SEYED MEHDI KAZEMI TORBAGHAN, MORTEZA MIRMOHAMMAD REZAII

Bulletn of Mathematcal Analyss and Applcatons ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 9 Issue 1(2017), Pages 19-30. f-harmonic MAPS FROM FINSLER MANIFOLDS SEYED MEHDI KAZEMI TORBAGHAN, MORTEZA MIRMOHAMMAD REZAII Abstract. In ths paper, the frst and second varaton formulas of the f- energy functonal for a smooth map from a Fnsler manfold to a Remannan manfold are obtaned. As an applcaton, t s proved that there exsts no non-constant stable f-harmonc map from a Fnsler manfold to the standard unt sphere S n (n > 2). 1. Introducton f-harmonc maps as a generalzaton of harmonc maps, geodescs and mnmal surfaces were frst studed by A. Lchnerowcz [9] n 1970. Recently, N. Course [6] studed the f-harmonc flow on surfaces. Y. Ou [14] analysed the f-harmonc morphsms as a subclass of harmonc maps whch pull back harmonc functons to f-harmonc functons. In [4], the researchers studed the stablty of harmonc and f-harmonc maps on spheres. Many scholars have studed and done researches on the f-harmonc maps, see for nstance, [3, 4, 5, 9, 10, 14, 15]. f-harmonc maps are appled n many branches of geometry and mathematcal physcs. In vew of Physcs, f-harmonc maps could be consdered as the statonary solutons of nhomogeneous Hesenberg spn system, see for nstance [5, 14]. Furthermore, the ntersecton of f-harmoncty wth curvature condtons justfes ther applcaton for gleanng valuable nformaton on weghted manfolds and gradent Rcc soltons, see [10, 15]. Let φ : (M, g) (N, h) be a smooth map between Remannan manfolds and f C (M) a postve smooth functon on M. The map φ s called f-harmonc f φ Ω s a crtcal pont of the f-energy functonal E f (φ) := 1 f dφ 2 dυ g, 2 Ω 2010 Mathematcs Subject Classfcaton. 53C60, 58E20. Key words and phrases. Fnsler manfolds; Harmonc maps; f-harmonc maps; Stablty; Varatonal problem. c 2017 Unverstet Prshtnës, Prshtnë, Kosovë. Correspondng author. Submtted Jun 28, 2016. Publshed January 2, 2017. supported by Amrkabr Unversty of Technology, Tehran, Iran. Communcated by Uday Chand De. 19

20 SEYED MEHDI KAZEMI TORBAGHAN, MORTEZA MIRMOHAMMAD REZAII for any compact sub-doman Ω M. Here dυ g s the volume element of M and dφ denotes the Hlbert-Schmdt norm of the dfferental dφ Γ(T M φ 1 T N). Let φ : (M, F ) (N, h) be a smooth map from a Fnsler manfold (M, F ) to a Remannan manfold (N, h) and f : (0, ) be a smooth postve functon on the projectve sphere bundle of M. In ths paper, the f-energy functonal of φ s ntroduced and the correspondng varaton formulas are obtaned. It can be seen that the frst and second varaton formulas of the f-energy functonal s consstent to that of Remannan case f M s Remannan and f s defned on M, see [4]. The concept of harmonc maps from a Fnsler manfold to a Remannan manfold was frst ntroduced by X. Mo, see [11]. On the workshop of Fnsler Geometry n 2000, Professor S. S. Chern conjectured that the fundamental exstence theorem of harmonc maps on Fnsler spaces s true. In [13], the researchers have proved ths conjecture and shown that any smooth map from a compact Fnsler manfold to a compact Remannan manfold of non-postve sectonal curvature could be deformed nto a harmonc map whch