\documentstyle[amssymbo,tr_def,12pt]{article} \sloppy \renewcommand{\baselinestretch}{1.15} \topmargin 1.3cm \oddsidemargin 1cm \parindent 0.5cm \parskip 0.25cm \textwidth 13.6cm \textheight 17.8cm \newenvironment{mini1}{\begin{minipage}[t]{5.5cm}}{\end{minipage}} \newcommand{\be}{\begin{equation}} \newcommand{\re}{\rangle} \newcommand{\ee}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\db}{\displaystyle} \newcommand{\mb}{\mbox{\ }} \newcommand{\vs}{\vspace} \newcommand{\bc}{\begin{center}} \newcommand{\ec}{\end{center}} \newcommand{\ben}{\begin{enumerate}} \newcommand{\een}{\end{enumerate}} \newcommand{\bfi}{\begin{figure}} \newcommand{\efi}{\end{figure}} \newcommand{\al}{\alpha} \newcommand{\sg}{\Sigma} \newcommand{\bt}{\beta} \newcommand{\gm}{\gamma} \newcommand{\dt}{\delta} \newcommand{\ep}{\epsilon} \newcommand{\vr}{\varphi} \newcommand{\pr}{\partial} \newcommand{\lam}{\lambda} \newcommand{\iy}{\infty} \newcommand{\si}{\sigma} \newcommand{\la}{\leftarrow} \newcommand{\ra}{\rightarrow} \newcommand{\ovl}{\overline} \newcommand{\unl}{\underline} \newcommand{\btrd}{\bigtriangledown} \newcommand{\R}{\Bbb{R}} \newcommand{\N}{\Bbb{N}} \newcommand{\M}{\Bbb{M}} \newcommand{\C}{\Bbb{C}} \newcommand{\A}{\Bbb{A}} \renewcommand{\P}{\Bbb{P}} \newcommand{\E}{\Bbb{E}} \renewcommand{\L}{\Bbb{L}} \renewcommand{\S}{\Bbb{S}} \newcommand{\bq}{\begin{quote}} \newcommand{\eq}{\end{quote}} \newcommand{\bqu}{\begin{quotation}} \newcommand{\equ}{\end{quotation}} \newenvironment{emphit}{\begin{itemize}}{\end{itemize}} \newcommand{\bemp}{\begin{emphit}} \newcommand{\eemp}{\end{emphit}} %a,b,c'li denklemler icin % \newcounter{sayi} %\renewcommand{\theequation}{\Alph{sayi}.\arabic{equation}} %\newcommand{\bseq}{\begin{subequations} \begin{eqalignno}} %\newcommand{\eseq}{\end{eqalignno} \end{subequations}} %\begin{document} %\setcounter{sayi}{1} %1.1., 2.1.'li denklem %\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} %\newcommand{\bseq}{\begin{subequations} \begin{eqalignno}} %\newcommand{\eseq}{\end{eqalignno} \end{subequations}} %\begin{list}{}{\usecounter{enumi}\setlength{\leftmargin}{1.5cm} %\setlength{\labelwidth}{1.2cm}} %\item[ ] %\end{list} %\pagestyle{empty} \begin{document} \bc {\bf PARACOMPLEX PROJECTVE MODELS\\ AND\\ HARMONIC MAPS INTO THEM}\\ \mb\vs{0.1cm} Sadettin ERDEM \ec {\bf Introduction} Since the first papers written on paracomplex geometry (a geometry related to the ring of para-complex numbers) by Rashevskij ${\bf [R]}$, Libermann ${\bf [L]}$ and Patterson ${\bf [P]}$, many other mathematicans and physicists have devoloped interests into it and used it (see ${\bf [C-G-M],[C-F-G]}$ and the references therein). The paracomplex geometry has its own characteristics: It first plance one can note that $S^{2n},n=1,2,\cdots,$ does not admint an almost paracomplex structure. (Thus $S^4$ can have neither ain almost paracomplex nor an almost complex structures). On the other hand every (almost) para-Kaehler manifold has naturally defined (almost) symplectic