document: the study of harmonic maps was initiated by f. b. fuller , j. nash and j. h. sampson @xcite while the first general result on the existence of harmonic maps is due to eells - sampson @xcite . 
   harmonic maps are extrema ( critical points ) of the energy functional defined on the space of smooth maps between riemannian ( pseudo - riemannian ) manifolds . the trace of the second fundamental form of such maps vanishes . 
   more precisely , let @xmath0 , @xmath1 be pseudo - riemannian manifolds and @xmath2 denotes the sections of the tangent bundle @xmath3 of @xmath4 , that is , the space of vector fields on @xmath4 . then _ 
   @xmath5 of a smooth map @xmath6 is defined by the formula @xmath7 where @xmath8 is the volume measure associated to the metric @xmath9 and the _ energy density _ 
   @xmath10 of @xmath11 is the smooth function @xmath12 given by @xmath13 for each @xmath14 . in the above equation @xmath15 is a linear map @xmath16 therefore it can be considered as a section of the bundle @xmath17 where @xmath18 is the pullback bundle having fibres @xmath19 , @xmath14 and @xmath20 is the pullback metric on @xmath21 . if we denote by @xmath22 and @xmath23 the levi - civita connections on @xmath24 and @xmath25 respectively , 
   then the second fundamental form of @xmath11 is the symmetric map @xmath26 defined by @xmath27 for any @xmath28 . where @xmath29 is the pullback of the levi - civita connection @xmath23 of @xmath25 to the induced vector bundle @xmath30 . 
   the section @xmath31 , defined by @xmath32 is called the _ tension field _ of @xmath11 and a map 
   is said to be harmonic if its tension field vanishes identically ( see @xcite ) .    if we consider @xmath33 a smooth two - parameter variation of @xmath11 such that @xmath34 and let @xmath35 be the corresponding variational vector fields then @xmath36 the _ hessian _ of a harmonic map @xmath11 is defined by : @xmath37    the index of a harmonic map @xmath38 is defined as the dimension of the tangent subspace of @xmath39 on which the hessian @xmath40 is negative definite . a harmonic map @xmath11 is said to be _ stable _ if morse index ( _ i.e. _ , the dimension of largest subspace of @xmath41 on which the hessian @xmath40 is negative definite ) of @xmath11 is zero and otherwise 
   , it is said to be _ unstable _ ( see @xcite ) . for a non - degenerate point @xmath42 
   , we decompose the space @xmath43 into its _ vertical space _ @xmath44 and its _ horizontal space _ 
   @xmath45 , that is , @xmath46 , so that @xmath47 . 
   the map is said to be horizontally conformal if for each @xmath14 either the rank of @xmath48 is zero ( that is , @xmath49 is a critical point ) , or the restriction of @xmath48 to the horizontal space @xmath50 is surjective and conformal ( here @xmath49 is a regular point ) @xcite . 
   the premise of harmonic maps has acknowledged several important contributions and has been successfully applied in computational fluid dynamics ( cfd ) , minimal surface theory , string theory , gravity and quantum field theory ( see @xcite ) . 
   most of works on harmonic maps are between riemannian manifolds @xcite . 
   the harmonic maps between pseudo - riemannian manifolds behave differently and their study must be subject to some restricted classes of pseudo - riemannian manifolds @xcite . 
   this paper is organized as follows . in sect . 
   [ prem ] , the basic definitions about almost para - hermitian manifolds , almost paracontact manifolds and normal almost paracontact manifolds are given . in sect . 
   [ paraholo ] , we define and study paraholomorphic map . we prove that the tension field of any @xmath51-paraholomorphic map between almost para - hermitian manifold and para - sasakian manifold lies in @xmath52 . 
