Biconservative quasi-minimal immersions into semi-Euclidean spaces
Rüya Yeğin Şen, Alev Kelleci, Nurettin Cenk Turgay, Elif Özkara Canfes
aa r X i v : . [ m a t h . G M ] J un Biconservative quasi-minimal immersions intosemi-Euclidean spaces
R. Ye˘gin S¸en, A. Kelleci, N. C. Turgay and E. ¨Ozkara Canfes
Abstract
In this paper we study biconservative immersions into the semi-Riemannianspace form R ( c ) of dimension 4, index 2 and constant curvature, where c ∈ { , − , } . First, we obtain a characterization of quasi-minimal properbiconservative immersions into R ( c ). Then we obtain the complete clas-sification of quasi-minimal biconservative surfaces in R (0) = E . We alsoobtain a new class of biharmonic quasi-minimal isometric immersion into E . Surfaces of semi-Riemannian manifolds with zero mean curvature is oneof mostly interested topics in differential geometry. When the ambient man-ifold ( N, ˜ g ) is Riemannian, a surface with zero mean curvature, called minimal-surface, arises as the solution of the variational problem of finding the surface in N with minimum area among all surfaces with the common boundary. On theother hand, if N is a semi-Riemannian manifold with positive index, it admitsan important class of surfaces whose mean curvature is zero. These surfaces arecalled quasi-minimal surfaces and they have no counter part on Riemannianmanifolds: By the definition, a submanifold M of ( N, ˜ g ) is said to be quasi-minimal if its mean curvature vector is light-like at every point. Quasi-minimalsubmanifolds play some fundamental roles in geometry as well as in physics andthey are also called as ‘ marginally trapped ’ in the physic literature when theambient manifold is a Lorentzian space-time, [19].Consider the bienergy integral E ( ψ ) = 12 Z M k τ ( ψ ) k v g (1.1)for a mapping ψ : (Ω , g ) → ( N, ˜ g ) between two semi-Riemannian manifolds,where v g is the volume element of g and τ ( ψ ) = − trace ∇ dψ is the tension of ψ . Let τ ( ψ ) stand for the bitension field of ψ defined by τ ( ψ ) = − ∆ τ ( ψ ) − trace (cid:16) ˜ R ( dψ, τ ( ψ )) dψ (cid:17) , where ∆ is the rough Laplacian defined on sections of ψ − ( T N ), i.e.,∆ = − trace (cid:16) ∇ ψ ∇ ψ − ∇ ψ ∇ (cid:17) R is the curvature tensor of ( N, ˜ g ).When (1.1) is assumed to define a functional from C ∞ (Ω , N ), it is namedas bi-energy functional. In this case, the critical points of E are called asbiharmonic maps, [6]. In [15,16], Jiang obtained the first and second variationalformulas for E and proved that ψ is biharmonic if and only if the fourth ordersystem of partial differential equations given by τ ( ψ ) = 0 (1.2)is satisfied. Biharmonic immersions particularly take interest of many geome-ters, [1, 8, 11].On the other hand, if ψ : M → ( N, ˜ g ) is a given smooth mapping, one canalso define a functional from the set of all metrics on M by using (1.1), [8].