structure, namely the para-Kaehler form. The physicist have mobilized this geometry in quite significant ways: The quantum field theory of gravitation based on a non-symmetric tensor such as a para-Hermitian form an developed by Kunstater, Moffat and Malzan ${\bf [K-M-M]}$ is known to have no inconsistancy problem concerning unitarity. Also Malzan, ${\bf [B]}$, has intereperet the paracomplex number $\varepsilon, (\varepsilon^2=1)$ as {\em fermion} number. Consequently this interpretation led to fundemental explanation of stability of fermionic matter. In ${\bf [L-K]}$, Lambert and Kibler used the paracomplex geometry to generate non-bijective canonical transformation. A part of this paper is devoted to descrebe a paracomplex projective model ${\P}_m({\A})$, a paracomplex analogue of complex projective space, as a differentiable manifold. And then two different extra structures are introduced so that $({\P}_m ({\A}), J,g)$ becomes a para-Kaehler space form and $({\P}_m ({\A}), P G^c)$ becomes and almost para-Keahler manifold. Some totally in particular, the real projective model $\ovl{({\P}_m ({\R})}, g)$ which is an analogue result of the fact that the real projective ${\P}_m ({\R})$ sits naturally in ${\P}_m ({\C})$ as a totally geodesic sulomanifold. The rest of the paper is devoted to desctibe and construct harmonic maps of manifolds with definite or neutral metrics into paracomplex projective models, example of a non-compact manifolds. One should note that harmonic maps of a manifold with neutral metric e.g Lorentz surfaces, are the solutions of a system of wave equations. While those of a Riemannian manifolds are the solutions of a system of elliptic ones. The physicists are calling the conformal harmonic maps of Lorentz surfaces strings. Harmonic maps between semi-Riemannian manifolds are connected to the hyperbolic systems. This connection is particularly significant in the areas of quantum field theory, solutions and scattering theory and general relativity, (see ${\bf [G]}$). This work also demonstraces all along how beautifull the paracomplex numbers fits to construct non-compact manifolds with neutral metrics and to attack problems of harmonic maps between then as the complex number do in the case of harmonic maps of Riemannian manifolds. {\bf 0 - Preliminaries} Let $\theta$ denote $\varepsilon$ with $\varepsilon^2=1$ or $i$ with $r^2=-1$ and then set \[ C_\theta=\{a+\theta b: a, b \in \Bbb{R}\} \] Clearly $C_i=\Bbb{C}$, the field of complex numbers. But $C_\varepsilon$, which we will denote by $\Bbb{A}$, forms a ring with zero divisors $k(1+ \varepsilon)$ and $k(1- \varepsilon), \;\; k \in \Bbb{R}$. The set $\Bbb{A}$ is called {\em paracomplex (or double or hyperbolic complex) numbers.