You are an expert at summarizing long articles. Proceed to summarize the following text: 
the study of submanifolds with constant mean curvature , i.e. , @xmath4 submanifolds , and , in particular , that of @xmath4 surfaces in @xmath0-dimensional spaces , represents a very active research topic in differential geometry for more than @xmath7 years . there are several ways to generalize these submanifolds . for example , keeping the @xmath4 hypothesis and adding other geometric hypotheses to the submanifold or , by contrast , in the particular case of hypersurfaces in space forms , studying the hypersurfaces which are `` highly non-@xmath4 '' . the biconservative submanifolds seem to be an interesting generalization of @xmath4 submanifolds . biconservative submanifolds in arbitrary manifolds ( and in particular , biconservative surfaces ) which are also @xmath4 have some remarkable properties ( see , for example @xcite ) . @xmath4 hypersurfaces in space forms are trivially biconservative , so more interesting is the study of biconservative hypersurfaces which are non-@xmath4 ; recent results in non-@xmath4 biconservative hypersurfaces were obtained in @xcite . the biconservative submanifolds are closely related to the biharmonic submanifolds . more precisely , let us consider the _ bienergy functional _ defined for all smooth maps between two riemannian manifolds @xmath8 and @xmath9 and given by @xmath10 where @xmath11 is the tension field of @xmath12 . a critical point of @xmath13 is called a _ biharmonic map _ and is characterized by the vanishing of the _ bitension field _ @xmath14 ( see @xcite ) . a riemannian immersion @xmath15 or , simply , a submanifold @xmath16 of @xmath17 , is called _ biharmonic _ if @xmath12 is a biharmonic map . now , if @xmath18 is a fixed map , then @xmath13 can be thought as a functional defined on the set of all riemannian metrics on @xmath16 . this new functional s critical points are riemannian metrics determined by the vanishing of the _ stress - bienergy tensor _ @xmath19 . this tensor field satisfies @xmath20 if @xmath21 for a submanifold @xmath16 in @xmath17 , then @xmath16 is called a biconservative submanifold and it is characterized by the fact that the tangent part of its bitension field vanishes . thus we can expect that the class of biconservative submanifolds to be much larger that the class of biharmonic submanifolds . the paper is organized as follows . after a section where we recall some notions and general results about biconservative submanifolds , we present in _ section @xmath0 _ the local , intrinsic characterization of biconservative surfaces . the local , intrinsic characterization theorem provides the necessary and sufficient conditions for an abstract surface @xmath22 to admit , locally , a biconservative embedding with positive mean curvature function @xmath23 and @xmath24 at any point . our main goal is to extend the _ classification results for biconservative surfaces in @xmath1 , with @xmath2 and @xmath3 , to _ global _ results , i.e. , we ask that biconservative surfaces to be _ complete _ , with @xmath25 everywhere and @xmath26 on an open dense subset .    in _ section @xmath27 _ we consider the global problem and construct complete biconservative surfaces in @xmath5 with @xmath25 on @xmath16 and @xmath24 at any point of an open dense subset of @xmath16 . we determine such surfaces in two ways . one way is to use the local , extrinsic characterization of biconservative surfaces in @xmath5 and `` glue '' two pieces together in order to obtain a complete biconservative surface . the other way is more analytic and consists in using the local , intrinsic characterization theorem in order to obtain a biconservative immersion from @xmath28 in @xmath5 with @xmath25 on @xmath29 and @xmath30 on an open dense subset of @xmath29 ( the immersion has to be unique ) ; here , @xmath31 is a positive constant and therefore we obtain a one - parameter family of solutions . it is worth mentioning that , by a simple transformation of the metric @xmath32 , @xmath33 is ( intrinsically ) isometric to a helicoid .    