Randers Space with Special Nonlinear Connection

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1 ISSN , obachevsk Journal of Mathematcs, 2008, Vol. 29, No. 1, pp c Pleades Publshng, td., Rers Space wth Specal Nonlnear Connecton H. G. Nagaraja * (submtted by M.A. Malakhaltsev) Department of Mathematcs, Bangalore Unversty, Central College Campus, Bangalore , Inda Receved July 31, 2007 Abstract Ths artcle characterzes Berwald, ocally sbergan Rers spaces wth specal non-lnear connecton Mathematcs Subject Classfcaton: 53C60, 53B40 DOI: /S Key words phrases: Rers metrc, weakly sberg, Berwald space. INTRODUCTION The concept of Rers metrc was proposed by physcst G. Rers n 1941 from the st pont of general relatvty [8]. Snce then many authors made nvestgatons on Rers spaces ther geometrcal propertes [1, 5, 10]. They have obtaned condtons for these spaces to be Berwald, sberg, locally sberg locally Berwald. In ths paper we cons der a non-lnear connecton constructed from the gven non-lnear connecton obtaned condtons for Rers space to be Berwald, locally sbergan a space of scalar curvature based on the new connecton. Rers metrcs are the Fnsler metrcs whch are expressed n the form = α + β, whereα =(a j (x)y y j ) 1 2 s a Remannan metrc β = b (x)y sa1-form. 1. NON-INEAR CONNECTION N j et F n =(M n,) be an n-dmensonal Fnsler space wth the metrc = α + β, (1.1) where α =(a j (x)y y j ) 1 2 s a Remannan metrc β = b (x)y s a 1-form. Then the space F n s called a Rers space RΓ =(γj k,γ0k, 0),where γ j k are Chrstoffel symbols constructed from a j (x), s a Fnsler connecton on F n.etfγ=(γ j k,n j,c j k ) be any Fnsler connecton on F n.we now consder a new non-lnear connecton N j(x, y) whch s defned by [2] N j(x, y) =N j + jy, (1.2) where j denote the covarant dfferentaton wth respect to F Γ. Thenwehave X k = k N r k r = k N r k r ky r r = X k ky r r. (1.3) Now from Γ j k = 1 2 gr (X j g rk + X k g jr X r g jk ) by usng ( j g k )y j =0,wehave et F Γ=( Γ j k, N k,c j k ). * E-mal address: hgnraj@yahoo.com Γ j k =Γ j k. (1.4) 27

2 28 NAGARAJA Takng h-covarant dervatve of wth respect to F Γ, weget k = X k = X k ky r r =0,.e. the new connecton F Γ s h-metrcal. Dfferentatng (1.2) by y k we get k Nj = k Nj +( k j )l + j h k. (1.5) Then from (1.5), hv-torson P j k = k N j Γ j k of F Γ s gven by P j k = P j k + l k jl + j h k, (1.6) snce the ordnary dervatve commutes wth the covarant dervatve for scalar functon. If Nj = γ 0 j f : denote the covarant dfferentaton wth respect to RΓ,thenweget :k = b :k y ( yj ) l j:k = α + b j = b j:k, :k snce y j:k =0 α :k =0. Therefore (1.6) becomes P j k = P j k + b k:jl + b r:j y r h k. (1.7) If b :j =0, then from (1.7), we have P j k = P j k. Conversely suppose P j k = P j k, then from (1.7) we get b k:j l + b r:j y r h k =0. Transvectng the above by y we get b :j =0. As t s well known that [5], a Rers space s Berwald f only f b :j =0. Hence we can state that Theorem 1.1. If n a Rers space F n admttng F Γ, Nj = γ 0 j, then F n s Berwald f only f the torson of F Γ concdes wth that of F Γ. Snce F Γ s h-metrcal, we have k Nk r r =0. (1.8) Suppose Nj = γ 0 j + Bj. From (1.1), we have k = k α + k β. Usng these n (1.8), we get b j:k y j = Bk r lr, snce α :k =0.Dfferentatng the above wth respect to y,weget b :k = Bk r l r + Bk r h r, where Bjk = j Bk. From the above equaton we get that b :j =0f only f Bj =0.Thuswehave Theorem 1.2. A Rers space F n admttng F Γ s a Berwald space f only f F Γ reduces to the connecton RΓ =(γj k,γ0k, 0). The (v)h-torson R j k of F Γ s gven by = X k Nj j/k, (1.9)