has mnmum energy n ts homotopy class. Y. Shen and Y. Zhang [16] extended Mo s work to Fnsler target manfold and obtaned the frst and second varaton formulas. As an applcaton, Q. He and Y. Shen [7] proved that any harmonc map from an Ensten Remannan manfold to a Fnsler manfold wth certan condtons s totally geodesc and there s no stable harmonc map from an Eucldean unt sphere S n to any Fnsler manfolds. Harmonc maps between Fnsler manfolds have been studed extensvely by varous researchers, see for nstance, [7, 8, 11, 12, 13, 16]. The current paper s organzed as follows: In the second secton, a few concepts of Fnsler geometry are revewed. In secton 3, the f-energy functonal of a smooth map from a Fnsler manfold to a Remannan manfold s ntroduced and the correspondng Euler-Lagrange equaton s obtaned va calculatng the frst varaton formula of the f-energy functonal. In secton 4, the second varaton formula of the f-energy functonal for an f-harmonc map s derved. Fnally, t s shown that there exsts no non-constant stable f-harmonc map from a Fnsler manfold to the standard sphere S n (n > 2). 2. Prelmnares and Notatons In ths secton, a few basc notons of Fnsler geometry are provded whch wll be used later. For more detals see ([1, 11, 12, 16]). Throughout ths paper, t s assumed that M s an m-dmensonal connected compact orented manfold wthout boundary and π : T M M be ts tangent bundle. Let (x ) be a local coordnates system wth the doman U M and (x, y ) the nduced standard local coordnates system on π 1 (U). A Fnsler manfold s a par (M, F ) ncludes a smooth manfold M and a Fnsler metrc F : T M [0, ) satsfes the followng propertes: ) F s smooth on T M \ 0, ) F (x, λy) = λf (x, y) for λ > 0, ) The fundamental quadratc form g := g j (x, y) dx dx j, g j := 1 2 F 2 2 y y j, (2.1)

f-harmonic MAPS FROM FINSLER MANIFOLDS 21 s postve defnte at every pont (x, y) T M \ 0. The Remannan manfolds and locally Mnkowsk manfolds are mportant examples of Fnsler manfolds. In the sequel, the followng conventon of ndex ranges are used 1, j, k,... m, 1 a, b, c,... m 1, 1 A, B, C,... 2m 1. The Fnsler structure F nduces two more sgnfcant quanttes as follows A := A jk dx dx j dx k, A jk := F 4 [F 2 ] y y j y k, η := η dx, η := g jk A jk, called Cartan tensor and Cartan form, respectvely. Let us denote the projectve sphere bundle of M by, where := x S x M. Almost every geometrc quanttes constructed by Fnsler structure are nvarant under rescalng y ty for t > 0, thus make sense on. The canoncal projecton p : M defned by (x, y) x pulls back the tangent bundle T M to the m-dmensonal vector bundle p T M over (2m 1) dmensonal manfold. The bundle p T M and ts dual p T M are sad to be the Fnsler bundle and dual Fnsler bundle, respectvely. At each pont (x, y), the fbre of p T M has a local bass { } and x k a metrc g defned by (2.1). Here and ts dual dx k stand for the sectons x k (x, y, ) Γ(p T M) and (x, y, dx k ) Γ(p T M), respectvely. The bundle x k p T M has a global secton l(x, y) := y F x whch s called the dstngushed secton. The dual of the former secton ω = [F ] y dx s called Hlbert form. Further- more, each fbre of the Remannan vector bundle (p T M, g) has an adapted frame {e := u j x },.e. g(e j, e