SEYED MEHDI KAZEMI TORBAGHAN, MORTEZA MIRMOHAMMAD REZAII

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1 Bulletn of Mathematcal Analyss and Applcatons ISSN: , URL: Volume 9 Issue 1(2017), Pages 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 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, Publshed January 2, supported by Amrkabr Unversty of Technology, Tehran, Iran. Communcated by Uday Chand De. 19

2 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)

3 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)

4 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)

5 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 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)

6 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

7 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

8 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 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

9 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)

10 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)

11 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 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, [2] D. Bao, Z. Shen, On the Volume of Unt Tangent Spheres n a Fnsler Manfold, Results n Mathematcs, 26 (1994) [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) [4] A. M. Cherf, M. Djaa, K. Zegga, Stable f-harmonc maps on sphere, Commun. Korean Math Soc, 30 4 (2015) [5] Y. J. Chang, f-bharmonc maps between Remannan manfolds, Journal of Geometry and Symmetry n Physcs, 27 (2012) [6] N. Course, f-harmonc maps, Ph.D thess, Unversty of Warwck, Coventry, England, [7] Q. He, Y. B. Shen, Some results on harmonc maps for Fnsler manfolds, Internatonal Journal of Mathematcs, 16 9 (2005) [8] J. L, Stable F-harmonc maps between Fnsler manfolds, Acta Mathematca Snca, 26 5 (2010) [9] A. Lchnerowcz, Applcatons harmonques et varétés kählerennes, Symposa Mathematca III, Academc Press London, (1970) [10] W. Lu, On f-b-harmonc maps and b-f-harmonc maps between Remannan manfolds, Scence Chna Mathematcs, 58 7 (2015) [11] X. Mo, Harmonc maps from Fnsler manfolds, Illnos Journal of Mathematcs, 45 4 (2001) [12] X. Mo, An Introducton to Fnsler Geometry, World Scentfc, Sngapore, [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) [14] Y. Ou, f-harmonc morphsms between Remannan manfolds, Chnese Annals of Mathematcs, 35B 2 (2014) [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) [16] Y. Shen, Y. Zhang, Second varaton of harmonc maps between Fnsler manfolds, Scence n Chna Ser. A Mathematcs, 47 1 (2004)

12 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: Morteza Mrmohammad Reza Faculty of Mathematcs and Computer Scences, Amrkabr Unversty of Technology, Tehran, Iran. E-mal address: