f-biharmonic AND BI-f-HARMONIC SUBMANIFOLDS OF PRODUCT SPACES

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1 SARAJEVO JOURNAL OF MATHEMATICS Vol.13 (5), No.1, (017), DOI: /SJM f-biharmonic AND BI-f-HARMONIC SUBMANIFOLDS OF PRODUCT SPACES FATMA KARACA AND CIHAN ÖZGÜR Abstract. We consider f-biharmonic bi-f-harmonic submanifolds of the product of two real space forms. We find the necessary sufficient conditions for a submanifold to be f-biharmonic bi-fharmonic in a product of two real space forms. 1. Introduction Let (M, g) (N, h) be two Riemannian manifolds. ϕ : M N is called a harmonic map if it is a critical point of the energy functional E(ϕ) = 1 dϕ dν g, Ω where Ω is a compact domain of M. The Euler-Lagrange equation of E(ϕ) is τ(ϕ) = tr( dϕ) = 0, where τ(ϕ) is the tension field of ϕ [3]. The map ϕ is said to be biharmonic if it is a critical point of the bienergy functional E (ϕ) = 1 τ(ϕ) dν g, Ω where Ω is a compact domain of M. In [4], Jiang obtained the Euler- Lagrange equation of E (ϕ). This gives us τ (ϕ) = tr( ϕ ϕ ϕ )τ(ϕ) tr(rn (dϕ, τ(ϕ))dϕ) = 0, (1.1) where τ (ϕ) is the bitension field of ϕ R N is the curvature tensor of N. The map ϕ is said to be f-biharmonic if it is a critical point of the 010 Mathematics Subject Classification. 53C40, 53C4. Key words phrases. f-biharmonic map, f-biharmonic submanifold, bi-f-harmonic map, bi-f-harmonic submanifold, product space. Copyright c 017 by ANUBIH.

2 116 FATMA KARACA AND CIHAN ÖZGÜR f-bienergy functional E,f (ϕ) = 1 Ω f τ(ϕ) dν g, where Ω is a compact domain of M [5]. The Euler-Lagrange equation for the f-bienergy functional is defined by τ,f (ϕ) = fτ (ϕ) + ( f) τ(ϕ) + ϕ gradf τ(ϕ) = 0, (1.) where τ,f (ϕ) is the f-bitension field of ϕ [5]. From the definition, it is trivial that any harmonic map is f-biharmonic. If the f-biharmonic map is neither harmonic nor biharmonic then we call it by proper f-biharmonic if f is a constant, an f-biharmonic map turns into a biharmonic map [5]. An f-harmonic map with a positive function f : M C R is a critical point of the f-energy E f (ϕ) = 1 f dϕ dν g, Ω where Ω is a compact domain of M. The Euler-Lagrange equation for the f-energy functional gives us the f-tension field τ f (ϕ) (see [1], [8]) by τ f (ϕ) = fτ(ϕ) + dϕ(gradf) = 0. (1.3) The map ϕ is said to be bi-f-harmonic if it is a critical point of the bi-f-energy functional Ef (ϕ) = 1 τ f (ϕ) dν g, Ω where Ω is a compact domain of M. The Euler-Lagrange equation gives the bi-f-harmonic map equation τ f (ϕ) = fj ϕ (τ f (ϕ)) ϕ gradf τ f (ϕ) = 0, (1.4) where τf (ϕ) is the bi-f-tension field of ϕ J ϕ is the Jacobi operator of the map defined by J ϕ (X) = [ T r g ϕ ϕ X ϕ X R N (dϕ, X) dϕ ] M [8]. It is trivial that any f-harmonic map is bi-f-harmonic [8]. Eells Sampson studied harmonic mappings of Riemannian manifolds [3]. Jiang defined biharmonic maps by using the first second variational formulas of bienergy functional [4]. In [5], Lu defined the notion of f-biharmonic maps. He obtained the f-biharmonic map equation studied f-biharmonicity of some special maps. In [7], Ou studied on some properties of f-biharmonic maps f-biharmonic submanifolds. Course defined f-harmonic maps in [1]. Later, Ouakkas, Nasri Djaa obtained some properties for f-harmonic maps between two Riemannian manifolds