j ) = δ j and e m := l. Denote ts dual by {ω := vj dxj }, ω (e j ) = δj. It s clear that ωm = ω. In the rest of ths paper, these abbrevatons wll be used. Accordng to the notatons above, t can be seen that x = v k e k and dx = u k ωk, where (u j ) and (v j ) are related by uk vj k = δj. More relatons among (u j ) s, (v j ) s and the quadratc form of F can be found n [1]. Let Nj := 1 G 2 y be the coeffcents of non-lnear connecton on T M, where G := j 1 4 gh ( 2 F 2 y j F 2 ). Consder the local orthogonal bass { δ y h x j x h δx, y } on T z T M, δ δx := x where N j y and dual bass as {dx, δy }, where δy := dy + N j j dxj. It can be shown that {ω := vj dxj, ω m+a := vj a δy j F } s a local bass for the tangent bundle T. Consder ω 2m = [F ] y δy F as dual to the vector y y. Therefore, ω 2m vanshes on. Based on the above notatons, the Sasak-type metrc, the volume element, the horzontal sub-bundle and the vertcal sub-bundle of are defned by G : = δ j ω ω j + δ ab ω m+a ω m+b, dv := ω 1 ω 2 ω 2m 1, H : = {v T, ω m+a (v) = 0}, V := x M T S x M, (2.2)

22 SEYED MEHDI KAZEMI TORBAGHAN, MORTEZA MIRMOHAMMAD REZAII respectvely, see [2]. Due to the fact that H s somorph wth p T M, H s also called the Fnsler bundle. In the sequel, for any X Γ(p T M) the correspondng horzontal lft of X s denoted by X H. As well-known, there exsts a lnear connecton on p T M called the Chern connecton and denoted by c. Its connecton forms are characterzed by the followng equatons d(dx ) dx k ωk = 0, (2.3) and dg j g k ωj k g jk ω k δy k = 2A jk F. (2.4) By takng the exteror dervatve of (2.3), the curvature 2 forms of the Chern connecton, Ω j := dω j ωk j ω k, have the followng structure Ω j = 1 2 R jkldx k dx l + P jkldx k δyl F. (2.5) By (2.5), the Landsberg curvature s defned as follows L := L jk dx dx j dx k, L jk := g l y m F P l mjk. It can be seen that L jk = A jk, where dot denotes the covarant dervatve along the Hlbert form, (see [16], p. 41). Let D denotes the Lev-Cvta connecton on (, G). The dvergence of a form ψ = ψ ω Γ(p T M) s dv G ψ := T r G Dψ. Note that the bundle p T M s somorph wth the horzontal sub-bundle of T. It can be shown that dv G ψ = ψ + a,b ψ a L bba = ( c e H ψ)(e ) + a,b ψ a L bba, (2.6) where denotes the horzontal covarant dfferental wth respect to the Chern connecton, {e } be the adapted frame wth respect to g and L abc = L(e a, e b, e c ), (see [8], Lemma 2.1). 3. The frst varaton formula Let φ : (M m, F ) (N n, h) be a smooth map from an m-dmensonal Fnsler manfold (M, F ) to an n-dmensonal Remannan manfold (N, h). Henceforth, the Chern connecton on p TM, the Lev-Cvta connecton on (N, h) and the pull-back connecton on p (φ 1 T N) are denoted by c, N and, respectvely. Let f C () be a smooth postve functon on. The f-energy densty of φ s a functon e f (φ) : R defned by e f (φ)(x, y) := 1 2 f(x, y)t r gh(dφ, dφ), (3.1) where T r g stands for takng the trace wth respect to g (the fundamental quadratc form of F ) at (x, y). In the local coordnates (x ) on M and ( x ) on N, the f-energy densty of φ can be wrtten as follows e f (φ)(x, y) = 1 2 f(x, y)δj h(dφ(e ), dφ(e j )) = 1 2 f(x, y)δj φ φ β j h β( x), (3.2)

f-harmonic MAPS FROM FINSLER MANIFOLDS 23 where {e = u k } s the adapted frame wth respect to g at (x, y), x = φ(x) x k and dφ(e ) = φ x φ. Defnton 3.1. A map φ : (M, F ) (N, h) s sad to be f-harmonc, f t s a crtcal pont of the f-energy functonal E f (φ) := 1 e f (φ)dv, (3.3) c m 1 where c m 1 denotes the volume of the standard (m 1) dmensonal sphere and dv s the canoncal volume element of defned by (2.2). Let φ t : M N ( ε < t < ε) be a smooth varaton of φ such that φ 0 = φ and set V = φ t t t=0 := V x φ. By (3.2), the f-energy densty of φ t can be wrtten as follows e f (φ t )(x, y) = 1 2 f(x, y)δj φ t φβ t j h β( x), (3.4) φ t x k where x = φ t (x), dφ t (e ) = u k x φ := φ t x φ. Due to the fact that {V } s ndependent of y and usng (3.4), t s obtaned that t e f (φ t ) t=0 = 1 2 t (δj fφ t φβ t j h β) t=0 = δ j {fu k V x k φβ j h β + 1 2 fφ φ β h β j x γ V γ } = {fu k δv δx k φβ h β + fφ φ β N Γ σ βγh σ V γ } = h( e H V, fdφ(e )), (3.5) where { N Γ βγ } denotes the coeffcents of the Lev-Cvta connectons on (N,h). Let ψ := h(v, fdφ(e ))ω Γ(p T M). Usng the fact that L bba = A bba and equaton (2.6), t follows that dv G ψ = ( c e H ψ)(e ) + a,b h(v, fdφ(e a ))L bba = {h( e H V, fdφ(e )) + h(v, ( e H fdφ)(e ))} a,b ( ) = h V, ft r g dφ + dφ p(grad H f) fdφ p(k H ) h(v, fdφ(e a )) A bba + h( e H V, fdφ(e )). (3.6) where T r g dφ = g j ( dφ( x x ) dφ( c j defned as follows K := a,b x x j )), A bba = A(e b, e b, e a ) and K s A bba e a Γ(p T M). (3.7)

24 SEYED MEHDI KAZEMI TORBAGHAN, MORTEZA MIRMOHAMMAD REZAII Combnng (3.5) and (3.6) and consderng the Green s theorem, t can be concluded that d dt E f (φ t ) t=0 = 1 h(τ f (φ), V )dv, c m 1 where τ f (φ) := ft r g dφ + dφ p(grad H f) fdφ p(k H ) Γ((φ p) T N), (3.8) here p : M s the canoncal projecton on, grad H f denotes the horzontal part of grad f Γ(T ) and K s defned by (3.7). The feld τ f (φ) s sad to be the f-tenson feld of φ. Theorem 3.2. Let φ : (M, F ) (N, h) be a smooth map from a Fnsler manfold to a Remannan manfold and f C () a smooth postve functon on. Then, φ s f-harmonc f and only f τ f (φ) 0 Due to the fact that the Landsberg curvature of locally Mnkowsk manfold vanshes and consderng Theorem 3.2 and equaton (3.8), the followng result s obtaned mmedately Corollary 3.3. Let φ : (M, F ) (N, h) be an mmerson harmonc map from a locally Mnkowsk manfold (M, F ) to an arbtrary Remannan manfold (N, h) and f C () a smooth postve functon on. Then, φ s f-harmonc f and only f f(x, y) = f(y) for any (x, y). Example 3.4. Assume that (R 2, F ) be a locally Mnkowsk manfold and (R 3,, ) be the three-dmensonal Eucldean space. Let φ : (R 2, F ) (R 3,, ) s defned by φ(x) := (x 2, x 1 + 2x 2, 3x 1 x 2 ), where x = (x 1, x 2 ) R 2. Let f(x, y) := exp( y1 (y 1 2y 2 ) (y 1 ) 2 +(y 2 ) ) be a postve smooth map on SR 2. By (3.8), t can be seen that φ 2 s f-harmonc. Remark 3.5. Let (M, F ) be a locally Mnkowsk manfold and (M, h) be a flat Remannan manfold. It s conspcuous that the dentty map Id : (M, F ) (M, h) s harmonc, (see [12], Proposton 9.5.1). By Corollary 3.3, t can be concluded that Id s f-harmonc f and only f f(x, y) = f(y) for all (x, y). Before proceedng, t s worth notng that f-harmonc maps shouldn t be confused wth F-harmonc maps and p-harmonc maps from a Fnsler manfolds