} {\bf Notations:} We frequently use one of the followings where it is convenient. \begin{eqnarray*} Z=(z_k)&=& (z_1, \cdots, z_N)=(x_1+\theta y_1, \cdots, x_N+\theta y_N)=(x_k)+\theta (y_k)\\ &=& ((x_k),(y_k))=(x+\theta y) \in C^N_\theta \end{eqnarray*} $\ovl{Z}=(\ovl{z}_k)$ and $\ovl{z}=x-\theta y, \theta$-complex conjugate of $z$ For $Z=(z_k), \;\; W=(w_k)=(u_k)+\theta (v_k),$ $r=0,1, \cdots , (N-1);$ let $$\langle Z , W) \rangle^\theta_r= -\db\sum^r_{k=1} z_k \ovl{w}_k+ \db\sum^N_{k=r+1} z_k \ovl{w}_k$$ the symmetric $C_\theta-$ bilinear form and $ \langle Z , W\rangle ^{\ovl{\theta}}_r=-\db \sum^r_{k=1} z_k \ovl{w}_k+ \db \sum^N_{k=r+1} z_k \ovl{w}_k$ \noindent the $C_\theta-$ Hermitian form. They are both nondegenerate on $C^N_\theta$. For $\theta=i$ we write $\langle , \rangle^A_r$ and $\langle ,\rangle^{\ovl{A}}_r$ for symmetric $\Bbb{A}$-bilinear (or paracomplex bilinear) and para-Hermitian forms respectively. For $Z, W \in \Bbb{R}^N$ we write \[G_r (Z, W)= \langle Z, W \rangle_r=\langle Z, W\rangle^R_r=\langle Z, W\rangle^\theta=\langle Z, W\rangle ^{\ovl{\theta}}\] Note that then $G = G_0= \langle , \rangle _0$ is the standart inner product on $\Bbb{R}^N$ Let $V$ be a real vector space of dimension $n$ and $J_\theta: V \rightarrow V$ be an endomorphism with $J_\varepsilon^2=I$ and $J^2_i=-I$, where $I$ is the identity. $J_i$ is called complex structure on $V$. Let $k_1, k_2$ be the dimensions of the eigenspaces of $J_\varepsilon$ corresponding to eigenvalues $1,-1$ respectively. Then $J_\varepsilon$ is called {\em product structure} if $k_1, k_2 \geq 1.$ A {\em paracomple\ structure} on $V$ is a product structure with $k_1=k_2$. A (semi-) Riemannian metric $g$ on $V$ is said to be {\em Hermitian} (resp: {\em para-Hermitian}) if \[ g(X,Y)=mg (J_\theta X, J_\theta Y) \;\; X,Y \in V. \] where $m=1$ when $J_\theta=J_i$ (resp: $m=-1$ when $J_\theta=J_\varepsilon)$ Note that if $g$ is para-Hermitian then it has to be an indefinite metric of signature $(s,s)$ on $V$ in which case $g$ is called {\em neutral metric}, where $dim V=n=2s$. (Almost) paracomplex, (almost) para-Hermitian, (almost) para-Kaehler manifolds and paracomplex, paraholomorphic, para-Hermition vector bundles are defined in the usual way with the difference that the defining elements are replaced with the correspondig ``para" ones where necessary. Let $f: (M,g,J) \rightarrow (M',g',J')$ be a differentiable map between (almost) para-Hermitian manifolds. The second fundemental form $\btrd df$ of $f$ is given by \[ \btrd df (u,v)=D^M_{df (u)} (df (v))-df (D^{M'}_u v) \] where $D^M$ and $D^{M'}$ are the metric connections of the indicated manifolds. Also the tension field $\tau (f)$ of $f$ is given by \[ \tau(f)=\db \sum^n_{i,j=1} g^{ij} \btrd df (e_i,e_j) \in T M' \] where $g^{ij}$ is the $(i,j)$ th entry of the matrix $(g_{i,j})^{-1}$, with $g_{ij}=g(e_i,e_j)$ and $\{e_j,\cdots , e_m\}$ is a basis for TM. {\bf Definitions}: {\em $f$ is said to be} i) {\em paraholomorphic if \quad $df \circ J=J' \circ df$} ii) {\em harmonic if \quad $\tau (f)=0$} iii) {\em totally geodesic if \quad $\btrd df=0$} \noindent we call i) a paracomplex manifold of dimension two a {\em para-Riemann surface} ii) an oriented, paracompact, connected surface with a neutral metric (and therefore