3 f-biharmonic AND BI-f-HARMONIC SUBMANIFOLDS OF PRODUCT SPACES117 defined bi-f-harmonic maps [8]. In [11], Zegga, Cherif Djaa considered bi-f-harmonic maps submanifolds. In [9], Roth studied biharmonic submanifolds of the product of two space forms. Motivated by the above studies, in this paper, we consider f-biharmonic bi-f-harmonic submanifolds of the product of two space forms obtain the necessary sufficient conditions for a submanifold to be f-biharmonic bi-f-harmonic in a product of two real space forms.. Preliminaries Let M n 1 (c 1 ) M n (c ) be two real space forms of constant curvatures c 1, c with dimensions n 1 n, respectively. Let us consider the product space (M n 1 (c 1 ) M n (c ), g). Assume that (M m, g) be a Riemannian manifold isometrically immersed into the product space (M n 1 (c 1 ) M n (c ), g). Denote by F the Levi-Civita connection of M m the product structure of the product space (M n 1 (c 1 ) M n (c ), g), respectively. The product structure F : T M n 1 (c 1 ) T M n (c ) T M n 1 (c 1 ) T M n (c ) is a (1, 1)-tensor field defined by F (X 1 + X ) = X 1 X for any vector field X = X 1 + X, X 1, X denote the parts of X tangent to the first second factors, respectively. It is easy to see that F satisfies F = I ( F I), (.1) g(f X, Y ) = g(x, F Y ), (.) F = 0, (.3) (see [10]). By an easy calculation, we obtain the curvature tensor of (M n 1 (c 1 ) M n (c ), g) as R(X, Y )Z = a[g(y, Z)X g(x, Z)Y + g(f Y, Z)F X g(f X, Z)F Y ] +b[g(y, Z)F X g(x, Z)F Y + g(y, F Z)X g(x, F Z)Y ] (.4) with a = c 1+c 4 b = c 1 c 4 []. Now let X T M m ξ T M m. The decompositions of F X F ξ into tangent normal components can be written as F X = kx + hx F ξ = sξ + tξ, (.5) where k : T M m T M m, h : T M m T M m, s : T M m T M m, t : T M m T M m are (1, 1)-tensor fields. From equations (.1) (.), it is easy to see that k t are symmetric satisfy the following properties: k X = X shx, (.6)

4 118 FATMA KARACA AND CIHAN ÖZGÜR (for more details see [10]). t ξ = ξ hsξ, (.7) ksξ + stξ = 0, (.8) hkx + thx = 0, (.9) g(hx, ξ) = g(x, sξ), (.10) 3. f-biharmonic submanifolds of product spaces Let ϕ : M m (M n 1 (c 1 ) M n (c ), g) be an isometric immersion from an m-dimensional Riemannian manifold (M m, g) into the product of two space forms M n 1 (c 1 ) M n (c ) of constant curvatures c 1, c with dimensions n 1 n. We shall denote by B, A, H, the second fundamental form, the shape operator, the mean curvature vector field, the Laplacian the Laplacian on the normal bundle of M m in N = (M n 1 (c 1 ) M n (c ), g), respectively. Firstly we have the following theorem: Theorem 3.1. Let M m be a Riemannian manifold isometrically immersed into the product space N = (M n 1 (c 1 ) M n (c ), g). Then M m is f-biharmonic if only if the following two equations hold: H + trb(, A H ( )) f f H grad ln f H = a(mh hsh + tr(k)th) + b(mth + tr(k)h) (3.1) m grad H + tr(a H ) + A Hgrad ln f = a( ksh + tr(k)sh) + b(m 1)sH. (3.) Proof. Let us denote by ϕ, the Levi-Civita connections on N M m, respectively. Let {e i }, 1 i m be a local geodesic orthonormal frame at p M m. Then τ(ϕ) = tr( dϕ) = mh. (3.3) From (1.1) (3.3), we have τ (ϕ) = tr( ϕ ϕ ϕ )τ(ϕ) tr(rn (dϕ, τ(ϕ))dϕ) = ( ϕ ei e i )mh R N (dϕ(e i ), mh)dϕ(e i ) = m { H + } R N (dϕ(e i ), H)dϕ(e i ). (3.4)