to a Remannan manfolds. Let F : [0, ) [0, ) be a C 2 strctly ncreasng functon on (0, ). The smooth map φ : (M, F ) (N, h) from a Fnsler manfold (M, F ) to a Remannan manfold (N, h) s called F-harmonc f t s a crtcal ponts of the F-energy functonal E F (φ) := F( dφ 2 2 )dv. (3.9) The noton of F harmonc maps was frst ntroduced by J. L [8]. F-energy functonal can be categorzed as energy, p-energy and exponental energy when F(t) s equal to t, (2t) p 2 \ p (p 4) and e t, respectvely. In terms of the Euler-Lagrange equaton, φ s F-harmonc f t satsfes the followng equaton τ F (φ) := T r g (F ( dφ 2 )dφ) F dφ 2 ( )dφ(k) = 0. (3.10) 2 2

f-harmonic MAPS FROM FINSLER MANIFOLDS 25 For more detals, see [8]. The feld τ F (φ) s called the F tenson feld of φ. Let φ : (M, F ) (N, h) be a non-degenerate smooth map (.e. dφ x 0 for all x M) from a Fnsler manfold to a Remannan manfold. By (3.8) and (3.10), the followng proposton s obtaned mmedately. Proposton 3.6. Let φ : (M, F ) (N, h) be a non-degenerate F-harmonc map from a Fnsler manfold to a Remannan manfold. Then, φ s an f-harmonc map wth f = F ( dφ 2 2 ). Partcularly, any non-degenerate p-harmonc map s an f-harmonc map wth f = dφ p 2. Remark 3.7. Ths result was obtaned by Y. Chang [5] n the Remannan case. 4. The second varaton formula In ths secton, the second varaton formula of the f-energy functonal for an f-harmonc map from a Fnsler manfold to a Remannan manfold s obtaned. As an applcaton, t s shown that any stable f-harmonc map φ from a Fnsler manfold to the standard sphere S n (n > 2) s constant. Theorem 4.1. (The second varaton formula). Let φ : (M, F ) (N, h) be an f-harmonc map from a Fnsler manfold (M, F ) to a Remannan manfold (N, h). Let φ t : M N ( ε < t < ε) be a smooth varaton such that φ 0 = φ and set V = φt t t=0. Then d 2 dt 2 E f (φ t ) t=0 = 1 h ( V, ft r g ( 2 V ) + ft r g R N (V, dφ)dφ c m 1 + gradh f V f K H V ) dv, (4.1) where K s defned by (3.7), R N s the curvature tensor on (N, h) and T r g ( 2 V ) = g j ( V x x j c V ). x x j Set Q φ d2 f (V ) := dt 2 E f (φ t ) t=0. An f-harmonc map φ s sad to be stable f-harmonc f Q φ f (V ) 0 for any vector feld V along φ. Proof. Let M denotes the product manfold ( ε, ε) M, Φ : M N s defned by Φ(t, x) := φ t (x) and p : S M M be the natural projecton on the sphere bundle S M. Denote the same notatons of c and for the Chern connecton on p T M and the pull-back connecton on p (Φ 1 T N), respectvely. By (3.1), t can

26 SEYED MEHDI KAZEMI TORBAGHAN, MORTEZA MIRMOHAMMAD REZAII be shown 2 t 2 e f (φ t ) = t h( dφ(e ), fdφ(e )) t = t h( e H dφ( t ), fdφ(e )) = h( t e H dφ( t ), fdφ(e )) + fh( e H dφ( t ), e H dφ( t )) = h( e H dφ( t t ), fdφ(e )) + fh( e H dφ( t ), e H dφ( t )) + fh(r N (dφ( t ), dφ(e ))dφ( t ), dφ(e )), (4.2) where t s used dφ(e ) t e H dφ( t ) = d(φ p)[ t, eh ] = 0, for the thrd and fourth equaltes n (4.2). By (4.2), t can be seen that 2 t 2 E f (φ t ) t=0 = 1 { c m 1 + + h( e H fh( e H dφ( t ), e H dφ( t )) t=0 dv t dφ( t ), fdφ(e )) t=0 dv fh(r N (dφ( t ), dφ(e ))dφ( t ), dφ(e )) t=0 dv } = I 1 + I 2 + I 3 (4.3) Now each term of the rght hand sde(rhs) of the above equaton s calculated. Frst, let ψ := fh( e H dφ( t ), dφ( t ))w Γ(p T M). By (2.6), t follows that dv G ψ = ( c e H ψ)(e ) + fh( e H a dφ( t ), dφ( t ))L bba a,b = { e H (f)h( e H dφ( t ), dφ( t )) + fh( e H e H dφ( t ), dφ( t )) + fh( e H dφ( t ), e H dφ( t )) + fh( e H dφ( j t ), dφ( } t ))(c e H w j )(e ) a,b fh( e H a dφ( t ), dφ( t )) A bba. (4.4) By (4.4) and Green s theorem, I 1 can be obtaned as follows I 1 = 1 ( ) h ft r g ( 2 V ) + c gradh f V f K H V, V dv. (4.5) m 1