3 RANDERS SPACE WITH SPECIA NONINEAR CONNECTION 29 where j/k denote the nterchange of j k substracton. Usng (1.2) (1.3) n (1.9), we get = R j k + B j k Br k r Bj j/k, (1.10) where Bj = jy. The formula (1.10) can be wrtten as = R j k + { j kl j k l j/k} or = R j k + { j k k j }l. (1.11) But by Rcc dentty, we have j k k j = ( m )R j mk. Therefore the equaton (1.11) becomes R j k = R j k l m m l. Transvectng the above by y j,weget If F n wth F Γ s of scalar curvature K, then R 0 k = K 2 h k. Substtutng ths n (1.11), we get R 0 k = K 2 h k, R 0 k = R 0 k + l m R 0 m k l. (1.12) snce l m h m k =0. Thus we can state that Theorem 1.3. If a Rers space F n s of scalar curvature K then the space F n admttng F Γ s also of scalar curvature K. 2. h-recurrent FINSER CONNECTION We are concerned wth The Fnsler connecton F Γ=(Γ j k,nj,c j k ) wth respect to whch s recurrent,.e. k = a k. Suppose recurrence vector a k = l k.thenl j = l j l [4]. Then Bj = l jy. Takng covarant dervatve of Bj wth respect to F Γ, thenweget Bj k = l j ky + l j y k. (2.1) Now consder r Bj = r (l j y )=h jr y + l j δr, whch then mples that r Bj Br k = l ky r (h jr y + l j δr )=l kl j y. Substtutng these n (2.1),we get R j k = + l j ky + l j y k + l kl j y j/k. (2.2) If Nj = γ 0 j,then = r = r R h jk y h,wherer h jk s a Remannan curvature tensor. Equaton (2.22) becomes R j k = r +(b j:k b k:j )y. Thus f b (x) s localy a gradent vector,.e. b :j =0,then R j k = r conversely. Thus we can state that Theorem 2.1. If n a Rers space admttng F Γ, (v)h-torson R j k of F Γ reduces to Remannan quantty r f only f b (x) s locally a gradent vector feld.

4 30 NAGARAJA 3. WEAKY ANDSBERGIAN AND BERWAD SPACES et : denote the covarant dfferentaton wth respect to the connecton RΓ =(γ j k,γ 0 k, 0). et r j = 1 2 (b :j + b j: ), s j = 1 2 (b :j b j: ), (3.1) r j = a r r rj s j = a r s rj, r j = b r r r j, s j = b r s r j, b = a r b r, b 2 = a rs b r b s, e j = r j + b s j + b j s, e 0 = e j y j, e 0j = e j y, e 00 = e j y y j. The E-curvature E j, the mean sberg curvature J are defned by E j = 1 2 m j G m J = 1 2 l rg jk j k G r. The space F n s weakly Berwaldan f E j =0 weakly sbergan f J =0. It s proved n [1] that for a Rers metrc = α + β, the mean sberg curvature J s gven by J = 1 4 (n +1)F 2 α 2 {2α(e 0 α 2 y e 00 ) 2β(s α 2 y s 0 )+s 0 (α 2 + β 2 ) (3.2) + α 2 (e 0 β b e 00 )+β(e 0 α 2 y e 00 ) 2(s α 2 y s 0 )(α 2 + β 2 )+4s 0 α 2 β}. Takng the covarant dervatve of b wth respect to F Γ,weget b k = k b b r Γ r k = k b b r Γ r k = b k. Therefore (a) r j = r j, (b) s j = s j, (c) J = J. (3.3) If b (x) s locally a gradent vector feld then s j =0. Whch then mples s =0, s j =0 s 0 =0. Substtutng these n (3.1), we get r j = b :j e j = b :j. Further We have e 0j = e j0 = β :j. Puttng these n (3.2), we get J = 1 4 (n +1)F 2 α 2 {2α(e 0 α 2 y e 00 )+α 2 (e 0 β b e 00 )+β(e 0 α 2 y e 00 )}. (3.4) If J =0, then equaton (3.4) reduces to (2α + β)(e 0 α 2 y e 00 )+α 2 (e 0 β b e 00 )=0. Transvectng the above by y,weget (2α + β)(e 00 (α 2 2 )) = 0 from whch we get that e 00 =0. Conversely suppose that e 00 =0.Dfferentatng the above wth respect to y k,weget k (e 00 )=β :k + b k:0. Snce e 0k = e k0,wehavee k0 =0. Puttng these n (3.4) we get J = J =0.Thuswehave Theorem 3.1. If n a Rers space F n admttng F Γ, b (x) s locally a gradent vector feld then F n becomes locally sbergan f only f b :0 =0. Conformal change of the Rers metrc = α + β. et us consder the conformal change = e c(x).

