Internatonal Journal of Trend n Research and Development, Volume 2(6), ISSN 2394-9333 www.jtrd.com Projectve change between two Specal (, β)- Fnsler Metrcs Gayathr.K 1 and Narasmhamurthy.S.K 2 1 Assstant Professor, Department of Mathematcs, S.I.T, Tumkur, Karnataka,Inda 2 Professor and Charman, Department of Mathematcs, Kuvempu Unversty, Shankaragatta, Karnataka. Inda. Abstract: In Fnsler space we see specal metrcs such as Randers metrc, Kropna metrc and Matsumoto metrc., etc. Projectve change between two Fnsler metrcs arse from Informaton Geometry. Such metrcs have specal geometrc propertes and wll play an mportant role n Fnsler geometry. In ths paper,we are gong to study class of Projectve change between two, β metrcs, whch are defned as the sum of a Remannan metrc and 1 form. Keywords: Fnsler metrc, Specal Fnsler metrc,, β metrc, Douglas Space, Geodesc, Spray coeffcents, Projectvely related metrc, Projectve change between two metrcs. I. INTRODUCTION The projectve change between two Fnsler spaces have been studed by many authors ( 1, 5, 6, 8 ). An nterestng result concerned wth the theory of projectve change was gven by Rapscak s paper. He proved necessary and suffcent condtons for projectve change. S. Bacso and M. Matsumoto 2 dscussed the projectve change between Fnsler spaces wth, β metrc. H. S. Park and Y. Lee 6 studed on projectve changes between a Fnsler space wth, β metrc and the assocated Remannan metrc. Recently some results on a class of, β metrcs wth constant flag curvature have been studed by Nngwe Cu, Y-Bng Shen 5, N. Cu and Z. Ln. II. SOME IMPORTANT DEFINITIONS 2.1 Defnton: A Fnsler geometry s just Remannan geometry wthout the quadratc restrcton. 2.2 Defnton: A Fnsler metrc s a scalar feld L(x, y) whch satsfes the followng three condtons:. It s defned and dfferentable at any pont of TM n \{0},. It s postvely homogeneous of frst degree n y, that s, L x, λy = λl(x, y), for any postve number λ,. It s regular, that s, g j x, y = 1 2 j L 2, consttute the regular matrx g j, where = y. The manfold M n equpped wth a fundamental functon L x, y s called Fnsler metrc F n = (M n, L). 2.3 Defnton: Two Fnsler metrcs L and L are projectvely related f and only f ther spray coeffcents have the relaton G = G + P y y (2. 1) 2.4 Defnton : A Fnsler metrc s projectvely related to another metrc f they have the same geodescs as pont sets. In Remannan geometry, two Remannan metrcs and are projectvely related f and only f ther spray coeffcents have the relaton G = G + λ x ky k y 2. 2 2.5 Defnton : Let φ s sφ s + b 2 s 2 φ s > 0, s b b 0. (2. 3) If = a j y y j s a Remannan metrc and β = b y s 1-form satsfyng β x < b 0 x M, then L = φ s, s = β/, s called an (regular), β metrc. In ths case, the fundamental Avalable Onlne@www.jtrd.com 43
Internatonal Journal of Trend n Research and Development, Volume 2(6), ISSN 2394-9333 www.jtrd.com form of the metrc tensor nduced by L s postve defnte. 2.6 Defnton: Let F n = (M n, L) and F n = (M n, L) be two Fnsler spaces on a common underlyng manfold M n. If any geodesc on F n s also a geodesc on F n and the converse s true, then the change L L of the metrc s called a projectve change. The relaton between the geodesc coeffcents G of L and geodesc coeffcents G of s gven by G = G + Qs 0 + 2Qs 0 + r 00 Ψb + Θ 1 y, (2. 4) where, Θ = φφ s(φφ +φ φ ) 2φ( φ sφ +(b 2 s 2 )φ ), Q = φ φ sφ, Ψ = 1 2 2.7 Defnton: Let D jkl 1 G m φ φ sφ +(b 2 s 2 )φ = 3 y j y k y l (G n+1 y m y ), where G are the spray coeffcents of L. The tensor D = D jkl dx j dx k dx l s called the Douglas tensor. A Fnsler metrc s called Douglas metrc f the Douglas tensor vanshes. Then there exsts a class of scalar functons H jk = H jk x, such that H 00 = T T 1 n+1 T m y m T m y m y, (2. 5) Where H 00 = H jk y j y k, T and T m y m are gven by the relatons T = Qs 0 + Ψ 2Qs 0 + r 00 b (2. 6) and T m y m = Q s 0 + Ψ 1 b 2 s 2 r 00 2Qs 0 + 2Ψ r 0 Q b 2 s 2 s 0 Qss 0 (2. 7) III. PROJECTIVE CHANGE BETWEEN TWO FINSLER METRICS In ths secton, we fnd the projectve relaton between two, β metrcs, that s, Specal, β metrc L = + β 2 and Randers metrc L = + β on a same underlyng manfold M of dmenson n > 2. From (2.3), L = + β 2 s a regular Fnsler metrc f and only f 1-form β satsfes the condton β x < 1 for any x M. The geodesc 2 coeffcents are gven by (2.4) wth, 2s 3 Θ =, Q = 1+2b 2 1+s 2 2s 2 3s 4 1 Ψ = 1 3s 2 + 2b 2 Substtutng (3.1) nto (2.4), we get G = G + 2s 1 s 2, (3. 1) 22 β 2 β 2 s 0 + 42 β 2 β 2 s 0 + r 00 Ψb 2β 3 + 4 + 2b 2 2 2 + β 2 2 2 β 2 3β 4, (3. 2) From (2.3), L = + β s a regular Fnsler metrc f and only f β x < 1 for any x M. The geodesc coeffcents are gven by (2.4) wth Θ = 1 2(1+s), Q = 1, Ψ = 0 (3. 3) Frst we prove the followng lemma: Lemma 3.1: Let L = + β2 and L = + β be two, β metrcs on a manfold M wth dmenson n > 2. Then they have the same Douglas tensor f and f both the metrcs L and L are Douglas metrcs. Proof: Frst, we prove the suffcent condton. Let L and L be Douglas metrcs and correspondng Douglas tensors be D jkl and D jkl. Then by the defnton of Douglas metrc, we have D jkl = 0 and D jkl = 0, that s, both L and L have same Douglas tensor. Next, we prove the necessary condton. If L and L have the same Douglas tensor, then (2.5) holds. Substtutng (3.1) and (3.3) n to (2.5), we obtan H 00 = A 5 +B 4 +C 3 +D 2 I 4 +J 2 s0 (3. 4) +K Where A = n + 1 b 2 (4s 0 β r 00 ), B = n + 1 2 1 + 2b 2 s 0 2s 0 b β + b r 00 2(1 + 2b 2 s 0 ), Avalable Onlne@www.jtrd.com 44
Internatonal Journal of Trend n Research and Development, Volume 2(6), ISSN 2394-9333 www.jtrd.com C = n + 1 r 00 (1 + b 2 )β 2, D = 6s 0 β 3 + r 00 b β 2 + 6s 0 β 2, λ = 1 n + 1 (3. 5) and I = 1 + 2b 2 (1 + n), J = 2β 2 (2 + b 2 )(n + 1), K = 3β 4 (n + 1) (3. 6) Then (3.4) equvalent to A 5 + B 4 + C 3 + D 2 = I 4 + J 2 + K H 00 + s0 (3. 7) Replacng y by y n (3.7) yelds A 5 + B 4 C 3 + D 2 = I 4 J 2 + K H 00 s0 (3. 8) Subtractng 3.8 from 3.7, we obtan A 5 + C 3 = H 00 2 I 2 + J + s0 (K) (3. 9) Now, we study two cases for Remannan metrc. Case (): If = μ(x), then (3.9) reduces to A 5 + C 3 = H 00 2 I 2 + J + μ x 2 s0 K above equaton can be wrtten as H = H 00 I 2 + J + μ x s0 (K) A 5 C 3 2 (3. 11) From (3.11), we can observe H has the factor 2, that s, 12λy r 00 β 4 has the factor 2. Snce β 2 s not havng 2 factor, the only possblty s that βr 00 has the factor 2. Then for each there exsts a scalar functon τ = τ(x) such that βr 00 = τ 2, whch s equvalent to b j r 0k + b k r 0j = 2τ jk. If n > 2 and assumng τ 0, then 2 rank b j r 0k + rank(b k r 0j ) = rank 2τ jk > 2, whch s mpossble unless τ = 0. Then βr 00 = 0. Snce β 0, we have r 00 = 0, mples that b j = 0. Case (): If μ(x), then from (3.9), observe H has the factor, that s, 12λy β 4 r 00 has the factor. Note that β 2 has no factor. Then the only possblty s that βr 00 has the factor 2. As n the case(), we have b j = 0 when n > 2. Specal, β metrc s a Douglas metrc f and only f b j = 0. Thus L s a Douglas metrc. Snce L s projectvely related to L, then both L and L are Douglas metrcs. Now, we prove the followng theorem: Theorem 3.1: The Fnsler metrc L = + β 2 / s projectvely related to f and only f the followng condtons are satsfed G = G + Py, b j = 0, dβ = 0, ( 3. 12) Where b = β, b j denote the coeffcents of the covarant dervatves of β wth respect to, P s a scalar functon. Proof: Let us prove the necessary condton. Snce Douglas tensor s an nvarant under projectve changes between two Fnsler metrcs, f L s projectvely related to L, then they have the same Douglas tensor. Accordng to lemma 3.1, we get both L and L are Douglas metrcs. Snce Randers metrc L = + β s a Douglas metrc f and only f β s closed, we have, dβ = 0 (3. 13) and L = + β 2 f b j = 0, s a Douglas metrc f and only (3. 14) Where b j denote the coeffcents of the covarant dervatves of β = b y wth respect to. In ths case, β s closed. Hence s j = 0, mples that b j = b j. Thus s 0 = 0, s 0 = 0. > rank (b j r 0k + b k r 0j ) Avalable Onlne@www.jtrd.com 45