5 f-biharmonic AND BI-f-HARMONIC SUBMANIFOLDS OF PRODUCT SPACES119 By (.4), we find (R N (dϕ(e i ), H)dϕ(e i )) = a [ mh + F (F H) T tr(k)f H ] + b [ mf H + (F H) T tr(k)h ]. Using (.5) in the above equality, we get (R N (dϕ(e i ), H)dϕ(e i ))=a [ mh + ksh +hsh tr(k)sh tr(k)th] + b [ msh mth + sh tr(k)h]. (3.5) By the use of the Gauss Weingarten formulas, we have H = ( H) = ( ( A H e i + e i H) = = { } ei A H e i B(e i, A H e i ) A ei He i + e i e i H ei A H e i + B(e i, A H e i ) + A ei He i e i e i H = tr( A H ) + trb(, A H ) + tr(a H ) + H. (3.6) Now we shall compute tr( A H ). In view of Gauss Weingarten formulas, we obtain ei A H e i = i,j g( ei A H e i, e j )e j = i,j e i g(a H e i, e j )e j = i,j = i,j e i g(b(e i, e j ), H)e j = e i g( ϕ e j e i, H)e j i,j } {g( ϕ ei ϕ ej e i, H)e j + g( ϕ ej e i, ϕ ei H)e j {g( ϕ ei ϕ ej e i, H)e j + g(b(e i, e j ), ei H)e j } = i,j = i,j g( ϕ e j e i, H)e j + i A ei H(e i ). (3.7)

6 10 FATMA KARACA AND CIHAN ÖZGÜR Then, using the definition of the curvature tensor of N we have g( ϕ e j e i, H)e j = g(r N (e i, e j )e i + ϕ e j e i + ϕ [e i,e j ] e i, H) i,j i,j = mg( ej H, H) = m grad H. (3.8) Substituting (3.8) into (3.7) then using (3.7) into (3.6), we get H = m grad H + trb(, A H ) + tr(a H ) + H. (3.9) In view of equations (3.9) (3.5) into (3.4), we obtain τ (ϕ) = m { m grad H + trb(, A H ) + tr(a H ) + H +a [ mh + ksh + hsh tr(k)sh tr(k)th] +b [ msh mth + sh tr(k)h]}. (3.10) Using Weingarten formula equation (3.3), we have ϕ gradf τ(ϕ) = ϕ gradf mh = m ( A H gradf + gradf H ). (3.11) Finally substituting equations (3.3), (3.10) (3.11) into equation (1.), we obtain fm { m grad H + trb(, A H ) + tr(a H ) + H +a [ mh + ksh + hsh tr(k)sh tr(k)th] +b [ msh mth + sh tr(k)h] f } f H + A Hgrad ln f grad ln f H = 0. Hence comparing the tangential the normal parts, we obtain the desired result. Corollary 3.. Let M m be a Riemannian manifold isometrically immersed into the product space N = (M n 1 (c 1 ) M n (c ), g). 1) If F H is tangent to M m, then M m is f-biharmonic if only if H + trb(., A H (.)) f f H grad ln f H [a(m 1) + btr(k)] H = 0, (3.1) m grad H + tr(a H ) + A Hgrad ln f [atr(k) + b(m 1)] F H = 0. (3.13)