of signature (1,1)) a {\em Lorentz surface} iii) a free module over the ring $\Bbb{A}$ a {\em paracomplex vector space} On a surface $(M,g)$ with its metric $g$, local coordinates $(x,y)$ are said to be {\em $g-$ isothermal} if \[ g(\partial_x,\partial_x)=k g (\partial_y,\partial_y) \quad \mbox{and} \quad g (\partial_x,\partial_y)=0\] where $\partial_x= \db \frac{\partial}{\partial x}$ e.c.t and $k=1$ if $g$ is positive definite and $k=-1$ if it is indefinite. Note that every Lorentz surface $({\L}, g)$ possesses $g-$ isothermal coordinates, ${\bf [E]_2}$, {\bf [We]}. Thus one obtains that, ${\bf [E]_2}$, every Lorentz surface is a para-Riemann surface. The $\Bbb{A}^N$ is an obvious example of a paracomplex vector space with a canonical paracomplex structure $J,\;\;J(v)=\varepsilon v \in \Bbb{A}^N=\Bbb{R}^{2N}$ {\bf 1. Paracomplex Projective Models} We set $n=N-1, \;\; \langle ,\rangle \; =\; \langle ,\rangle ^{\ovl{A}}_0,\;\; g = Re (\langle , \rangle )$ which is the real part of $\langle , \rangle $ and \[ \Bbb{A}^N_0=\{Z \in \Bbb{A}^n: \langle Z,Z \rangle > 0\} \] Note that $g$ is the neutral metric (of signature $(N,N)$) on $\Bbb{R}^{2N}$ and $\langle Z , Z\rangle =g (Z,Z).$ Let $\Bbb{P}_n (\Bbb{A})$ denote the quotient of $\Bbb{A}^N_0$ under the equivalence relation: $Z \simeq W$ if $W=zZ$ for some $z \in \Bbb{A}_+$, where \[ \Bbb{A}_+=\{z=x+\varepsilon y \in \Bbb{A}: z \ovl{z} > 0, \quad x > 0\} \] Then the set $\Bbb{P}_n (\Bbb{A})$ may be endowed with a differentiable manifold structure ${\bf ([G-A]_1, [G-A]_2,[C-G-M])}$. We shall endow this differentiable manifold with two different metrics and paracomplex structures in order to obtain two different para-Hermitian manifolds: ${\bf A}: (c . f. {\bf [G-A]_1, [G-A]_2)}$ Let $\pi: \Bbb{A}^N_0 \rightarrow \Bbb{P}_n (\Bbb{A})$ be the natural projection. Then obviously $\pi$ is a submersion. Set \begin{eqnarray*} {\bf S}=S^{2N-1}_N &=& \{Z=((x_k), (y_k))=(x,y) \in \Bbb{A}^N_0: \langle Z , Z\rangle =1\}\\ &=& \{Z \in \Bbb{A}^N_0: \langle x , x\rangle - \langle y , y\rangle =g (z,z)=1\} \end{eqnarray*} and put $\tilde{\pi}=\pi \mid_S $, the restriction of $\pi$ to due pseudo-sphere ${\bf S} $ (whose metric induced from that on $\Bbb{R}^{2N}$ is of signature $(N, N-1))$. Note that \[ \pi:\Bbb{A}^N_0 \rightarrow \Bbb{P}_n (\Bbb{A}) \quad \mbox{and} \quad \tilde{\pi}: {\bf S} \rightarrow \Bbb{P}_n (\Bbb{A}) \] are principal bundles with structure groups $\Bbb{A}_+$ and \[ {\bf R_+}=\{\frac{1}{2}[(m+\frac{1}{m})+ \epsilon(m-\frac{1}{m})]:m \in \Bbb{R},\;\; m > 0\} \subseteq \Bbb{A}_+ \] respectively. Consider the tangent space \[ T_z=T_z {\bf S} =\{W \in \Bbb{A}^N: g (Z,W)=0\} \] and a subspace $\tilde{T}_Z$ of $T_Z$, \[ \tilde{T}_Z=\{W \in \Bbb{A}^N: \langle Z,W \rangle =0\} \] which is the $g-$ orthogonal complement of $\varepsilon Z$ in $T_Z$. The projection $\pi$ induces a linear isomorphism \[ \tilde{\pi}_\ast: \tilde{T}_Z \rightarrow T_p (\Bbb{P} (\Bbb{A})) \quad \mbox{with}\quad p=\tilde{\pi} (Z)\] The