(Ω , g ) is said to be a biconservative submanifold if g is a critical point of thisfunctional and ψ : (Ω , g ) ֒ → ( N, ˜ g ) is an isometric immersion. Note that criticalpoints of this functional is characterized by the equation h τ ( ψ ) , dψ i = 0 , (1.3)[8] (See also [14]).It is obvious that any biharmonic immersion is also biconservative. Becauseof this reason, biconservative submanifolds have been studied in many papers sofar to understand geometry of biharmonic immersions, [8,12,13,20]. In [20], thethird named author studied biconservative hypersurfaces in Euclidean spaceswith three distinct principal curvatures. Also, classification results on biconser-vative hypersurfaces in 3-dimensional semi-Riemannian space forms have beenappeared in some papers, [12, 13]. Most recently, biconservative surfaces in4-dimensional Euclidean space have been studied in [10] and [17].In [3, 5], all flat biharmonic quasi-minimal surfaces in the 4-dimensionalpseudo-Euclidean space E with neutral metric were obtained. Furthermore,in [3] the complete classification of flat quasi-minimal surfaces is given. More-over, Chen and Garay studied quasi-minimal surfaces with parallel mean cur-vature vector in the pseudo-Euclidean space E in [4]. In this paper, we studyquasi-minimal biconservative immersions into E and complete the study of bi-conservative quasi-minimal surfaces initiated in [3–5]. In Sect. 2, we give basicdefinitions and equations on isometric immersions into semi-Riemannian spaceforms after we describe the notation used in the paper. In Sect. 3, we ob-tained a characterization of biconservative immersions into space forms of index2. Finally in Sect. 4, we obtain our main result which is the complete localclassification of biconservative surfaces of E . We are going to denote the n -dimensional semi-Riemannian space form ofindex s and constant curvature c ∈ {− , , } by R ns ( c ), i.e., R ns ( c ) = S ns if c = 1, E ns if c = 0, H ns if c = − h· , ·i stand for its metric tensor. When c = 0, we define the light-cone of E ns by L C = { p ∈ E ns |h p, p i = 0 } . On the other hand, a non-zero vector w in a finite dimensional non-degeneratedinner product space W is said to be space-like, light-like or time-like if h w, w i > h w, w i = 0 or h w, w i <
0, respectively. We are going to use the followingwell-known lemma later (see, for example, [18, Lemma 22, p. 49])
Lemma 2.1. [18] Let V be a subspace of W and V ⊥ its orthogonal complement.Then, dim V + dim V ⊥ = dim W . Consider an isometric immersion f : (Ω , g ) ֒ → R ns ( c ) from an m -dimensionalsemi-Riemannian manifold (Ω , g ) with the Levi-Civita connection ∇ . Let T Ωand N f Ω stand for the tangent bundle of Ω and the normal bundle of f , re-spectively. If e ∇ denote the Levi-Civita connection of R ns ( c ), then the Gauss andWeingarten formulas are given, respectively, by e ∇ X Y = ∇ X Y + α f ( X, Y ) , (2.1) e ∇ X ξ = − A fξ ( X ) + ∇ ⊥ X ξ, (2.2)for any vector fields X, Y ∈ T Ω and ξ ∈ N f Ω, where α f and ∇ ⊥ are the secondfundamental form and the normal connection of f , respectively, and A fξ standsfor the shape operator of f along the normal direction ξ . A f and α f are relatedby h A fξ X, Y i = h α f ( X, Y ) , ξ i . (2.3)On the other hand, the second fundamental form α f of f , the curvature ten-sor R of (Ω , g ) and the normal curvature tensor R ⊥ of f satisfies the integrabilityconditions R ( X, Y ) Z = c ( X ∧ Y ) Z + A fα f ( Y,Z ) X − A fα f ( X,Z ) Y, (2.4a)( ¯ ∇ X α f )( Y, Z ) = ( ¯ ∇ Y α f )( X, Z ) , (2.4b) R ⊥ ( X, Y ) ξ = α f ( X, A fξ Y ) − α f ( A fξ X, Y ) , (2.4c)called Gauss, Codazzi and Ricci equations, respectively, where, by the definition,we have ( X ∧ Y ) Z = h Y, Z i X − h X, Z i Y, ( ¯ ∇ X α f )( Y, Z ) = ∇ ⊥ X α f ( Y, Z ) − α f ( ∇ X Y, Z ) − α f ( Y, ∇ X Z ) . The mean curvature vector field of the isometric immersion f is defined