7 f-biharmonic AND BI-f-HARMONIC SUBMANIFOLDS OF PRODUCT SPACES11 ) If F H is normal to M m, then M m is f-biharmonic if only if H + trb(., A H (.)) f f H grad ln f H [am + btr(k)] H [atr(k) + bm] F H = 0, (3.14) m grad H + tr(a H ) + A Hgrad ln f = 0. (3.15) Proof. 1) If F H is tangent to M m, then by the use of (.5) we have F H = sh th = 0. So from (.7), we have hsh = H by Theorem 3.1 we find (3.1) (3.13). ) If F H is normal to M m, then sh = 0 th = F H. Hence from Theorem 3.1 we get (3.14) (3.15). Corollary 3.3. Let M m be a submanifold of S p (r) S n p (r) of dimension m with non-zero constant mean curvature such that F H is tangent to M m. 1) If M m is proper f-biharmonic, then { 1 0 < H (m 1) + f } r inf f. (3.16) m ) Assume that f is an eigenfunction of the Laplacian corresponding to real eigenvalue λ. Hence the equality in (3.16) occurs M m is proper f-biharmonic if only if M m is pseudo-umbilical, H = 0, A H grad ln f 1 r tr(k)f H = 0 [ ] 1 trb(, A H ) = (m 1) + λ H. r Proof. We assume that F H is tangent to M m, then in view of (.5) we have F H = sh th = 0. Hence by (.7), we have hsh = H. By the use of (3.1) we have H + trb(., A H (.)) f f H grad ln f H 1 (m 1)H = 0. (3.17) r Then taking the scalar product of (3.17) with H, we find g( H, H) + g(trb(., A H (.)), H) f g(h, H) f ) g ( grad ln f H, H 1 (m 1)g(H, H) = 0. r

8 1 FATMA KARACA AND CIHAN ÖZGÜR Since H is a constant, we have g( H, H) = f f H A H + 1 r (m 1) H. Using the Bochner formula, we get [ H + A H f = f + 1 ] (m 1) H. (3.18) r By the use of Cauchy-Schwarz inequality, we have A H m H 4 (see [9]). Hence we find [ f f + 1 ] (m 1) H m H 4 + r H m H 4. (3.19) Since H is a non-zero constant, we can write [ ] f 0 < H f + 1 (m 1) r inf m. (3.0) Now, if f is an eigenfunction of the Laplacian corresponding to the real eigenvalue λ, then f f = λ. We can write [ λ + H 1 (m 1) ] r =. (3.1) m Assume that M m is proper f-biharmonic. From (3.19), first we have H = 0. Moreover substituting the equation (3.1) into (3.19) we find [ λ + A H 1 (m 1) ] r =. m That is, M m is pseudo-umbilical. Then from (3.13) we have A H grad ln f 1 tr(k)f H = 0. r In this case (3.1) turns into This completes the proof. [ trb(, A H ) = λ + 1 ] (m 1) H. r Now we consider f-biharmonic hypersurface M m of S p (r) S n p (r) such that F H is tangent to M m. Firstly we have:

9 f-biharmonic AND BI-f-HARMONIC SUBMANIFOLDS OF PRODUCT SPACES13 Proposition 3.4. Let M n 1 be a hypersurface of S p (r) S n p (r) with non-zero constant mean curvature such that F H is tangent to M n 1. Then M n 1 is f-biharmonic if only if A H grad ln f = 1 4r tr(k)f H B = n r + f f. Proof. Assume that M n 1 is a hypersurface of S p (r) S n p (r) with nonzero constant mean curvature such that F H is tangent to M n 1. Then from (3.1) we get [ f trb(, A H ) = f + 1 ] (n ) H. r Then, by taking the scalar product with H, we have n 1 [ f g (B(e i, A H e i ), H) = f + 1 ] (n ) H, r n 1 [ f g (A H e i, A H e i ) = f + 1 ] (n ) H r [ A H f = f + 1 ] (n ) H. r From ([10], page 71), we know that A H = B. Hence we find B = n r So the equation (3.13) is reduced to A H grad ln f = 1 tr(k)f H. 4r This completes the proof of the proposition. + f f. (3.) Proposition 3.5. Let M n 1 be a proper f-biharmonic hypersurface of S p (r) S n 1 (r) with non-zero constant mean curvature such that F H is tangent to M n 1. Then the scalar curvature of M n 1 is given by Scal M n 1 = 1 r { (n 1)(n 3) (n ) + tr(k) } + (n 1) H f f.

