it is clear that the deformed Berwald-Matsumoto hypersuface is a hyperplane of first kind if and only if H 0 = 0. Thus from (31) it is obvious that F (n 1) F n 1 is a hyperplane of first kind if and only if b i j y i y j = 0. This b i j being the covariant deriva ve with respect to CΓ of F n defined on y i, but b ij = j b i is the covariant deriva ve with respect to Riemannian connec on { i jk } constructed from a ij(x). Hence b ij does not depend on y i. We shall consider the difference b i j b ij where b ij = j b i in the following. The difference tensor D i jk = Γ i jk {i jk } is given by (11). Since b i is a gradient vector, then from (10) we have Thus (11) reduces to where E ij = b ij F ij = 0 and F i j = 0 D i jk = Bi b jk + B i jb 0k + B i k b 0j b 0m g im B jk C i jma m k (33) C i km Am j + C jkm A m s g is + λ s (C i jmc m sk + C i km Cm sj C m jk Ci ms) B i = 17b i + 6α 1 Y i, B i 2 3 = ( )bi + 2α(1 + 2 ) yi, (34) λ m = B m b 00, B ij = 3 α (a ij Y iy j α 2 ) + 13 α b ib j, Bj i = 3 4α (δi j α 2 y i 7 Y j ) + 4α(1 + 2 ) bi b j (9 + 39b2 ) 8α 2 (1 + 2 ) b jy i, A m k = Bm k b 00 + B m b k0

9 Hypersurface of Finsler space with In view of (21) and (22), the rela on in (12) becomes equa on (33). Further we have B i 0 = 0, B i0 = 0 which leads A m 0 = Bm b 00. Now contrac ng (32) by y k we get D i j0 = Bi b j0 + B i j b 00 B m C i jm b 00 Again contrac ng the above equa on with respect to y j and we have D i 00 = Bi b 00 = {( )b i + 3 2α(1+2 ) yi }b 00 Paying a en on to (19), along F (n 1), we get b i D i j0 = 2 (1 + 2 ) b j0 + (3 + 13b2 ) 4α(1 + 2 ) jb 00 + (1 + 2 ) b ib m Cjmb i 00 (35) Now we contract (34) by y j we have b i D i 00 = From (15), (23), (24), (27) and M α = 0, we have 2 (1 + 2 ) b 00 (36) b i b m C i jm Bj α = M α = 0 Thus the rela on b i j = b ij b r Dij r equa ons (34) and (35) give b i j y i y j = b 00 b r D r 00 = b 00 Consequently (30) and (31) may be wri en as 4(1 + 2 ) H α + 4(1 + 2 ) H b i0b i α = 0, (37) b 00 = 0 Thus condi on H 0 = 0 is equivalent to b 00 = 0. Using the fact that β = b i y i = 0 the condi on b 00 = 0 can be wri en as b ij y i y j = b i y i b j y j for some c j (x). Thus we can write, Now from (19) and (37) we get 2b ij = b i c j + b j c i (38) b 00 = 0, b ij B i αb j β = 0, b ijb i αy j = 0

10 46 V. K. Chaubey and Brijesh Kumar Tripathi Hence from (36) we get H α = 0, again from (37) and (33) we get b i0 b i = c 0 2, λm = 0, A i j Bj β = 0 and B ijb i αb j β = 3 α h αβ. Now we use equa ons (15), (22), (23), (24), (27) and (32) then we have Thus equa on (29) reduces to Hence the hypersurface F n 1 b r D r ijb i αb j β = 3c 0 8α(1 + 2 ) h αβ (39) 4(1 + 2 ) H 3 αβ + 8α(1 + 2 ) h αβ = 0 (40) is umbilic. Theorem 3. The necessary and sufficient condi on for a deformed Berwald-Matsumoto hypersurface F (n 1) to be a hyperplane of first kind is (37). In this case the second fundamental tensor of F n 1 is propor onal to its angular metric tensor. Now from lemma (3), F (n 1) is a hyperplane of second kind if and only if H α = 0 and H αβ = 0. Thus from (38), we get c 0 = c i (x)y i = 0 Therefore there exists a func on ψ(x) such that Therefore from (37) we get This can also be wri en as c i (x) = ψ(x)b i (x) 2b ij = b i (x)ψ(x)b j (x) + b j (x)ψ(x)b i (x) b ij = ψ(x)b i b j Theorem 4. The necessary and sufficient condi on for a deformed Berwald-Matsumoto hypersurface F (n 1) to be a hyperplane of second kind is (39). Again lemma (4) and with equa on (27) and M α = 0 show that F n 1 does not become a hyperplane of third kind. Theorem 5. The deformed Berwald-Matsumoto hypersurface F (n 1) is not a hyperplane of the third kind.

11 Hypersurface of Finsler space with References [1] Aikou, T., Hashiguchi, M. and Yamauchi, K.: On Matsumoto s Finsler space with me measure, Rep. Fac. Sci., Kagoshima Univ., (Math., Phys. and Chem.) 23, (1990), [2] Berwald, L.: Über die n-dimensionalen Geometrien konstanter Krümmung, in denen die Geraden die kürzesten sind. Math. Z. 30, (1929), [3] Chaubey, V. K. and Mishra, A.: Hypersurface of a Finsler Space with Special (α, β)- metric, Journal of Contemporary Mathema cal Analysis 52 (2017), 1-7. [4] Chaubey, V. K. and Tripathi, B. K..: Finslerian Hypersurface of a Finsler Spaces with Special (γ, β)-metric, Journal of Dynamical System and Geometric Theories 12 (2014), no. 1, [5] Kitayama, M.: On Finslerian hypersurfaces given by β change, Balkan Journal of Geometry and Its Applica ons 7 (2002), no. 2, [6] Lee, I. Y., Park, H. Y. and Lee, Y. D.: On a hypersurface of a special Finsler spaces with a metric (α + β2 α ), Korean J. Math. Sciences 8 (2001), [7] Matsumoto, M.: Theory of Finsler spaces with (α, β)-metric, Rep. on Math, Phys. 31 (1992), [8] Matsumoto, M.: The induced and intrinsic Finsler connec ons of a hypersurface and Finslerian projec ve geometry, J. Math. Kyoto Univ. 25 (1985), [9] Matsumoto, M.: A slope of Mountain is a Finsler surface with respect to me measure, J. Math. Kyoto Univ. 29 (1989), no. 1, [10] Pandey, T. N. and Tripathi, B. K. : On a hypersurface of a Finsler Space with Special (α, β)-metric, Tensor, N. S., 68 (2007), [11] Singh, U. P. and Kumari, Bindu: On a hypersurface of a Matsumoto space, Indian J. pure appl. Math. 32 (2001), [12] Shen, Z. M. and Yu, C. T.: On Einstein square metrics, Publ. Math. Debr. 85 (2014), no. 3-4,

12 48 V. K. Chaubey and Brijesh Kumar Tripathi