by H f = 1 m trace α f . (2.5) f is said to be quasi-minimal if H f is light-like at every point of Ω, i.e, h H f , H f i =0 and H f = 0. In this case, M = f (Ω) is called a quasi-minimal submanifold(quasi-minimal surface if m = 2) of R ns ( c ).Further, we are going to denote the kernel of the shape operator along H f by T f , i.e., T f = { X ∈ T M | A fH f ( X ) = 0 } . .1 Lorentzian surfaces in R ( c ) Let (Ω , g ) be a 2-dimensional semi-Riemannian manifold. Consider an iso-metric immersion f : (Ω , g ) ֒ → R ( c ) and let the surface M be the image of f ,i.e., M = f (Ω). Then, the Gaussian curvature K of Ω is defined by K = R ( X, Y, Y, X ) h f ∗ X, f ∗ X ih f ∗ Y, f ∗ Y i − h f ∗ X, f ∗ Y i , (2.6)where X and Y span the tangent bundle of Ω. Ω, and thus M , is said to be flatif K vanishes identically.If g has index 1, then M is said to be a Lorentzian surface. In this case,for any m ∈ Ω there exists a local coordinates system ( N m , ( u , u )), calledisothermal coordinate system of Ω, such that m ∈ N m and g | N m = ˜ m ( u, v )( du ⊗ du − du ⊗ du )for a positive function ˜ m ∈ C ∞ (Ω). By defining a new local coordinate system( u, v ) by u = u + u √ and v = u − u √ , we obtain ( [2]) g | N m = − ˜ m ( u, v )( du ⊗ dv + dv ⊗ du ) . It is well-known that a light-like vector w tangent to M is propositonal to either f u = df ( ∂ u ) or f v = df ( ∂ v ).Note that the light-like curves u = const and v = const are pre-geodesics of M . In other words, there exists a re-parametrization of the curve u = c (or v = c ) which is a geodesic of M . Therefore, by defining a new local coordinatesystem ( s, t ) on M by s = s ( u, v ) = Z uu ˜ m ( ξ, v ) dξ, t = v and letting m ( u, v ) = ∂∂v (cid:18)Z uu ˜ m ( ξ, v ) dξ (cid:19) , we obtain a semi-geodesic coordi-nate system on M (see, for example, [7]). Proposition 2.2.
Let M be a Lorentzian surface with the metric tensor g .Then, there exists a local coordinate system ( s, t ) such that g = g m := − ( ds ⊗ dt + dt ⊗ ds ) + 2 mdt ⊗ dt. (2.7) Furthermore, the Levi-Civita connection of M satisfies ∇ ∂ s ∂ s = 0 , ∇ ∂ s ∂ t = ∇ ∂ t ∂ s = − m s ∂ s , ∇ ∂ t ∂ t = m s ∂ t + (2 mm s − m t ) ∂ s and the Gaussian curvature of M is K = m ss . (2.8)4 .2 Biharmonic immersions First, we would like to recall a necessary and sufficient condition for anisometric immersion to be biharmonic. In this case by splitting τ ( f ) into itsnormal and tangential part and employing (1.2), one can obtain the followingwell-known result. Proposition 2.3.
An isometric immersion f : (Ω , g ) ֒ → ( N, ˜ g ) is biharmonicif and only if the equations m grad (cid:0) ˜ g ( H f , H f ) (cid:1) + 4 trace A f ∇ ⊥· H f ( · ) + 4 trace (cid:0) ˜ R ( · , H f ) · (cid:1) T = 0 (2.9) and trace α f ( A fH f ( · ) , · ) − ∆ ⊥ H f + 2 trace (cid:0) ˜ R ( · , H f ) · (cid:1) ⊥ = 0 (2.10) are satisfied, where m is the dimension of Ω , ∆ ⊥ denote the Laplace operatorassociated with the normal connection of f . On the other hand, if ψ = f is an isometric immersion, then (1.3) is equiva-lent to ( τ ( f )) T = 0. Therefore, by using Proposition 2.3 we have Proposition 2.4.
An isometric immersion f : (Ω , g ) ֒ → ( N, ˜ g ) between semi-Riemannian manifolds is biconservative if and only if the equation (2.9) is sat-isfied. We immediately have the following result of Propositon 2.4 for the case( N, ˜ g ) = R ns ( c ). Corollary 2.5.