f-harmonic MAPS FROM FINSLER MANIFOLDS 27 Smlarly, let Ψ := h( dφ( t t ), fdφ(e ))w Γ(p T M). It can be seen that dv G Ψ = { h( e H dφ( t t ), fdφ(e )) + h ( dφ( } t t ), ( e H fdφ)(e )) f a,b h( dφ( t t ), A bba dφ(e a )). (4.6) By (4.6) and consderng the Green s Theorem and the f-harmoncty condton of φ, I 2 s gven by I 2 = 1 h ( dφ( c m 1 t t ) t=0, τ f (φ))dv = 0. (4.7) Substtutng the formulas (4.5) and (4.7) nto equaton (4.3) yelds the formula (4.1) and hence completes the proof. 5. Stablty of f-harmonc maps to S n Consder the unt sphere S n as a submanfold of the Eucldean space (R n+1,, ). At each pont x S n any vector feld V n R n+1 can be decomposed as V = V + V, where V s the component of V tangent to S n and V = V, x x s the component of V normal to S n. Let R be the Lev-Cvta connecton on R n+1, S be the Lev-Cvta connecton on S n and B be the second fundamental form of S n n R n+1. We have the followng relaton R X Y = S X Y + B(X, Y ), (5.1) where X and Y are smooth vector felds on S n. The shape operator wth respect to any normal vector feld W on S n s defned by A W (X) := ( R X W ), (5.2) for any smooth vector feld X on S n. At any pont of x S n, the tensors A and B are related by A W (X), Y = B(X, Y ), W = X, Y x, W, (5.3) where X and Y are vector felds on S n and W s a normal vector feld on S n. Theorem 5.1. Any stable f-harmonc map φ : (M, F ) S n manfold (M, F ) to the standard sphere S n (n > 2) s constant. from a Fnsler Proof. The above notatons are used to prove ths theorem. Choose an arbtrary pont z. Then, set φ = φ p and x = φ(z), where p s the canoncal projecton on. Let R S denotes the curvature tensor of S n and {Λ 1,, Λ n+1 } a constant orthonormal bass n R n+1. By (4.1), t follows that n+1 Q φ f (Λ ) = 1 n+1 c m 1 =1 =1 ( h grad H f Λ f K H Λ + ft r g ( 2 Λ ) + ft r g R S (Λ, dφ)dφ, Λ ) dv. (5.4)

28 SEYED MEHDI KAZEMI TORBAGHAN, MORTEZA MIRMOHAMMAD REZAII Snce Λ s parallel n R n+1 and consderng (5.2), we obtan grad H f Λ = S d φ(gradh f)λ = ( R d φ(gradh f)λ ) = ( R d φ(gradh f)(λ Λ )) = ( R d φ(gradh f)λ ) = A Λ (d φ(grad H f)). (5.5) Let λ : S n R defned by λ (x) := Λ, x for all x S n. One can easly check that A Λ (X) = λ X, (5.6) for every vector feld X on S n. By means of (5.3) and (5.5) at x, t follows that grad H f Λ, Λ = A Λ (d φ(grad H f)), Λ = d φ(grad H f), Λ x, Λ = d φ(grad H f), Λ x, Λ = λ ( x) d φ(grad H f), Λ. (5.7) Thus, the frst term of RHS of (5.4) s obtaned as follows grad H f Λ, Λ = λ φ d φ(grad H f), Λ. (5.8) Smlarly, the second term of RHS of (5.4) s gven by f K H Λ, Λ = fλ φ d φ(k H ), Λ. (5.9) Due to the fact that e H Λ = A Λ (d H φ(e )) from (5.5) and consderng (5.6), t can be concluded that e H e H Λ = e H A Λ (d H φ(e )) = e H (λ φ d φ(e H )) = d φ (grad λ φ) λ φ e H dφ(e ). (5.10) Snce grad λ = Λ and usng defnton of gradent operator, t can be seen that d φ(grad λ φ) = d φ(e H ), (grad λ ) φ d φ(e H ) = d φ(e H ), Λ φ d φ(e H ). (5.11) By means of (5.10) and (5.11), the thrd term of RHS of (5.4) has the followng expresson f T r g ( 2 Λ ), Λ = λ φ ft r g dφ, Λ f dφ 2. (5.12)

f-harmonic MAPS FROM FINSLER MANIFOLDS 29 Fnally, snce the sphere S n has constant curvature, t can be shown that f T r g R S (Λ, dφ)dφ, Λ = (n 1)f dφ 2. (5.13) Replacng (5.8), (5.9), (5.12) and (5.13) n (5.4) and usng the f-harmoncty condton of φ, t follows Q φ f (Λ ) = 2 n f dφ 2 dv c m 1 + 1 λ c φ τ f (φ), Λ dv m 1 = 2 n f dφ 2 dv 0, (5.14) c m 1 by means of (5.14) and the stable f-harmoncty condton of φ, t can be concluded that φ s constant. Ths completes the proof. Acknowledgments. The authors would lke to express ther thanks to Professor Hans-Bert Rademacher who always helped us durng the research and who was always helpful for hs constructve comments. References [1] D. Bao, S. S. Chern, Z. Shen, An Introducton to Remann-Fnsler Geometry, Sprnger, New york, 2000. [2] D. Bao, Z. Shen, On the Volume of Unt Tangent Spheres n a Fnsler Manfold, Results n Mathematcs, 26 (1994) 1 17. [3] A. Boulal, N. Djaa, M. Djaa, S. Ouakkas. Harmonc maps on generalzed warped product manfolds, Bulletn of Mathematcal Analyss and Applcatons, 4 1 (2012) 1256 1265. [4] A. M. Cherf, M. Djaa, K. Zegga, Stable f-harmonc maps on sphere, Commun. Korean Math Soc, 30 4 (2015) 471 479. [5] Y. J. Chang, f-bharmonc maps between Remannan manfolds, Journal of Geometry and Symmetry n Physcs, 27 (2012) 45 58. [6] N. Course, f-harmonc maps, Ph.D thess, Unversty of Warwck, Coventry, England, 2004. [7] Q. He, Y. B. Shen, Some results on harmonc maps for Fnsler manfolds, Internatonal Journal of Mathematcs, 16 9 (2005) 1017-1031. [8] J. L, Stable F-harmonc maps between Fnsler manfolds, Acta Mathematca Snca, 26 5 (2010) 885 900. [9] A. Lchnerowcz, Applcatons harmonques et varétés kählerennes, Symposa Mathematca III, Academc Press London, (1970) 341 402. [10] W. Lu, On f-b-harmonc maps and b-f-harmonc maps between Remannan manfolds, Scence Chna Mathematcs, 58 7 (2015) 1483-1498. [11] X. Mo, Harmonc maps from Fnsler manfolds, Illnos Journal of Mathematcs, 45 4 (2001) 1331 1345. [12] X. Mo, An Introducton to Fnsler Geometry, World Scentfc, Sngapore, 2006. [13] X. Mo, Y. Yang, The exstence of harmonc maps from Fnsler manfolds to Remannan manfolds, Scence n Chna Ser. A Mathematcs, 48 1 (2005) 115 130. [14] Y. Ou, f-harmonc morphsms between Remannan manfolds, Chnese Annals of Mathematcs, 35B 2 (2014) 225 236. [15] M. Rmold, G. Veronell, Topology of steady and expandng gradent Rcc soltons va f- harmonc maps, Dfferental Geometry and ts Applcatons, 31 5 (2013) 623 638. [16] Y. Shen, Y. Zhang, Second varaton of harmonc maps between Fnsler manfolds, Scence n Chna Ser. A Mathematcs, 47 1 (2004) 39 51.

30 SEYED MEHDI KAZEMI TORBAGHAN, MORTEZA MIRMOHAMMAD REZAII Seyed Mehd Kazem Torbaghan Faculty of Mathematcs and Computer Scences, Amrkabr Unversty of Technology, Tehran, Iran. E-mal address: mehdkazem@aut.ac.r Morteza Mrmohammad Reza Faculty of Mathematcs and Computer Scences, Amrkabr Unversty of Technology, Tehran, Iran. E-mal address: mmreza@aut.ac.r