5 RANDERS SPACE WITH SPECIA NONINEAR CONNECTION 31 By ths conformal change, we have (a) b = e c b, α = e c α, a j = e 2c a j, β = e c β, (3.5) The non lnear connecton Nj s gven by snce j y (b) N j = N j + δ jc + c j y c y k, (c) Γ j k =Γ j k + δ jc k + δ k c j c g jk. N j = N j + jy, (3.6) = jy. From (1.4), t follows that Γ j k =Γ j k + δ jc k + δk c j c g jk. (3.7) Takng covarant dervatve of b wth respect to F Γ, weget b j = e c [b j b j c + b r c r g j ]. Suppose F n wth F Γ s Berwald..e. b j =0.Thenwehave b j = b j c + b r c r g j. (3.8) Substtutng ths n (3.1),we get s j = 1 2 [b c j + b j c ] r j = b r c r g j s j. If b (x) s locally a gradent vector feld, then we have s j =0 r j = b r c r g j. Ths then mples that e j = bg j,whereb = b r c r, e 0j = by j, e 00 = b 2. Substtutng these n (3.4), we get J = 1 4 (n +1)F 2 α 2 b{2α(y α 2 y 2 )+α 2 (y β b 2 )+β(y α 2 y 2 )}. (3.9) As the space F n wth F Γ s Berwald, t s locally sberg snce J = J,wehaveJ =0. Now transvectng (3.9) by y, after smplfcaton we get that b =0. From whch we get that r j =0. Ths mples that b j = b j. Snce b s locally a gradent vector feld, we get b j =0,.e. the space F n s Berwald. Hence we can state that Theorem 3.2 If n a Rers space F n admttng F Γ, b (x) s locally a gradent vector feld then F n s Berwald f only f F n wth respect to F Γ s Berwald. REFERENCES 1. X. Chen Z. Shen, Rers Metrcs wth Specal Curvature Propertes, Osaka Journal of Mathematcs 40, 87 (2003). 2. Hong-Suh Park, Ha-Yong Park, Byung-Doo Km, On a Fnsler Space wth (α, β)-metrc Certan Metrcal Non-near Connecton, Commun. Korean Math. Soc. 21(1), 177 (2006). 3. M. Matsumoto, Rers Space of Constant Curvature, Reports on Math. Phys. 28, 249 (1989). 4. M. Matsumoto, On h-recurrent Fnsler Connectons Conformally Mnkowsk Spaces, Tensor, N.S. 49, 18 (1990). 5. M. Matsumoto, TheBerwaldConnectonsofaFnslerSpacewth (α, β)-metrc, Tensor, N.S. 50, 18 (1991). 6. H. G. Nagaraja, C. S. Bagewad, H. Izum, On Infntesmal h-conformal Motons of Fnsler Metrc, Proc.Indan Acad. Sc. (Math. Sc.) 105(1), 33 (1995). 7. H. G. Nagaraja, On Projectve Changes of Fnsler Spaces, Tensor, N.S. 65, 104 (2004). 8. G. Rers, On an Asymmetrc Metrc n the Four-Space of General Relatvty, Phys. Rev. 59, 195 (1941). 9. Shun-ch Hojo, M.Matsumoto, K.Okubo, Theory of Conformally Berwald Fnsler Spaces Its Applcatons to (α, β)-metrcs, Balkan J. Geom. Its Applcatons 5(1), 107 (2000). 10. H. Yasuda H. Shmada, On Rers Spaces of Scalar Curvature, Reports on Math. Phys. 11, 347 (1997).