An isometric immersion f : (Ω , g ) ֒ → R ns ( c ) is biconservativeif its mean curvature vector is parallel on the normal bundle.Remark . Because of Corollary 2.5, we are going to call a biconservativeisometric immersion f from (Ω , g ) into R ns ( c ) as proper if ∇ ⊥ H f = 0 at anypoint of Ω. Moreover, we would like to refer to [4] for classification of quasi-minimal surfaces with parallel mean curvature vector in E (See also [9]). In this section, we consider quasi-minimal biconservative immersions into R ( c ) for c ∈ {− , , } . Consider a 2-dimensional semi-Riemannian manifold(Ω , g ), where g is a Lorentzian metric. Let f : (Ω , g ) ֒ → R ( c ) be a quasi-minimalisometric immersion and put M = f (Ω).We choose two vector fields e , e tangent to M such that h e i , e j i = 1 − δ ij , i, j = 1 ,
2. Then, there exist smooth functions φ , φ such that ∇ e i e = φ i e , (3.1a) ∇ e i e = − φ i e . (3.1b)Put e = − H f ∈ N f Ω and let e ∈ N f Ω be the unique light-like vector fieldsatisfying h e , e i = −
1. On the other hand, if we define smooth functions h αij by h αij = h α f ( e i , e j ) , e α i , i, j, = 1 , , α = 3 , , α f ( e i , e i ) = − h ii e − h ii e , (3.2a) α f ( e , e ) = e , (3.2b)where (3.2b) follows from H f = − α f ( e , e ). Note that we also have h αij = h A fe α e i , e j i because of (2.3). Therefore, the shape operators of f satisfies A fe e = − h e , A fe e = − h e , (3.3a) A fe e = e − h e , A fe e = − h e + e . (3.3b)On the other hand, the Laplace operator ∆ ⊥ associated with the normal con-nection of f takes the form∆ ⊥ = ∇ ⊥ e ∇ ⊥ e − ∇ ⊥∇ e e + ∇ ⊥ e ∇ ⊥ e − ∇ ⊥∇ e e . Furthermore, one can define smooth functions ξ , ξ by ∇ ⊥ e i e = ξ i e and ∇ ⊥ e i e = − ξ i e . (3.4)We obtain the following characterization of proper biconservative immer-sions. Proposition 3.1.
Let (Ω , g ) be a 2-dimensional semi-Riemannian manifold and f : (Ω , g ) ֒ → R ( c ) a quasi-minimal isometric immersion. Then, f is properbiconservative if and only if for any point p such that A fH f ( p ) = 0 , there existsa neighborhood N p such that T F is a degenerated distribution along which H F is parallel, where F = f | N p .Proof. Since the ambient space is R ( c ), we have trace (cid:0) ˜ R ( · , H f ) · (cid:1) T = 0. Fur-thermore, being quasi-minimal of f implies grad (cid:0) ˜ g ( H f , H f ) (cid:1) = 0. Therefore, f is biconservative if and only iftrace A f ∇ ⊥· H f ( · ) = 0 (3.5)because of Proposition 2.4. Note that (3.5) is equivalent to A f ∇ ⊥ e e ( e ) + A f ∇ ⊥ e e ( e ) = 0in terms of vector fields e , e , e defined above. By considering (3.3a) and (3.4),we conclude that f is biconservative if and only if ξ h e + ξ h e = 0 . (3.6)Note that being proper of the biconservative immersion f implies ξ ( q ) = 0 or ξ ( q ) = 0 at any point q ∈ Ω.Now, in order to prove the necessary condition, assume that f is a properbiconservative immersion and let A fH f ( p ) = 0 at a point p of Ω. Then, withoutloss of generality, we may assume h ( p ) = 0 on a neighboorhood N p of Ω. Inthis case, because of (3.6), we have ξ = 0 on N p which implies ξ ( q ) = 0 forany q ∈ N p . Thus, (3.6) implies h = 0 on N p . Put F = f | N p . Then, we6ave T F = span { e } which is a degenerated distribution. Moreover, since ξ vanishes identically on N p , we have ∇ ⊥ X H F = 0 whenever X ∈ T F . Hence, wehave completed the proof of the necessary condition.For the proof of the sufficient condition, we consider the following two casesseparately. If A fH f = 0, then (3.3a) implies h = h = 0. Therefore (3.6) issatisfied. On the other hand, consider the case A fH f ( p ) = 0 on N p . Assume T F = span { e } for a light-like vector field e and let ∇ ⊥ e e = 0. Then, wehave h = ξ = 0. Therefore (3.6) is satisfied again. Hence, the proof of thesufficient condition is completed.Now, we study the case c = 0. Let f : (Ω , g m ) ֒ → E be a proper biconser-vative quasi-minimal immersion, where Ω = I × J and g m is the metric definedby (2.7) for a m ∈ C ∞ (Ω). Assume that the Gaussian curvature K of Ω doesnot vanish. Note that, because of the Gauss equation (2.4a), if A fH f = 0 at apoint p ∈ Ω, then the Gaussian curvature K ( p ) = 0 which is a contradiction.Therefore, we have A fH f ( q ) = 0 for all q ∈ Ω . Hence, Proposition 3.1 impliesthat T f is a degenerated distribution along which H f is parallel. In terms ofa local pseudo-orthonormal frame field { e , e ; e , e } , we have A fe e = 0 and ∇ ⊥ e e = 0, or, equivalently, h = ξ = 0. Now, f is biharmonic if and only if(2.10) is satisfied. However, since ˜ R = 0, (2.10) becomes ∇ ⊥ e ∇ ⊥ e e − ∇ ⊥∇ e e e = α f ( A fe e , e )which is equivalent to the Ricci equation (2.4c) for X = e , Y = e and ξ = e .Hence, we have the following result. Theorem 3.2.
Let (Ω , g ) be a Lorentzian surface with the Gaussian curvature K and f : (Ω , g ) ֒ → E a quasi-minimal isometric immersion. Assume that K does not vanish. If f is a proper biconservative immersion, then it is biharmonic. E E with neutral metric. We get the complete local classification of quasi-minimal,biconservative surfaces.First, we consider flat surfaces and get the following classification of biconser-vative surfaces. We want to note that the proof of this proposition immediatelyfollows from the proof of [3, Theorem 4.1]. Proposition 4.1.
A flat surface in E is quasi-minimal and biconservative ifand only if locally congruent to one of the following surfaces:(i) The surface given by f ( s, t ) = ( ψ ( s, t ) , s − t √ , s + t √ , ψ ( s, t )) , ( s, t ) ∈ U , where ψ : U → R is a smooth function and U is open in R ,(ii) The surface given by f ( s, t ) = z ( s ) t + w ( s ) , where z ( s ) is a light-like curvein the light-cone L C and w is a light-like curve satisfying h z ′ , w ′ i = 0 and h z, w ′ i = − .Proof. A direct computation shows that the above surfaces are flat, quasi-minimal and biconservative. Conversely assume that M is a flat quasi-minimal7iconservative surface and p ∈ M . Consider a frame field { e , e ; e , e } de-scribed in Sect. 3 and let h αii are functions defined by (3.2) and (3.4), re-spectively. Then, by Proposition 3.1, we have two cases: h = h = 0 and h = h = 0. By considering the proof of [3, Theorem 4.1], one can concludethat f is congruent to one of these two surfaces given in the proposition.Now, we are going to consider non-flat quasi-minimal surfaces with non-parallel mean curvature vector. First we define an intrinsic L : Ω → R of(Ω , g m ) by L = − K t + mK s + 3 m s KK . (4.1)Then, we construct the following example of biconservative immersion from anon-flat two-dimensional Lorentzian manifold into E . We would like to notethat this immersion is also biharmonic because of Theorem 3.2. Proposition 4.2.
Let
Ω = I × J for some open intervals I, J and m ∈ C ∞ (Ω) and assume that the intrinsic L : Ω → R of (Ω , g m ) satisfies L = L ( t ) . Considera light-like curve α : J ֒ → E lying on L C such that V t = span { α ( t ) , α ′ ( t ) } istwo dimensional for all t ∈ J . Assume that η : J → R satisfies the conditions h η ′ , η ′ i = 0 , (4.2a) h α, η ′ i = 0 , (4.2b) h η ′ , α ′ i = − a , (4.2c) h η ′ , α ′′ i = 2 a ′ − aLa (4.2d) for a function a ∈ C ∞ ( J ) . Then, the mapping f : (Ω , g m ) −→ E f ( s, t ) = η ( t ) + (cid:16) sa ′ ( t ) − a ( t )( m ( s, t ) + sL ( t )) (cid:17) α ( t )+ sa ( t ) α ′ ( t ) (4.3) is a quasi-minimal, proper biconservative isometric immersion.Proof. Since the light-like curve α lies on L C , we have h α, α i = h α ′ , α ′ i = 0 (4.4)which implies h· , ·i| V t = 0. Thus, we have V t ⊂ V ⊥ t . Therefore, Lemma 2.1implies V t = V ⊥ t . Note that (4.4) also gives α ′′ ( t ) ∈ V ⊥ t = V t . Thus, we have α ′′ ( t ) = A ( t ) α ( t ) + a ( t ) L ( t ) − a ′ ( t ) a ( t ) α ′ (4.5)for a smooth function A because of (4.2b)-(4.2d). By a direct computationconsidering (4.3) and (4.5) we obtain h f s , f s i = 0, h f s , f t i = − h f t , f t i =2 m which yields that f is an isometric immersion.By a further computation, we get e = α f ( e , e ) = − H f = B ( t ) α ( t ) and α f ( e , e ) = a ( t ) m ss ( s, t ) α ( t ) . Therefore, we have h = 0 and ∇ ⊥ e e = 0 which yields that T f = span { ∂ s } and H f is parallel along T f . Hence, Proposition 3.1 yields that f is biconservative.8n the remaining part of this section we are going to show that the converseof Proposition 4.2 is also true. Lemma 4.3.
Let f : (Ω , g m ) ֒ → E be a quasi-minimal immersion and assumethat the Gaussian curvature of (Ω , g m ) does not vanish. Consider the pseudo-orthonormal frame field { e , e ; e , e } such that e = f ∗ ∂ s , e = f ∗ ( m∂ s + ∂ t ) and e = − H f . If f is proper biconservative and T f = span { ∂ s } , then the LeviCivita connection of E satisfies e ∇ e e = − am ss e , e ∇ e e = − m s e + e , (4.6a) e ∇ e e = e , e ∇ e e = m s e + ( m + bs − z ) e − a e , (4.6b) e ∇ e e = 0 , e ∇ e e = 1 a e + ( m s + b ) e , (4.6c) e ∇ e e = − e + am ss e , e ∇ e e = ( z − m − bs ) e − e − ( m s + b ) e for some smooth functions a, b, z such that e ( a ) = e ( b ) = e ( z ) = 0 and b + a ′ a = L. (4.6d) Proof.
Assume that f is proper biconservative. Then, we have h = ξ = 0because of Proposition 3.1. Thus, we have e ∇ e e = φ e − h e , e ∇ e e = φ e + e , e ∇ e e = − φ e + e , e ∇ e e = − φ e − h e − h e , e ∇ e e = 0 , e ∇ e e = h e + ξ e , e ∇ e e = − e + h e , e ∇ e e = h e − e − ξ e . (4.7)Note that Proposition 2.2 implies φ = 0 and φ = − m s . (4.8)On the other hand, Codazzi equation (2.4b) for X = Z = e , Y = e gives e ( h ) = − ξ , e ( h ) = 0 , (4.9)Furthermore, by using Gauss equation (2.4a) and Ricci equation (2.4c), we get h h = K = m ss (4.10) e ( ξ ) = K = m ss . (4.11)By taking into account e = f ∗ ∂ s , e = f ∗ ( m∂ s + ∂ t ), we consider (4.9), (4.10)and (4.11) to get h ( s, t ) = 1 a ( t ) , h ( s, t ) = a ( t ) m ss ( s, t ) ,h ( s, t ) = − m ( s, t ) − b ( t ) s + z ( t ) , ξ ( s, t ) = m s ( s, t ) + b ( t ) (4.12)for some a, b, z ∈ C ∞ ( J ). Finally, by combining (4.8) and (4.12) with (4.7), weobtain (4.6).On the other hand, Codazzi equation (2.4b) for X = Z = e , Y = e gives K ( a ′ + ab ) + ( K t + mK s + 3 m s K ) a = 0 . By combining this equation with (4.1), we get (4.6d).9ext, we get a necessary and sufficient condition for the existence of bicon-servative immersions from a Lorentzian surface (Ω , g m ). Proposition 4.4.
Let m ∈ C ∞ (Ω) and Ω = I × J for some open intervals I, J and consider the Lorentzian surface (Ω , g m ) with non-vanishing Gaussian cur-vature, where g m is the metric defined by (2.7) . Then, (Ω , g m ) admits a quasi-minimal, proper biconservative isometric immersion with non-parallel mean cur-vature vector such that T f = span { ∂ s } if and only if L = L ( t ) .Proof. In order to prove necessary condition, we assume the existence of suchimmersion f . Then, the Levi-Civita connection e ∇ satisfies (4.6) because ofLemma 4.3. (4.6d) yields ∂ s ( L ) = 0. Conversely, if L = L ( t ), the immersion f described by (4.3) is proper biconservative by Proposition (4.2). Hence, theproof is completed. Theorem 4.5. If M is a proper biconservative, quasi-minimal surface with non-vanishing Gaussian curvature, then it is locally congruent to the image f (Ω) ofthe isometric immersion f given in Proposition 4.2.Proof. Let (Ω , g m ) has non-vanishing Gaussian curvature. Consider a quasi-minimal isometric immersion f : (Ω , g m ) ֒ → R ( c ) and put M = f (Ω). Assumethat f is proper biconservative. Then, e ∇ satisfies (4.6) because of Lemma 4.3.The first equation in (4.6c) gives ∂e ∂s = 0 which implies e ( s, t ) = α ( t ) (4.13)for a mapping α : J → E . Also the second equation in (4.6c) and (4.13) give α ′ ( t ) = 1 a ( t ) f s ( s, t ) + ( m s ( s, t ) + b ( t )) α ( t ) . (4.14)By considering (4.13) and (4.14) one can see that α is a light-like curve lyingon L C because K does not vanish.On the other hand, the first equation in (4.6a) turns into f ss ( s, t ) = − a ( t ) m ss ( s, t ) α ( t )whose solution is f ( s, t ) = − a ( t ) m ( s, t ) α ( t ) + sξ ( t ) + η ( t ) (4.15)for some functions ξ, η : J → E . By using (4.15) and considering (4.6d) in thisequation, we obtain ξ = ( a ′ − aL ) α + aα ′ . (4.16)By combining (4.15) and (4.16) we get (4.3).Now, since f is an isometric immersion, we have h f s , f t i = − h f t , f t i =2 m . By a direct computation using h f s , f t i = − − am s h α, η ′ i + a ′ h α, η ′ i − aL h α, η ′ i + a h α ′ , η ′ i = − K = m ss does notvanish. On the other hand, h f t , f t i = 2 m and (4.3) imply2 m + 2 s (cid:18) − a ′ a + L + a h α ′′ , η ′ i (cid:19) + h η ′ , η ′ i = 2 m which gives the second equation in (4.2b) and (4.2d). Hence f is as given inProposition 4.2 which completes the proof.10y combining Proposition 4.1 and Theorem 4.5, we obtain the followingcomplete classification of quasi-minimal, proper biconservative surfaces in E . Theorem 4.6.
A surface M in E is quasi-minimal and proper biconservativeif and only if it is congruent to one of the following surfaces:(i) The surface given by f ( s, t ) = ( ψ ( s, t ) , s − t √ , s + t √ , ψ ( s, t )) , ( s, t ) ∈ U , where ψ : U → R is a smooth function and U is open in R ,(ii) The surface given by f ( s, t ) = z ( s ) t + w ( s ) , where z ( s ) is a light-like curvein the light-cone L C and w is a light-like curve satisfying h z ′ , w ′ i = 0 and h z, w ′ i = − ,(iii) The surface given in Proposition 4.2. Finally, by combining [3, Theorem 5.1] with Theorem 3.2 and Theorem 4.5,we get
Theorem 4.7.
A surface M in E is quasi-minimal and biharmonic if and onlyif it is congruent to one of the following surfaces:(i) The surface given by f ( s, t ) = ( ψ ( s, t ) , s − t √ , s + t √ , ψ ( s, t )) , ( s, t ) ∈ U , for asmooth function ψ : U → R satifying f st = 0 and f sstt = 0 , where U isopen in R ,(ii) The surface given by f ( s, t ) = z ( s ) t + w ( s ) , where z ( s ) is a light-like curvein the light-cone L C and w is a light-like curve satisfying h z ′ , w ′ i = 0 and h z, w ′ i = − ,(iii) The surface given in Proposition 4.2. Acknowledgements
This work was obtained during the ITU-GAP project
ARI2Harmoni (ProjectNumber: TGA-2017-40722).
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