in the _ last section _ we consider the global problem of biconservative surfaces in @xmath6 with @xmath25 on @xmath16 and @xmath24 at any point of an open dense subset of @xmath16 . as in the @xmath5 case , we use the local , extrinsic classification of biconservative surfaces in @xmath6 , but now the `` gluing '' process is not as clear as in @xmath5 . further , we change the point of view and use the local , intrinsic characterization of biconservative surfaces in @xmath6 . we determine the complete riemannian surfaces @xmath34 which admit a biconservative immersion in @xmath6 with @xmath25 everywhere and @xmath26 on an open dense subset of @xmath29 and we show that , up to isometries , there exists only a one - parameter family of such riemannian surfaces indexed by @xmath35 . we end the paper with some figures , obtained for particular choices of the constants , which represent the non-@xmath4 complete biconservative surfaces in @xmath5 and the way how these surfaces can be obtained in @xmath6 . throughout this work , all manifolds , metrics , maps are assumed to be smooth , i.e. in the @xmath36 category , and we will often indicate the various riemannian metrics by the same symbol @xmath37 . all surfaces are assumed to be connected and oriented . a _ harmonic map _ @xmath38 between two riemannian manifolds is a critical point of the _ energy functional _ @xmath39 and it is characterized by the vanishing of its _ tension field _ @xmath40    the idea of the stress - energy tensor associated to a functional comes from d. hilbert ( @xcite ) . given a functional @xmath41 , one can associate to it a symmetric 2-covariant tensor field @xmath42 such that @xmath43 at the critical points of @xmath41 . when @xmath41 is the energy functional , p. baird and j. eells ( @xcite ) , and a. sanini ( @xcite ) , defined the tensor field @xmath44 and proved that @xmath45 thus , @xmath42 can be chosen as the stress - energy tensor of the energy functional . it is worth mentioning that @xmath42 has a variational meaning . indeed , we can fix a map @xmath46 and think @xmath41 as being defined on the set of all riemannian metrics on @xmath16 . the critical points of this new functional are riemannian metrics determined by the vanishing of their stress - energy tensor @xmath42 . more precisely , we assume that @xmath16 is compact and denote @xmath47 for a deformation @xmath48 of @xmath49 we consider @xmath50 . we define the new functional @xmath51 and we have the following result . let @xmath52 and assume that @xmath16 is compact . then @xmath53 therefore @xmath49 is a critical point of @xmath54 if and only if its stress - energy tensor @xmath42 vanishes . we mention here that , if @xmath55 is an arbitrary isometric immersion , then @xmath43 . a natural generalization of harmonic maps is given by biharmonic maps . a _ biharmonic map _ @xmath38 between two riemannian manifolds is a critical point of the _ bienergy functional _ @xmath56 and it is characterized by the vanishing of its _ bitension field _ @xmath57 where @xmath58 is the rough laplacian of @xmath59 and the curvature tensor field is @xmath60 } z , \quad \forall x , y , z\in c(tm).\ ] ]    we remark that the _ biharmonic equation _ @xmath61 is a fourth - order non - linear elliptic equation and that any harmonic map is biharmonic . a non - harmonic biharmonic map is called proper biharmonic .    in @xcite , g. y. jiang defined the stress - energy tensor @xmath19 of the bienergy ( also called _ stress - bienergy tensor _ ) by @xmath62 as it satisfies @xmath20    the tensor field @xmath19 has a variational meaning , as in the harmonic case . we fix a map @xmath46 and define a new functional @xmath63 then we have the following result . let @xmath46 and assume that @xmath16 is compact . then @xmath64 so @xmath49 is a critical point of @xmath65 is and only if @xmath66 . we mention that , if @xmath55 is an isometric immersion then @xmath67 does not necessarily vanish .    _ a submanifold of a given riemannian manifold @xmath9 _ is a pair @xmath68 , where @xmath69 is a manifold and @xmath70 is an immersion . we always consider on @xmath16 the induced metric @xmath71 , thus @xmath72 is an isometric immersion ; for simplicity we will write @xmath70 without mentioning the metrics . also , we will write @xmath73 , or even @xmath16 , instead of @xmath74 .    a submanifold @xmath75 is called _ biharmonic _ if the isometric immersion @xmath12 is a biharmonic map from @xmath8 to @xmath9 .    even if the notion of biharmonicity may be more appropriate for maps than for submanifolds , as the domain and the codomain metrics are fixed and the variation is made only through the maps , the biharmonic submanifolds proved to be an interesting notion ( see , for example , @xcite ) .    