10 14 FATMA KARACA AND CIHAN ÖZGÜR Proof. By the use of the Gauss equation, we can write Scal M n 1 = g ( R N ) m (e i, e j )e j, e i + g (B(e i, e i ), B(e j, e j )) g (B(e j, e i ), B(e j, e i )). Then we compute Scal M n 1 = g ( R N ) (e i, e j )e j, e i + (n 1) H B. (3.3) Using (.4) we write g ( { R N ) 1 m (e i, e j )e j, e i = r g(e j, e j )g(e i, e i ) + g(f e j, e j )g(f e i, e i ) g(e i, e j )g(e i, e j ) } g(f e i, e j )g(f e i, e j ). Hence we find g ( R N ) 1 { (e i, e j )e j, e i = (n 1)(n 3) + tr(k) } r. (3.4) Finally, in view of equations (3.4) (3.) into (3.3), we get Scal M n 1 = 1 r { (n 1)(n 3) (n ) + tr(k) } + (n 1) H f f. This proves the proposition. 4. Bi-f-harmonic submanifolds of product spaces In this section, we consider bi-f-harmonic submanifolds of product of two real space forms. We firstly state the following theorem: Theorem 4.1. Let M m be a Riemannian manifold isometrically immersed into the product space N = (M n 1 (c 1 ) M n (c ), g). Then M m is bi-fharmonic if only if the following two equations hold: ( mf ) ( H) + ( mf ) trb(, A H ( )) fm ( f) H (3mf) gradf H ftrb (, gradf) ftr B (, gradf) m gradf H B (gradf, gradf) = ( mf ) {a [mh hsh + tr(k)th + tr(k)hgradf + hkgradf] +b [mth + tr(k)h + (m 1)hgradf]}

11 f-biharmonic AND BI-f-HARMONIC SUBMANIFOLDS OF PRODUCT SPACES15 (mf) grad H + ( mf ) tr(a H ) + 3 (mf) A Hgradf +fricci M (gradf) + fgrad ( f) + ftra B(,gradf) ( ) 1 ( grad gradf ) = ( mf ) { a [ ksh + tr(k)sh + (m 1)gradf + tr(k)kgradf k gradf ] +b [(m 1)sH + m (kgradf) + tr(k)gradf]}. Proof. Let us denote by ϕ, the Levi-Civita connections on N M m, respectively. Let {e i }, 1 i m be a local geodesic orthonormal frame at p M m. From the equations (1.3) (3.3), we find Then, we can write τ f (ϕ) = fmh + dϕ (gradf) = fmh + gradf. (4.1) (R N (τ f (ϕ), dϕ(e i ))dϕ(e i )) = mf Using the equation (.4), we obtain (R N (gradf, dϕ(e i ))dϕ(e i )) + (R N (H, dϕ(e i ))dϕ(e i )) (R N (gradf, dϕ(e i ))dϕ(e i )). (4.) = a [(m ] 1) (gradf) + tr(k) (F gradf) F (F gradf) T [ ] + b (m 1) (F gradf) + tr(k) (gradf) (F gradf) T. Using (.5) in the above equality, we get (R N (gradf, dϕ(e i ))dϕ(e i )) = a [(m 1) (gradf) + tr(k) (kgradf) +tr(k) (hgradf) k gradf + hkgradf ] + b [m (kgradf) + (m 1) (hgradf) + tr(k)gradf]. (4.3) In view of equations (3.5) (4.3) into equation (4.), we obtain (R N (τ f (ϕ), dϕ(e i ))dϕ(e i )) = a [mh ksh hsh + tr(k)sh + tr(k)th +(m 1) (gradf) + tr(k) (kgradf) + tr(k) (hgradf)