   [ parap ] deals with parapluriharmonic map in which we obtain the necessary and sufficient condition for a @xmath53-paraholomorphic map between para - sasakian manifolds to be @xmath54-parapluriharmonic and give an example for its illustrations . 
   a smooth manifold @xmath55 of dimension @xmath56 is said to be an almost product structure if it admits a tensor field @xmath57 of type @xmath58 satisfying : @xmath59 in this case the pair @xmath60 is called an almost product manifold . 
   an almost para - complex manifold is an almost product manifold @xmath61 such that the eigenbundles @xmath62 associated with the eigenvalues @xmath63 of tensor field @xmath57 have the same rank @xcite . 
   an almost para - hermitian manifold @xmath64 is a smooth manifold endowed with an almost para - complex structure @xmath57 and a pseudo - riemannian metric @xmath65 compatible in the sense that @xmath66 it follows that the metric @xmath65 has signature @xmath67 and the eigenbundles @xmath62 are totally isotropic with respect to @xmath65 . 
   let @xmath68 be an orthonormal basis and denote @xmath69 : @xmath70 for @xmath71 and @xmath72 for @xmath73 . 
   the fundamental @xmath74-form of almost para - hermitian manifold is defined by @xmath75 and the co - differential @xmath76 of @xmath77 is given as follows @xmath78 an almost para - hermitian manifold is called para - khler if @xmath79 @xcite . 
   a @xmath80 smooth manifold @xmath81 of dimension @xmath82 is said to have a triplet @xmath83-structure if it admits an endomorphism @xmath84 , a unique vector field @xmath85 and a @xmath86-form @xmath87 satisfying : @xmath88 where @xmath89 is the identity transformation ; and the endomorphism @xmath84 induces an almost paracomplex structure on each fibre of @xmath90 the contact subbundle , _ i.e. _ , eigen distributions @xmath91 corresponding to the characteristic values @xmath63 of @xmath84 
   have equal dimension @xmath92 . 
   + from the equation ( [ eta ] ) , it can be easily deduced that @xmath93 this triplet structure @xmath83 is called an almost paracontact structure and the manifold @xmath81 equipped with the @xmath83-structure is called an almost paracontact manifold ( see also @xcite ) . if an almost paracontact manifold admits a pseudo - riemannian metric @xmath94 satisfying : @xmath95 where signature of @xmath94 is necessarily @xmath96 for any vector fields @xmath97 and @xmath98 
   ; then the quadruple @xmath99 is called an almost paracontact metric structure and the manifold @xmath81 equipped with paracontact metric structure is called an almost paracontact metric manifold . with respect to @xmath94 , @xmath87 is metrically dual to @xmath85 , that is @xmath100 also , equation ( [ gphi ] ) implies that @xmath101 further , in addition to the above properties , if the structure-@xmath99 satisfies : @xmath102 for all vector fields @xmath103 on @xmath81 , then the manifold is called a paracontact metric manifold and the corresponding structure-@xmath99 is called a paracontact structure with the associated metric @xmath94 @xcite . for an almost paracontact metric manifold , there always exists a special kind of local pseudo - orthonormal basis @xmath104 ; where @xmath105 @xmath85 and @xmath106 s are space - like vector fields and @xmath107 s are time - like . 
   such a basis is called a @xmath84-basis . 
   hence , an almost paracontact metric manifold @xmath108 is an odd dimensional manifold with a structure group @xmath109 , where @xmath110 is the para - unitary group isomorphic to @xmath111 . 
   an almost paracontact metric structure-@xmath99 is para - sasakian if and only if @xmath112 from eqs . , and , it can be easily deduced for a para - sasakian manifold that @xmath113 in particular , a para - sasakian manifold is @xmath114-paracontact @xcite .      on an almost paracontact metric manifold , one defines the @xmath115-tensor field @xmath116 by @xmath117 - 2\,d\eta\otimes\xi,\end{aligned}\ ] ] where @xmath118 $ ] is the nijenhuis torsion of @xmath84 . if @xmath116 vanishes identically , then we say that the manifold @xmath81 is a normal almost paracontact metric manifold @xcite . 
   the normality condition implies that the almost paracomplex structure @xmath57 defined on @xmath119 by @xmath120 is integrable . here 
   @xmath97 is tangent to @xmath81 , @xmath121 is the coordinate on @xmath122 and @xmath123 is a @xmath124 function on @xmath119 . 
   now we recall the following proposition which characterized the normality of almost paracontact metric @xmath125-manifolds :    @xcite for an almost paracontact metric @xmath125-manifold @xmath126 , the following three conditions are mutually equivalent    * @xmath126 is normal , * there exist smooth functions @xmath127 on @xmath126 such that @xmath128 * there exist smooth functions @xmath127 on @xmath126 such that @xmath129    where @xmath22 is the levi - civita connection of the pseudo - riemannian metric @xmath94 . 
   the functions @xmath127 appearing in eqs . ( [ nablaxphiy ] ) and ( [ nablaxxi ] ) are given by @xmath130 a normal almost paracontact metric @xmath125-manifold is called paracosymplectic if @xmath131 and para - sasakian if @xmath132 @xcite . 
   one can look structure preserving mapping between almost para - hermitian and almost paracontact manifolds as analogous of the well - known holomorphic mappings in complex geometry @xcite . 
   +    let @xmath133 , @xmath1 be almost paracontact metric manifolds and @xmath64 be an almost para - hermitian manifold . then a smooth map    * @xmath134 is @xmath135-paraholomorphic map if @xmath136 . 
   for such a map @xmath137 . 
   * @xmath138 is @xmath139-paraholomorphic map if @xmath140 . here 
   * @xmath142 is @xmath53-paraholomorphic map if @xmath143 . 
   in particular , @xmath144 and @xmath145 .    when @xmath15 interwines the structures upto a minus sign , we say about @xmath135-anti paraholomorphic , @xmath139-anti paraholomorphic and @xmath53-anti paraholomorphic mappings .    now , we prove the following result . 
   let @xmath11 be a smooth @xmath53-paraholomorphic map between para - sasakian manifolds @xmath146 , @xmath1 . 
   then @xmath147 where @xmath148 .    since @xmath15 has values in @xmath149 so that @xmath150 and @xmath151 have values in @xmath149 . 
   thus , we have @xmath152 in the last equality , we have used . on the other hand , we obtain @xmath153 from eqs . and , 
   we have @xmath154 let @xmath155 be a local orthonormal frame for @xmath156 . taking the trace in and using the fact that @xmath157 is symmetric , we have . 
   this completes the proof .    following the proof of the above proposition 
   , we can give the following remarks :    for a para - sasakian manifold @xmath158 and a para - hermitian manifold @xmath64 . 
   if    * @xmath134 be a @xmath135-paraholomorphic map then we have @xmath159 where @xmath160 . + 
   * @xmath138 be a @xmath139-paraholomorphic map then we have @xmath161 where @xmath162 .    [ t1 ] 
   let @xmath11 be a @xmath135-paraholomorphic map between para - sasakian manifold @xmath163 and para - k@xmath164hler manifold @xmath64 . 
   then @xmath11 is harmonic . 
   let @xmath165 be a local orthonormal adapted basis on @xmath166 , then from eqs . and 
   , we have @xmath167 ( since for a @xmath135-paraholomorphic map @xmath137 ) . 
   it follows by the use of equation that @xmath168 as @xmath55 is a para - k@xmath169hler manifold . 
   therefore , @xmath170 and @xmath11 is harmonic . 
   this completes the proof of the theorem .    for @xmath1 , 
   let @xmath171 be real distributions , respectively , on para - sasakian manifolds @xmath172 of rank @xmath173 then it admits globally defined @xmath86-form @xmath174 such that @xmath175 . 
   clearly , @xmath176 , where @xmath177 is the real distribution of rank one defined by @xmath178 @xcite . 
   now , we prove :    [ hk1 ] for any @xmath139-paraholomorphic map @xmath11 between almost para - hermitian manifold @xmath64 and para - sasakian manifold @xmath179 , the tension field @xmath180 .    before going to proof of this theorem , 
   we first prove the following proposition :    for an almost para - hermitian manifold @xmath64 , we have @xmath181\bigg\}\end{aligned}\ ] ] where @xmath182 is a local orthonormal frame on @xmath183 . 
   it is straightforward to calculate @xmath184-j(\nabla_{e'_{i}}e'_{i})+j(\nabla_{je'_{i}}je'_{i})\big\ } \ ] ] and the result follows from and . 
   this completes the proof . 
   _ proof of theorem [ hk1 ] . _ since @xmath185 , @xmath186 therefore for any local orthonormal frame @xmath182 on @xmath183 
   , we obtain by using eqs . 
   , , and that @xmath187 employing eq . , the above equation reduces to @xmath188\bigg),\xi_{1}\bigg).\end{aligned}\ ] ] reusing eq . in 
   , we get @xmath189 which shows that @xmath180 . 
   this completes the proof of the theorem .    by the consequence of the above theorem we can state the following result as a corollary of the theorem [ hk1 ] . 
   let @xmath64 and @xmath163 be para - k@xmath164hler and para - sasakian manifolds respectively . 
   then for any @xmath139-paraholomorphic map @xmath138 , the tension field @xmath180 . 
   in this section we define the notion of @xmath54-parapluriharmonic map which is similar to the notion of @xmath84-pluriharmonic map between almost contact metric manifold and riemannian manifold , for @xmath84-pluriharmonic map see : @xcite . 
   a smooth map @xmath11 between almost paracontact metric manifold @xmath179 and pseudo - riemannian manifold @xmath190 is said to be @xmath54-parapluriharmonic if @xmath191 where the second fundamental form @xmath192 of @xmath11 is defined by . 
   in particular , @xmath193 for any tangent vector @xmath97 . 
   we recall that @xmath137 for a @xmath135-paraholomorphic map and @xmath200 is para - k@xmath201hler , and that from eq . for any vectors @xmath202 
   tangent to @xmath203 , we have @xmath204 using equation for a given map , we obtain @xmath205 replacing @xmath98 by @xmath206 and employing eqs . and , the above equation reduces to @xmath207 by the virtue of the fact that @xmath157 is symmetric , we obtain from above equation that @xmath208 the above expresion implies that @xmath11 is @xmath54-parapluriharmonic and thus harmonic from the proposition [ pro1 ] . 
   this completes the proof of the theorem . 
   let @xmath198 be a normal almost paracontact metric 3-manifold with @xmath210constant , @xmath199 be a para - k@xmath164hler manifold and @xmath211 be a smooth @xmath135-paraholomorphic map . 
   then @xmath212 is paracosymplectic manifold .        since @xmath11 is a @xmath53-paraholomorphic map then for all @xmath214 there exists a function @xmath123 on @xmath215 such that @xmath216 for any @xmath217 , we have from eqs . , and that @xmath218 from above equation and the fact that @xmath157 is symmetric , we obtain that @xmath219 replacing @xmath98 by @xmath220 in above expression and using eqs . and , we find @xmath221 this implies that @xmath222 if and only if @xmath213 
   . this completes the proof of the theorem . 
   let @xmath223 be @xmath125-dimensional manifolds with standard cartesian coordinates . 
   define the almost paracontact structures @xmath224 respectively on @xmath225 by @xmath226 where @xmath227 , @xmath228 , @xmath229 , @xmath230 , @xmath231 and @xmath232 . by direct calculations , one verifies that the nijenhuis torsion of @xmath233 for @xmath1 vanishes , which implies that the structures are normal . 
   let the pseudo - riemannian metrics @xmath234 are prescribed respectively on @xmath225 by @xmath235=\begin{bmatrix}-x&0&0\\0&x^4+x&x^2\\0&x^2&1\end{bmatrix},\ \left[g_2\left(e_s',e_t'\right)\right]=\begin{bmatrix}v^4+v&0&v^2\\0&-v&0\\v^2&0&1\end{bmatrix},\end{aligned}\ ] ] for all @xmath236 . 
   for the levi - civita connections @xmath237 with respect to metrics @xmath238 respectively , we obtain @xmath239 @xmath240 from above expressions and equation , we find @xmath241 , @xmath242 . 
   hence the @xmath203 and @xmath243 are para - saakian manifolds with invariant distributions @xmath244 and @xmath245 respectively . 
   let @xmath246 be a mapping defined by @xmath247 . 
   then @xmath143 , _ i_.e . , 
   @xmath11 is a @xmath53-paraholomorphic map between para - sasakian manifolds . for any @xmath248 and @xmath249 , 
   it is not hard to see that @xmath250 , @xmath251 and @xmath252 thus theorem [ main ] is verified .                                            : 
   harmonic maps in general relativity and quantum field theory , in : gauduchon , p. , editor , harmonic mappings , twistors , and @xmath253-models ( luminy , 1986 ) . 
   advanced series in math . 
   physics * 4 * , 270 - 305 , world scientific publishing , singapore ( 1988 ) .
summary: the purpose of this paper is to study the harmonicity of maps to or from para - sasakian manifolds . 
   we derive the condition for the tension field of paraholomorphic map between almost para - hermitian manifold and para - sasakian manifold . the necessary and sufficient condition for a paraholomorphic map between para - sasakian manifolds to be parapluriharmonic 
   are shown and a non - trivial example is presented for its illustrations . 
   dst / inspire/03/2014/001552 . ]