in order to fix the notations , we recall here only the fundamental equations of first order of a submanifold in an riemannian manifold . these equations define the second fundamental form , the shape operator and the connection in the normal bundle . let @xmath55 be an isometric immersion . for each @xmath76 , @xmath77 splits as an orthogonal direct sum @xmath78 and @xmath79 is referred to as the normal bundle of @xmath12 , or of @xmath16 , in @xmath17 . denote by @xmath80 and @xmath81 the levi - civita connections on @xmath16 and @xmath17 , respectively , and by @xmath82 the induced connection in the pull - back bundle @xmath83 . taking into account the decomposition in , one has @xmath84 where @xmath85 is called the second fundamental form of @xmath16 in @xmath17 . here @xmath86 denotes the cotangent bundle of @xmath16 . the mean curvature vector field of @xmath16 in @xmath17 is defined by @xmath87 , where the @xmath88 is considered with respect to the metric @xmath49 . furthermore , if @xmath89 , then @xmath90 where @xmath91 is called the shape operator of @xmath16 in @xmath17 in the direction of @xmath92 , and @xmath93 is the induced connection in the normal bundle . moreover , @xmath94 , for all @xmath95 , @xmath89 . in the case of hypersurfaces , we denote @xmath96 , where @xmath97 and @xmath92 is the unit normal vector field , and we have @xmath98 ; @xmath23 is the _ ( @xmath99 times ) mean curvature function_.    a submanifold @xmath16 of @xmath17 is called @xmath100 if @xmath101 is parallel in the normal bundle , and @xmath4 if @xmath102 is constant .    when confusion is unlikely we identify , locally , @xmath16 with its image through @xmath12 , @xmath103 with @xmath104 and @xmath105 with @xmath106 . with these identifications in mind , we write @xmath107 and @xmath108    if @xmath21 for a submanifold @xmath16 in @xmath17 , then @xmath16 is called _ biconservative_. thus , @xmath16 is biconservative if and only if the tangent part of its bitension field vanishes . we have the following characterization theorem of biharmonic submanifolds , obtained by splitting the bitension field in the tangent and normal part . [ biharmonic_submanifold ] a submanifold @xmath69 of a riemannian manifold @xmath109 is biharmonic if and only if @xmath110 and @xmath111 where @xmath112 is the laplacian in the normal bundle .    various forms of the above result were obtained in @xcite . from here we deduce some characterization formulas for the biconservativity . let @xmath69 be a submanifold of a riemannian manifold @xmath109 . then @xmath16 is a biconservative submanifold if and only if :    1 . @xmath113 ; 2 . @xmath114 ; 3 . @xmath115 . the following properties are immediate . let @xmath69 be a submanifold of a riemannian manifold @xmath109 . if @xmath116 then @xmath16 is biconservative . let @xmath69 be a submanifold of a riemannian manifold @xmath109 . assume that @xmath17 is a space form , i.e. , it has constant sectional curvature , and @xmath16 is @xmath100 . then @xmath16 is biconservative . let @xmath69 be a submanifold of a riemannian manifold @xmath109 . assume that @xmath16 is pseudo - umbilical , i.e. , @xmath117 , and @xmath118 . then @xmath16 is @xmath4 . if we consider the particular case of hypersurfaces , then theorem @xmath119 becomes    if @xmath69 is a hypersurface in a riemannian manifold @xmath120 , then @xmath16 is biharmonic if and only if @xmath121 and @xmath122 where @xmath92 is the unit normal vector field of @xmath16 in @xmath17 . a hypersurface @xmath69 in a space form @xmath123 is biconservative if and only if @xmath124    any @xmath4 hypersurface in @xmath123 is biconservative . we are interested to study biconservative surfaces which are non-@xmath4 . we will first look at them from a local , extrinsic point of view and then from a global point of view . while by `` local '' we will mean the biconservative surfaces @xmath125 with @xmath25 and @xmath24 at any point of @xmath16 , by `` global '' we will mean the _ complete _ biconservative surfaces @xmath125 with @xmath25 at any point of @xmath16 and @xmath24 at any point of an open and dense subset of @xmath16 .    in this section , we consider the local problem , i.e. , we take @xmath125 a biconservative surface and assume that @xmath25 and @xmath24 at any point of @xmath16 . let @xmath126 and @xmath127 two vector fields such that @xmath128 is a positively oriented orthonormal basis at any point @xmath76 . in particular , we obtain that @xmath16 is parallelizable . if we denote by @xmath129 the eigenvalues functions of the shape operator @xmath130 , since @xmath131 and @xmath132 , we get @xmath133 and @xmath134 . thus the matrix of @xmath130 with respect to the ( global ) orthonormal frame field @xmath135 is @xmath136 we denote by @xmath137 the gaussian curvature and , from the gauss equation , @xmath138 , we obtain @xmath139 thus @xmath140 on @xmath16 .    