12 16 FATMA KARACA AND CIHAN ÖZGÜR k gradf + hkgradf ] + b [msh + mth sh + tr(k)hm (kgradf) +(m 1) (hgradf) + tr(k)gradf]. (4.4) By the use of the Gauss Weingarten formulas, we have ( τ f (ϕ) ϕ [ ei e i τ f (ϕ)) = ϕ ei (fmh + gradf) ] = m = m + ϕ ( e i ei (f) H + f H ) + ( ei gradf + B (e i, gradf)) { ei (e i (f)) H + e i (f) H + f H } + ei ei gradf { } B (e i, ei gradf) A B(ei,gradf) (e i ) + e i B (e i, gradf). (4.5) Using the equation (3.9), we can write ( H) = m grad H trb(, A H ) tr(a H ) H. (4.6) In view of equation (4.6) into (4.5), we obtain ( τ f (ϕ) ϕ ei e i τ f (ϕ))=m ( f) H + m ϕ gradf H m f grad H (mf) trb(, A H ) (mf) tr(a H ) (mf) H + ei ei gradf + B (e i, ei gradf) A B(ei,gradf) (e i ) + e i B (e i, gradf). (4.7) Using Gauss Weingarten formulas, we have ϕ gradf τ f (ϕ) = ϕ gradf (fmh + gradf) = m gradf H (mf) A H gradf + (mf) gradf H + 1 grad ( gradf ) + B (gradf, gradf). (4.8) Finally substituting (4.4), (4.7) (4.8) into equation (1.4) comparing the tangential the normal parts, we obtain the desired result. Corollary 4.. Let M m be a Riemannian manifold isometrically immersed into the product space N = (M n 1 (c 1 ) M n (c ), g).

13 f-biharmonic AND BI-f-HARMONIC SUBMANIFOLDS OF PRODUCT SPACES17 1) If F H F gradf are tangent to M m, then M m is bi-f-harmonic if only if ( mf ) ( H) + ( mf ) trb(, A H ( )) fm ( f) H (3mf) gradf H ftrb (, gradf) ftr B (, gradf) m gradf H B (gradf, gradf) = ( mf ) {a [(m 1) H] + b [tr(k)h]} (mf) grad H + ( mf ) tr(a H ) + 3 (mf) A Hgradf +fricci M (gradf) + fgrad ( f) + ftra B(,gradf) ( ) 1 ( grad gradf ) = ( mf ) {a [tr(k) (F gradf) + (m )gradf + tr(k)f H] +b [(m 1)F H + m (F gradf) + tr(k)gradf]}. ) If F H is tangent to M m F gradf is normal to M m, then M m is bi-f-harmonic if only if ( mf ) ( H) + ( mf ) trb(, A H ( )) fm ( f) H (3mf) gradf H ftrb (, gradf) ftr B (, gradf) m gradf H B (gradf, gradf) = ( mf ) {a [(m 1) H + tr(k) (F gradf)] +b [tr(k)h + (m 1) (F gradf)]} (mf) grad H + ( mf ) tr(a H ) + 3 (mf) A Hgradf +fricci M (gradf) + fgrad ( f) + ftra B(,gradf) ( ) 1 ( grad gradf ) = ( mf ) {a [tr(k)f H + (m 1)gradf] + b [(m 1)F H + tr(k)gradf]}. 3) If F H is normal to M m F gradf is tangent to M m, then M m is bi-f-harmonic if only if ( mf ) ( H) + ( mf ) trb(, A H ( )) fm ( f) H (3mf) gradf H ftrb (, gradf) ftr B (, gradf) m gradf H B (gradf, gradf) = ( mf ) {a [mh + tr(k)f H] + b [mf H + tr(k)h]} (mf) grad H + ( mf ) tr(a H ) + 3 (mf) A Hgradf +fricci M (gradf) + fgrad ( f) + ftra B(,gradf) ( ) 1 ( grad gradf ) = ( mf ) {a [(m )gradf + tr(k) (F gradf)]