Internatonal Journal of Trend n Research and Development, Volume 2(6), ISSN 2394-9333 www.jtrd.com By usng (3.14), we have r 00 = r j y y j = 0. Substtutng all these n (3.2) G = G, we obtan (3. 15) Snce L s projectve to L, ths s a Randers change between L and. Snce β s closed, then L s projectvely related to. Thus there s a scalar functon P = P(y) on TM\{0} such that G = G + Py (3. 16) From (3.15) and (3.16), we have G = G + Py (3. 17) Equatons (3.13) and (3.14) together wth (3.17) complete the proof of the necessary condton. Snce β s closed, t s suffcent to prove that L s projectvely related to. Substtutng (3.14) n to (3.2) yelds (3.15). From (3.15) and (3.17), we have G = G + Py That s, L s projectvely related to. From the prevous theorem, we get the followng corollares. Corollary 3.1: The Fnsler metrc L = + β2 s projectvely related to L = + β f and only f they are Douglas metrcs and the spray coeffcents of and have the followng relaton G = G + Py Where P s a scalar functon. In ths, we assume that the Randers metrc L = + β s locally Mnkowskan, where s an Eucldean metrc and β = b y s a one form wth b = constants. Then (3.12) can be wrtten as G = Py, b j = 0 (3. 18) Thus, we state Corollary 3.2: The Fnsler metrc L = + β2 s projectvley related to L = + β f and only f L s projectvely flat, n other words, L s projectvely flat f and only f (3.18) holds. CONCLUSION 1. The Fnsler metrc L = + β2 s projectvely related to L = + β f and only f L s projectvely flat, n other words, L s projectvely flat f and only f G = Py, b j = 0 holds true. 2. The Fnsler metrc L = + β2 s projectvely related to L = + β f and only f G = G + Py, b j = 0 and dβ = 0 are satsfed, where b = β, b j denote the coeffcents of the covarant dervatves of β wth respect to, P s a scalar functon. 3. Let L = + β2 and L = + β be two (, β)- metrcs on a manfold M wth dmenson n > 2. Then they have the same Douglas tensor f and only f both the metrcs L and L are Douglas metrcs. References [1] S. Bacso and M. Matsumoto, Projectve change between Fnsler spaces wth, β metrc, Tensor N. S., 55(1994), 252-257. [2] S. Bacso, X. Cheng and Z. Shen, Curvature Propertes of, β metrcs, Adv. Stud. Pure Math. Soc.,Japan (2007). [3] Chern S.S and Z.Shen, Remann-Fnsler Geometry, World Scentfc Publshng Co. Pte. Ltd., Hackensack,NJ, 2005. [4] X.Cheng and Z. Shen, A Class of Fnsler metrcs wth sotropc S-curvature, Israel J. Math., 169(2009), 317-340. [5] Nngwe Cu and Y-Bng Shen, Projectve change between two classes of, β metrcs, Dff. Geom. And ts Applcatons, 27 (2009), 566-573. [6] H. S. Park and II-Yong Lee, Projectve changes between a Fnsler space wth, β metrc and assocated Remannan metrc, Canad. J. Math., 60(2008), 443-456. Avalable Onlne@www.jtrd.com 46
Internatonal Journal of Trend n Research and Development, Volume 2(6), ISSN 2394-9333 www.jtrd.com [7] B.Nafaj, Z.Shen and A. Tayeb, Fnsler metrcs of scalar flag curvature wth specal non-remannan curvature propertes, J.Geom. Dedcata., 131 (2008), 87-97. [8] M. Matsumoto and X. We, Projectve changes of Fnsler spaces of constant curvature, Publ. Math. Debreen, 44(1994), 175-181. [9] M.Rafe-rad, B.Rezae, On Ensten-Matsumoto metrc, J. Nonlnear Anal. RealWorld Appl., 13(2012), 882-886. Avalable Onlne@www.jtrd.com 47