from the definitions of @xmath141 and @xmath127 , we find that @xmath142 using the connection @xmath143-forms , the codazzi equation and then the extrinsic and intrinsic expression for the gaussian curvature , we obtain the next result which shows that the mean curvature function of a non - cmc biconservative surface must satisfy a second - order partial differential equation . more precisely , we have the following theorem . let @xmath125 a biconservative surface with @xmath25 and @xmath24 at any point of @xmath16 . then we have @xmath144 where @xmath145 is the laplace - beltrami operator on @xmath16 . in fact , we can see that around any point of @xmath16 there exists @xmath146 local coordinates such that @xmath147 and @xmath148 is equivalent to @xmath149 i.e. , @xmath23 must satisfy a _ second - order ordinary differential equation_.    indeed , let @xmath150 be an arbitrary fixed point of @xmath16 and let @xmath151 be an integral curve of @xmath141 with @xmath152 . let @xmath153 the flow of @xmath127 and @xmath146 local coordinates with @xmath154 such that @xmath155 we have @xmath156 and @xmath157 if we write the riemannian metric @xmath49 on @xmath16 in local coordinates as @xmath158 we get @xmath159 , and @xmath141 can be expressed with respect to @xmath160 and @xmath161 as @xmath162 where @xmath163 , @xmath164 . let @xmath165 . since @xmath166 , we find that @xmath167 it can be proved that @xmath168=\frac{3\left(x_1 f\right)}{4f}x_2,\ ] ] and thus @xmath169f=0 $ ] . on the other hand we have @xmath170 we recall that @xmath171 at any point of @xmath172 , and then @xmath173 at any point of @xmath172 . therefore , from , @xmath174 , i.e. , @xmath175 . since @xmath176 , and @xmath177 , we have @xmath178 , i.e. , @xmath179 in @xcite it was found an equivalent expression for @xmath148 , i.e. , @xmath180 therefore , using , relation is equivalent to . if @xmath125 is a non-@xmath4 biharmonic surface , then , there exists an open subset @xmath172 such that @xmath25 , @xmath24 at any point of @xmath172 , and @xmath23 satisfies the following system @xmath181 as we have seen , this system implies @xmath182 which , in fact , is a ode system . we get @xmath183 as an immediate consequence we obtain @xmath184 and combining it with the first integral @xmath185 of the first equation from , where @xmath186 is a constant , we obtain @xmath187 if we denote @xmath188 , we get @xmath189 . thus , @xmath190 satisfies a polynomial equation with constant coefficients , so @xmath190 has to be a constant and then , @xmath23 is a constant , i.e. , @xmath191 on @xmath172 ( in fact , @xmath23 has to be zero ) . therefore , we have a contradiction ( see @xcite for @xmath2 and @xcite , for @xmath192 )    we can also note that relation @xmath148 , which is an extrinsic relation , together with , allows us to find an _ intrinsic relation _ that @xmath193 must satisfy . more precisely , the gaussian curvature of @xmath16 has to satisfy @xmath194 and the hypotheses @xmath25 and @xmath24 , at any point , are equivalent with the intrinsic conditions @xmath140 and @xmath195 . formula is very similar to the ricci condition . further , we will briefly recall the ricci problem . given an abstract surface @xmath22 , we want to find the conditions that have to be satisfied by @xmath16 such that , locally , it admits a minimal embedding in @xmath1 . it was proved ( see @xcite ) that if @xmath22 is an abstract surface such that @xmath140 at any point of @xmath16 , where @xmath196 is a constant , then , locally , it admits a minimal embedding in @xmath1 if and only if @xmath197 condition is called the _ ricci condition with respect to @xmath198 _ , or simply the _ ricci condition_. if @xmath199 holds , then , locally , @xmath16 admits a one - parameter family of minimal embeddings in @xmath1 . we can see that relations @xmath200 and @xmath199 are very similar and , in @xcite , the authors studied the link between them . thus , for @xmath2 , it was proved that if we consider a surface @xmath22 which satisfies @xmath201 and @xmath202 , then there exists a very simple conformal transformation of the metric @xmath49 such that @xmath203 satisfies @xmath199 . a similar result was also proved for @xmath204 , but in this case , the conformal factor has a complicated expression ( and it is not enough to impose that @xmath22 satisfy , but we need the stronger hypothesis of it to admit a biconservative immersion in @xmath1 ) . unfortunately , condition @xmath201 does not imply , locally , the existence of a biconservative immersion in @xmath1 , as in the minimal case . we need a stronger condition . it was obtained the following local , intrinsic characterization theorem . [ thm : char ] let @xmath22 be an abstract surface and @xmath205 a constant . then , locally , @xmath16 can be isometrically embedded in a space form @xmath1 as a biconservative surface with positive mean curvature having the gradient different from zero at any point if and only if the gaussian curvature @xmath137 satisfies @xmath206 , @xmath207 , for any point @xmath76 , and its level curves are circles in @xmath16 with curvature @xmath208    if the surface @xmath16 in theorem [ thm : char ] is simply connected , then the theorem holds globally , but , in this case , instead of a local isometric embedding we have a global isometric immersion . we remark that unlike in the minimal immersions case , if @xmath16 satisfies the hypotheses from theorem [ thm : char ] , then there exists a _ unique _ biconservative immersion in @xmath1 ( up to an isometry of @xmath1 ) , and not a one - parameter family . the characterization theorem can be equivalently rewritten as below . [ thm : char - local ] let @xmath22 be an abstract surface with gaussian curvature @xmath137 satisfying @xmath206 and @xmath207 at any point @xmath76 , where @xmath196 is a constant . let @xmath209 and @xmath210 be two vector fields on @xmath16 such that @xmath128 is a positively oriented basis at any point of @xmath76 . then , the following conditions are equivalent :    * the level curves of @xmath137 are circles in @xmath16 with constant curvature @xmath211 * @xmath212 * locally , the metric @xmath49 can be written as @xmath213 , where @xmath214 are local coordinates positively oriented , @xmath215 , and @xmath216 ; * locally , the metric @xmath49 can be written as @xmath217 , where @xmath214 are local coordinates positively oriented , and @xmath218 satisfies the equation @xmath219 and the condition @xmath220 ; moreover , the solutions of the above equation , @xmath221 , are @xmath222 where @xmath12 is in some open interval @xmath223 and @xmath224 are constants ; * locally , the metric @xmath49 can be written as @xmath217 , where @xmath214 are local coordinates positively oriented , and @xmath218 satisfies the equation @xmath225 and the conditions @xmath220 and @xmath226 ; moreover , the solutions of the above equation , @xmath221 , are @xmath227 where @xmath12 is in some open interval @xmath223 and @xmath228 are constants , @xmath229 . the proof follows by direct computations and by using remark 4.3 in @xcite and proposition 3.4 in @xcite .    from the above theorem we have the following remarks .    * if condition @xmath230 is satisfied , i.e. , the integral curves of @xmath127 are circles in @xmath16 with a precise constant curvature , then the integral curves of @xmath141 are geodesics of @xmath16 . * if condition @xmath231 is satisfied , then @xmath137 has to be a solution of the equation @xmath232 * if condition @xmath231 is satisfied and @xmath233 , then @xmath234 is a flat surface and , trivially , a ricci surface with respect to @xmath198 . * let @xmath218 be a solution of the equation . we consider the change of coordinates @xmath235 where @xmath236 is a positive constant and @xmath237 , and define @xmath238 then @xmath239 and @xmath153 also satisfies equation . if @xmath218 satisfies the first integral @xmath240 where @xmath229 , then , for @xmath241 , @xmath242 satisfies @xmath243 from here , as the classification is done up to isometries , we note that the parameter @xmath244 in the solution of is not essential and only the parameter @xmath245 counts . thus we have a one - parameter family of solutions . * if @xmath12 is a solution of , for some @xmath198 , then @xmath246 , where @xmath247 is a real constant , is a solution of for @xmath248 . * if @xmath2 , we note that if @xmath12 is a solution of , then also @xmath249 is a solution of the same equation , i.e , the condition @xmath230 from theorem @xmath250 is invariant under the homothetic tranformations of the metric @xmath49 . then , we see that equation is invariant under the affine change of parameter @xmath251 , where @xmath252 . therefore , we must solve equation up to this change of parameter and an additive constant of the solution @xmath12 . the additive constant will be the parameter that counts .    in the @xmath2 case , the solutions of equation , are explicitly determined in the next proposition . [ prop - c=0 ] the solutions of the equation @xmath253 which satisfy the conditions @xmath220 and @xmath254 , up to affine transformations of the parameter with @xmath252 , are given by @xmath255    we note that , when @xmath2 , we have a one - parameter family of solutions of equation , i.e. , @xmath256 , @xmath31 being a positive constant . if @xmath257 , then we can not determine explicitly @xmath218 . another way to see that in the @xmath204 case we have only a one - parameter family of solutions of equation is to rewrite the metric @xmath49 in certain non - isothermal coordinates . further , we will consider only the @xmath3 case and we have the next result . let @xmath22 be an abstract surface with @xmath258 , where @xmath221 satisfies @xmath259 where @xmath12 is in some open interval @xmath223 , @xmath260 are positive constants , and @xmath261 is a constant . then @xmath22 is isometric to @xmath262 where @xmath263 , @xmath264 is a positive constant , and @xmath265 and @xmath266 are the positive vanishing points of @xmath267 , with @xmath268 . let us consider @xmath269 and @xmath270 the surfaces @xmath271 and @xmath272 are isometric if and only if @xmath273 and the isometry is @xmath274 . therefore , we have a one - parameter family of surfaces . we note that the expression of the gaussian curvature of @xmath271 does not depend on @xmath35 . more precisely , @xmath275 but , if we change further the coordinates @xmath276 , then we `` fix '' the domain , i.e. , @xmath271 is isometric to @xmath277 and @xmath35 appears in the expression of @xmath278 . in this section we consider the global problem and construct complete biconservative surfaces in @xmath5 with @xmath25 everywhere and @xmath24 at any point of an open dense subset . or , from intrinsic point of view , we construct a complete abstract surface @xmath22 with @xmath202 everywhere and @xmath279 at any point of an open dense subset of @xmath16 , that admits a biconservative immersion in @xmath5 , defined on the whole @xmath16 , with @xmath25 on @xmath16 and @xmath26 on the open dense subset . first , we recall a local extrinsic result which provides a characterization of biconservative surfaces in @xmath5 . let @xmath280 be a surface in @xmath5 with @xmath281 and @xmath282 for any @xmath76 . then , @xmath16 is biconservative if and only if , locally , it is a surface of revolution , and the curvature @xmath283 of the profile curve @xmath175 , @xmath284 , is a positive solution of the following ode @xmath285    in @xcite there was found the local explicit parametric equation of a biconservative surface in @xmath5 . let @xmath280 be a biconservative surface in @xmath5 with @xmath281 and @xmath282 for any @xmath76 . then , locally , the surface can be parametrized by @xmath286 where @xmath287 with @xmath288 a positive constant and @xmath289 . we note that any two such surfaces are not locally isometric , so we have a one - parameter family of biconservative surfaces in @xmath5 . these surfaces are not complete . we denote by @xmath290 the image of @xmath291 . the `` boundary '' of @xmath290 , i.e. , @xmath292 , is the circle @xmath293 , which lies in the @xmath294 plane . at a boundary point , the tangent plane to the closure @xmath295 of @xmath290 is parallel to @xmath296 . moreover , along the boundary , the mean curvature function is constant @xmath297 and @xmath298 . thus , in order to obtain a complete biconservative surface in @xmath5 , we can expect to `` glue '' along the boundary two biconservative surfaces of type @xmath290 corresponding to the same @xmath288 ( the two constants have to be the same ) and symmetric to each other , at the level of @xmath36 smoothness .    in fact , it was proved that we can glue two biconservative surfaces @xmath290 and @xmath299 , at the level of @xmath36 smoothness , only along the boundary and , in this case , @xmath300 .    if we consider the symmetry of @xmath301 , with respect to the @xmath302 axis , we get a smooth , complete , biconservative surface @xmath303 in @xmath5 . moreover , its mean curvature function @xmath304 is positive and @xmath305 is different from zero at any point of an open dense subset of @xmath303 . the profile curve @xmath306 can be repara - metrized as @xmath307 and now @xmath308 . the homothety of @xmath5 , @xmath309 , renders @xmath310 onto @xmath311 .    in @xcite , there were also found the complete biconservative surfaces in @xmath5 with @xmath25 at any point and @xmath24 at any point of an open dense subset , but there , the idea was to use the intrinsic characterization of the biconservative surfaces . more precisely , we have the next global result .    [ main_th1 ] let @xmath312 be a surface , where @xmath313 is a positive constant . then we have :    * the metric on @xmath29 is complete ; * the gaussian curvature is given by @xmath314 and therefore @xmath315 at any point of @xmath316 ; * the immersion @xmath317 given by @xmath318 is biconservative in @xmath5 , where @xmath319    the first two items follow by standard arguments . for the last part , we note that choosing @xmath320 in and using the change of coordinates @xmath321 , where @xmath322 , the metric induced by @xmath323 coincides with @xmath32 . then , we define @xmath324 as : for @xmath322 , @xmath325 is obtained by rotating the profile curve @xmath326 and for @xmath327 , @xmath325 is obtained by rotating the profile curve @xmath328    we consider the surface @xmath28 . then @xmath33 satisfies the ricci condition and can be minimally immersed in @xmath5 as a helicoid or a catenoid .    concerning the biharmonic surfaces in @xmath5 we have the following non - existence result . there exists no proper biharmonic surface in @xmath5 . as in the previous section , we consider the global problem for biconservative surfaces in @xmath6 , i.e. , our aim is to construct complete biconservative surfaces in @xmath6 with @xmath25 everywhere and @xmath24 at any point of an open and dense subset . let @xmath280 be a biconservative surface in @xmath6 with @xmath281 i @xmath329 at any point @xmath76 . then , locally , the surface , viewed in @xmath330 , can be parametrized by @xmath331 where @xmath332 is a positive constant ; @xmath333 are two constant orthonormal vectors ; @xmath334 is a curve parametrized by arclength that satisfies @xmath335 and , as a curve in @xmath336 , its curvature @xmath283 is a positive non constant solution of the following ode @xmath337 such that @xmath338      we consider @xmath345 and @xmath346 and change the coordinates @xmath214 in @xmath347 . then , we get @xmath348 where @xmath349 , @xmath350 and @xmath351 are positive solutions of @xmath352 and @xmath353 with @xmath354 a constant and @xmath355 . we note that an alternative expression for @xmath356 was given in @xcite . if we denote @xmath360 the image of @xmath356 , then we note that the boundary of @xmath360 is made up from two circles and along the boundary , the mean curvature function is constant ( two different constants ) and its gradient vanishes . more precisely , the boundary of @xmath360 is given by the curves @xmath361 and @xmath362 these curves are circles in affine planes in @xmath330 parallel to the @xmath363 plane and their radii are @xmath364 and @xmath365 , respectively . at a boundary point , using the coordinates @xmath366 , we get that the tangent plane to the closure of @xmath360 is spanned by a vector which is tangent to the corresponding circle and by @xmath367 where @xmath368 or @xmath369 . thus , in order to construct a complete biconservative surface in @xmath6 , we can expect to glue along the boundary two biconservative surfaces of type @xmath360 , corresponding to the same @xmath339 . in fact , if we want to glue two surfaces corresponding to @xmath339 and @xmath370 along the boundary , then these constants have to coincide and there is no ambiguity concerning along which circle of the boundary we should glue the two pieces . but this process is not as clear as in @xmath5 since we should repeat it infinitely many times . the first item follows by standard arguments . for the second item , we note that choosing @xmath380 in and using the change of coordinates @xmath381 , the metric induced by @xmath382 coincides with @xmath383 . the immersion @xmath384 depends on the sign @xmath389 and on the constant @xmath390 in the expression of @xmath391 . as the classification is up to isometries of @xmath6 , the sign and the constant are not important , but they will play an important role in the gluing process .    the key idea in our construction of complete biconservative surface in @xmath6 is to notice that @xmath271 is , locally and intrinsically , isometric to a surface of revolution in @xmath5 . the first step is to construct a complete surface of revolution in @xmath5 which on an open dense subset is locally isometric to @xmath271 . we start with the next result . [ theorem3.18 ] let us consider @xmath271 as above . then @xmath271 is the universal cover of the surface of revolution in @xmath5 given by @xmath392 where @xmath393 , @xmath394 @xmath395 is a positive constant and @xmath396 is constant . we note that the boundary of @xmath400 is given by the curves @xmath403 and @xmath404 these curves are circles in affine planes in @xmath5 parallel to the @xmath405 plane and their radii are @xmath406 and @xmath407 , respectively .    