indefinite metric on ${\bf S}$ is invariant by the structure group ${\bf R_+}$ and therefore it defines a metric on $\Bbb{P}_n (\Bbb{A})$ of signature $(n,n)-$ which will be denoted by the same letter $g-$ in the standart way so that $\tilde{\pi}$ becomes a Riemannian submersion. On the other hand the canonical paracomplex structure $J'=\varepsilon$ on $\Bbb{A}^N$ leaves $\tilde{T}_Z$ invariant and satisfies: \[ J'(z W)=z J' (W) \quad \mbox{for all} \quad W \in \tilde{T}_Z,\;\; z \in {\bf R_+}\] Thus $J'$ induces an almost paracomplex structure $J$ on $\Bbb{P}_n(\Bbb{A})$: \[ J(X)=J'(X'), \quad \mbox{where} \quad X=\tilde{\pi}_\ast (X') \] The metric $g$ and the paracomplex structure $J$ just described on $\Bbb{P}_n(\Bbb{A})$ coincides with the ones described in $e. g. {\bf [G-A]_1}$, thus $(\Bbb{P}_n (\Bbb{A}), g, J)$ becomes a para-Kaehler manifold of constant (nonzero) paraholomorphic sectional curvature. So it is a para-Kaehler space form. The letters $g$ and $J$ will be reserved to denote the metric and the paracomplex structure on $\Bbb{P}_n (\Bbb{A})$ respectively obtained in the way just described. The para-Kaehler manifold $(\Bbb{P}_n (\Bbb{A}), g , J)$ has the following properties: ${\bf ([G-A]_1, [C-G-M],[G-A]_2)}$ It is i) complete since $({\bf S} , g)$ is complete and $\tilde{\pi}$ is totally geodesic ii) simply connected for $n > 1$ and connected for $n=1$ iii) a symmetric and homogeneous space. {\bf B:} We shall now put other structures on $\Bbb{P}_n (\Bbb{A})$: Let $(M,\beta)$ be a (semi-) Riemannian manifold of dimension $m$ and the signature of its metric $\beta$ be $(k, m-k),\;\; k=1,2,\cdots,m$. Consider the tangent bundle \[ \Pi:TM \rightarrow M \] Local coordinates $(U,x)$ on $M$ give rise local coordinates $(\Pi^{-1} (U),x,y)$ on $TM$; $i=1,2,\cdots,m; \;\;x=(x_i),\;\;y=(y_i)$. The subbundles of $T(TM)$ \[ V(TM)=Ker (d \Pi)=Span \{\partial_{yi}: i=1,\cdots,m\} \] and \[ H(TM)=span \{\varrho_{x_i}: i=1,\cdots,n\} \] are called {\em vertical} and {\em horizontal} vector bundles respectively. Here $\partial_{yi}=\db \frac{\partial}{\partial y_i}$ and \[ \varrho_{x_i} (x,y)=\partial_{x_i} (x,y)-\db \sum^n_{k=1} \biggl( \db \sum^n_{j=1} y_i \Gamma^k_{ij} (x)\biggr) \partial_{y_k} (x,y) \in T_{(x,y)} (TM) \] and \[ y=y(x)=\db \sum y_i \partial_{x_i} (x) \in T_x M \] where $\Gamma^k_{ij}$'s are the Christofell symbols of the metric $\beta$. One sees that \[ T(TM)=V (TM) \oplus H(TM) \] The vertical and horizontal lifts of a vector field $\quad Y=\db \sum Y_i \partial_{X_i} \in C(TM)$ \quad given by \[ Y^V (x,y)=\db \sum Y_i (x) \partial_{y_i} (x,y) \] \[ Y^H (x,y)=\db \sum Y_i (x) \varrho_{x_i} (x,y) \] respectively. Now we define on the manifold $TM$: i) a neutral metric $\beta^c$ (i.e. a metric of signature $(m,m)$) by \[ \left \{ \begin{array}{l} \beta^c (X^V,Y^V)=\beta^c (X^H,Y^H)=0\\ \mbox{and}\\ \beta^c (X^H,Y^V)=\beta^c (X^V,Y^H)=\beta (X,Y) \quad \end{array} \cdots \cdots(1.1) \right. \] The neutral metric $\beta^c$ will be called the {\em