14 18 FATMA KARACA AND CIHAN ÖZGÜR +b [m (F gradf) + tr(k)gradf]}. 4) If F H F gradf are normal to M m, then M m is bi-f-harmonic if only if ( mf ) ( H) + ( mf ) trb(, A H ( )) fm ( f) H (3mf) gradf H ftrb (, gradf) ftr B (, gradf) m gradf H B (gradf, gradf) = ( mf ) {a [mh + tr(k)f H + tr(k) (F gradf)] +b [mf H + tr(k)h + (m 1)F gradf]} (mf) grad H + ( mf ) tr(a H ) + 3 (mf) A Hgradf +fricci M (gradf) + fgrad ( f) + ftra B(,gradf) ( ) 1 ( grad gradf ) = ( mf ) {a [(m 1)gradf] + b [tr(k)gradf]}. Proof. 1) If F H F gradf are tangent to M m, then by the use of (.5) we have F H = sh, th = 0, F gradf = kgradf hgradf = 0. So from equations (.6), (.7) (.9), we have hsh = H, k gradf = gradf hkgradf = 0. By Theorem 4.1 we find the result. ) If F H is tangent F gradf is normal to M m, then th = 0, sh = F H, H = hsh, ksh = 0, kgradf = 0 F gradf = hgradf. Hence from Theorem 4.1, we get the result. 3) If F H is normal F gradf is tangent to M m, then sh = 0, F H = th, F gradf = kgradf, hgradf = 0, k gradf = gradf hkgradf = 0. Using Theorem 4.1, we obtain the result. 4) If F H F gradf are normal to M m, then sh = 0, F H = th, kgradf = 0 F gradf = hgradf. By Theorem 4.1 we find the result. Acknowledgement. The authors would like to thank the reviewer for his/her helpful constructive comments that greatly contributed to improving the paper. References [1] N. Course, f-harmonic maps, Thesis, University of Warwick, 004. [] F. Dillen D. Kowalczyk, Constant angle surfaces in product spaces, J. Geom. Phys., 6 (6) (01), [3] J. Eells J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), [4] G. Y. Jiang, -harmonic maps their first second variational formulas, Chinese Ann. Math. Ser. A, 7 (4) (1986), [5] W-J. Lu, On f-bi-harmonic maps bi-f-harmonic maps between Riemannian manifolds, Sci. China Math., 58 (015),

15 f-biharmonic AND BI-f-HARMONIC SUBMANIFOLDS OF PRODUCT SPACES19 [6] Y-L. Ou, On f-biharmonic maps f-biharmonic submanifolds, Pacific J. Math., 71 () (014), [7] Y-L. Ou, f-harmonic morphisms between Riemannian manifolds, Chin. Ann. Math. Ser. B, 35 () (014), [8] S. Ouakkas, R. Nasri M. Djaa, On the f-harmonic f-biharmonic maps, JP J. Geom. Topol., 10 (1) (010), [9] J. Roth, A note on biharmonic submanifolds of product spaces, J. Geom., 104 () (013), [10] K. Yano M. Kon, Structures on Manifolds, Series in Pure Mathematics, 3. World Scientific Publishing Co., Singapore, [11] K. Zegga, A. M. Cherif M. Djaa, On the f-biharmonic maps submanifolds, Kyungpook Math. J., 55 (1) (015), (Received: January 11, 016) (Revised: April 19, 016) Department of Mathematics Balıkesir University Campus of Çağış Balıkesir Turkey fatmagurlerr@gmail.com cozgur@balikesir.edu.tr