at a boundary point , using the coordinates @xmath408 , we get that the tangent plane to the closure of @xmath400 is spanned by a vector which is tangent to the corresponding circle and by the vector @xmath409 . thus , the tangent plane is parallel to @xmath296 axis . geometrically , we start with a piece of type @xmath400 and by symmetry to the planes where the boundary lie , we get our complete surface @xmath410 ; the process is periodic and we perform it along the whole @xmath296 axis . analytically , we fix @xmath35 and @xmath411 , and alternating the sign and with appropriate choices of the constant @xmath398 , we can construct a complete surface of revolution @xmath410 in @xmath5 which on an open subset is locally isometric to @xmath412 . in fact , these choices of @xmath413 and @xmath414 , and of the constants @xmath398 are uniquely determined by the `` first '' choice of @xmath413 , or of @xmath414 , and of the constant @xmath398 . we start with @xmath413 and @xmath415 . the profile curve of @xmath400 can be seen as the graph of a function depending on @xmath399 and this allows us to obtain a function @xmath416 such that the profile curve of @xmath410 to be the graph of the function @xmath417 depending on @xmath399 and defined on the whole @xmath296 ( or @xmath418 ) . the function @xmath419 $ ] is periodic and at least of class @xmath420 . [ main_th3 ] the surface of revolution given by @xmath421 is complete and , on an open dense subset , it is locally isometric to @xmath371 . the induced metric is given by @xmath422 @xmath423 . moreover , @xmath279 at any point of that open dense subset , and @xmath424 everywhere . the universal cover of the surface of revolution given by @xmath426 is @xmath29 endowed with the metric @xmath427 . it is complete , @xmath424 on @xmath29 and , on an open dense subset , it is locally isometric to @xmath271 and @xmath279 at any point . moreover any two surfaces @xmath428 and @xmath429 are isometric . the second step is to construct effectively the biconservative immersion from @xmath428 in @xmath6 , or from @xmath430 in @xmath6 . the geometric ideea of the construction is the following : from each piece @xmath400 of @xmath410 we `` go back '' to @xmath271 and then , using @xmath384 and a specific choice of @xmath413 or @xmath414 and of the constant @xmath390 , we get our biconservative immersion @xmath431 . again , the choices of @xmath413 and @xmath414 , and of the constant @xmath390 are uniquely determined ( modulo @xmath432 , for @xmath390 ) by the `` first '' choice of @xmath413 , or of @xmath414 , and of the constant @xmath390 ( see @xcite for all details ) .              in the case @xmath2 , the idea was to construct , by symmetry , a complete biconservative surface in @xmath5 starting with a piece of a biconservative surface . we illustrate this in the following figure obtained for @xmath439 .      in the case @xmath3 , the construction of a complete biconservative surface in @xmath6 can be summarized in the next diagram , obtained for @xmath440 , @xmath415 and we started with @xmath413 in the expression of @xmath399 . ( manifold ) at ( 3.6,0 ) ; ( 4.3,-2 ) node[below ] @xmath22 ; ( band ) at ( 12,0 ) ; ( 10,0 )  ( 14,0 ) ; ( 10.5,-2.5 )  ( 10.5,2.5 ) ; ( 11.2,0 ) node[below ] @xmath265 ; ( 12.9,0 ) node[below ] @xmath266 ; ( 14.1,0 ) node[below ] @xmath441 ; ( 10.5,2.6 ) node[left ] @xmath442 ; ( 12,-2.5 ) node[below ] @xmath271 ; ( manifold) node[midway , fill = none , above ] isometry(9.5,0 ) ; ( 11,-3.3)-| ( 11,-4 ) ; ( 11,-4)-| ( 13,-4 ) ; ( 13,-4 )  ( 13,-8 ) node[midway , sloped , fill = none , below ] @xmath443 node[midway , sloped , fill = none , above ] biconservative ; at ( 13,-9.3)@xmath6 ; ( 10.7,-3.3)-| ( 10.7,-4 ) ; ( 10.7,-4 ) (3.5,-4 ) node[midway , sloped , fill = none , below ] @xmath444 node[midway , sloped , fill = none , above ] isometry ; at ( 2,-4 ) ; ( 2,-5.5 ) node[below ] @xmath445 ; ( graph6 ) at ( 5.3,-9.3 ) ; ( 5.5,-10 ) node[below ] @xmath446 ; ( 2,-6.2)(5,-8 ) node[pos=0.5,sloped , fill = none , above ] playing with the node[pos=0.5,sloped , fill = none , below ] constant @xmath398 and @xmath389 ;
we survey some recent results on biconservative surfaces in @xmath0-dimensional space forms @xmath1 with a special emphasis on the @xmath2 and @xmath3 cases . 
 we study the local and global properties of such surfaces , from extrinsic and intrinsic point of view . 
 we obtain all non-@xmath4 complete biconservative surfaces in @xmath5 and @xmath6 .