complete lift} of $\beta, \quad {\bf ([Y-I], \;\;[B-B])}$. ii) an almost paracomplex structure $P$ by $$ P(X^V)=X^V, \quad P(X^H)=-X^H \quad \cdots \cdots \eqno{(1.2)} $$ where $X,Y$ are vector fields on $M$. Then $(TM,\beta^c,P)$ becomes an almost para-Hermitian manifold, ${\bf [B-B],[0]}$. A {\em regular Lagrangean} on $M$ is a smooth real valued function $L: TM \ra {\R}$ such that the matrix with entries \[ \gamma_{ij}= \frac{\partial^2 L}{\partial y_i \partial y_j} \] is everywhere nondegenerate, where $((x_i),(y_i))$ is a local coordinate chart on $TM$ induced by a local coordinate chart $(x_i)$ on $M$. A{\em Lagrange manifold} is a smooth manifold having a regular Lanrangean. Note that for a Riemannian manifold $(M^n, h)$, a function $L:TM \ra {\R}$ defined by \[ L(x,y)=\sum^n_{i,j=1} g_{ij}(x) y_iy_j, \qquad y=(y_i) \] is a regular Lagrangean on $M^n$. So every Riemannian manifold, in particular the unit sphere, is a Lagrange manifold. Let us now take $M={\S}^n$, the unit sphere in $(\Bbb{R}^N,G)$ with its standart Riemannian metric which will be denoted by the same letter $G$. Viewing the tangent bundle $T {\S}^n$ as a set \[ T {\S}^n= \{ (x,y) \in \Bbb{R}^N \times \Bbb{R}^N: G(x,x)=1, \;\; G(x,y)=0\} \] we define a map \[ \varphi: T{\S}^n \stackrel{\psi}{\rightarrow} {\bf S} \stackrel{\tilde{\pi}} {\rightarrow} \Bbb{P}_n (\Bbb{A}) \] where \begin{eqnarray*} \psi (x,y) &=& \frac{1}{2} (2x+y, \; y) \cong \frac{1}{2} \biggl((2x+y)+\varepsilon y \biggr) \\ &=& \frac{1}{2} \biggl( (2x_1+y_1)+\varepsilon y_1, (2x_2+y_2)+\varepsilon y_2, \cdots, (2x_N+y_N)+ \varepsilon y_N \biggr) \end{eqnarray*} Note that $\psi (x,y) \in {\bf S}= S^{2N-1}_N \subseteq \Bbb{R}^{2N}$ since $g (\psi (x,y), \psi (x,y))=1$ $\quad$ (Recall that $g=Re(\langle, \rangle^{\ovl{A}}_0)$ which is a neutral metric on $\Bbb{R}^{2N}). $ It is not difficult to see that $(c.f. {\bf [G-A]_1})$ the map $\varphi=\psi \circ \tilde {\pi}$ is a diffeomorphism of $T {\S}^n$ onto ${\P}_n ({\A})$. Thus ${\P}_n ({\A})$ and $T {\S}^n$ are diffeomorphic as differentiable manifolds. We now give $\Bbb{P}_n (\Bbb{A})$ the metric $G^c$ and paracomplex structure $P$ on $T {\S}^n$ described by ${\bf (1.1)}$ and ${\bf (1.2)}$ via the diffeomorphism $\varphi$ so that $(\Bbb{P}_n( \Bbb{A}), G^c,P)$ becomes an almost para-Hermitian manifold. Since ${\S}^n$ is a Lagrange manifold, ${\P}_n({\A}), G^c,P)$ is in fact almost para-Kaehler, ${bf [0]}$. The letters $G^c$ and $P$ will be reserved to denote the metric and almost paracomplex structure on $\Bbb{P}_n (\Bbb{A})$ respectively obtained in the way just described. This manifold $(\Bbb{P} (\Bbb{A}), G^c, P)$ has the following properties: i) It is locally conformally flat, locally symmetric space since ${\S}^n$ is symmetric, connected space of constant curvature, ${\bf [C-G-V]}$. ii) Unlike the $(\Bbb{P}_n (\Bbb{A}), g, J) \quad $, this is not a space of constant curvature since ${\S}^n$ is not locally flat. Nevertheless it has a vanishing scalar curvature, ${\bf [C-G-V]}$. Furthermore, ${\bf [G-K]}$, it is null isotropic, that is, for every null vector $u \in T_p=T_p (\Bbb{P}_n (\Bbb{A}))$ the function \[ c_u=T_p \rightarrow \Bbb{R}, \quad c_u (X)=\frac{R(X,u,u,X)}{G^c(X,X)} \] is constant for all non-null $X$ orthogonal to $u$. Where $R$ is the Riemannian curvature tensor field. Infact the null isotropy is equivalent to , when $n \geq 2$, a$^\circ ) \quad R(u,v,u,v)=0$ for every orthogonal pair of null vectors $u,v$ b$^\circ ) \quad$ locally conformally flatness {\bf C: Some totally geodesic submanifolds of $(\Bbb{P}_n (\Bbb{A}),g)$} For the pseudo-sphere \[ S^n_r=\{u=(u_1, \cdots,u_N) \in \Bbb{R}^N: G_r (u,u)=1\} \] \[ G_r(u,u)=- \db \sum^r_{i=1} u^2_i + \db \sum^N_{i=r+1} u^2_i \] consider the map $$ \ovl{\varphi}_r: S^n_r \stackrel{\ovl{\psi}_{N-r}}{\rightarrow} {\bf S}=S^{2N-1}_n \stackrel{\tilde{\pi}}{\rightarrow} \Bbb{P} (\Bbb{A})\quad \cdots \cdots \eqno{(1.3)} $$ given by \begin{eqnarray*} \ovl{\psi}_t (u) &=& (u_1,\cdots,u_t,0,\cdots,0; 0,\cdots, 0, \; \; u_{t+1},\cdots,u_N) \in {\bf S}\\ &\cong &(u_1, \cdots,u_t, \varepsilon u_{t+1}, \cdots, \varepsilon u_N) \in \Bbb{A}^N \end{eqnarray*} One can show that, $(c.f.{\bf [G-A]_1},$ thm 5.3) for each $r=0,1,\cdots,n$ the map $\ovl{\varphi}_r$ is an isometric imbedding and $\ovl{\varphi}_r (S^n_r)$ is a totally geodesic submanifold of $(\Bbb{P}_n ({\A}),g)$. In particular, when $r=0$, we have \[ \ovl{\varphi}=\ovl{\varphi}_0: {\S}^n=S^n_0 \stackrel{\ovl{\psi}_N}{\rightarrow} {\bf S} \stackrel{\tilde{\pi}}{\rightarrow} {\Bbb{P}}_n ({\Bbb{A}}) \] with \[ \ovl{\varphi} (u)=\tilde{\pi} (u,\ovl{0}) \cong span_{\Bbb{A}_+} \{(u,\ovl{0})\} \] which may be identified with $span_{\Bbb{R}_+}\{u,\ovl{0}\}$ where $(u,\ovl{0}) \in \Bbb{R}^N \times \Bbb{R}^N \cong \Bbb{A}^N, \quad \ovl{0}=(0, \cdots, 0) \in \Bbb{R}^N$. Thus $\ovl{\varphi} (\Bbb{S}^n)$ constitutes the set of all real rays in $\Bbb{R}^N$ which will be denoted by $\ovl{\Bbb{P}_n} (\Bbb{R})$ and called the {\em real projective model}. Obviously the $\ovl{\Bbb{P}}_n (\Bbb{R})$ is a double covering of the real projective space $\Bbb{P}_n (\Bbb{R})$. It is also a totally geodesic submanifold of $(\Bbb{P}_n (\Bbb{A}),g)$. This is an analogue result of the fact that the real projective space $\Bbb{P}_n (\Bbb{R})$ is a totally geodesic submanifold of the complex projective space $\Bbb{P}_n (\Bbb{C})$ with their standart metrics. {\bf The star condition} On an almost para-Hermitian manifold $(M,h,Q,\btrd)$ with its indicated metric, paracomplex structure and the Levi-Civita connection we set, ${\bf [B-B]:}$ \[ {\bf (\star)} \quad \cdots \cdots \quad \left \{ \begin{array}{ll} (\btrd_X Q) \Gamma (T^+ M)=0 & \forall X \in \Gamma (T^- M)\\ \mbox{and}&\\ (\btrd_Y Q) \Gamma (T^-M)=0 & \forall Y \in \Gamma (T^+ M) \end{array} \right. \] where \[ T^\pm M=\{X \in TM: Q (X)=\pm X \} \] which is called the {\em eigenbundles with respect to $Q$ } over $M$ {\bf Lemma (1.1)}: ${\bf [B-B]}$, Let $(M,h)$ be a (semi-) Riemannian manifold. Then $(TM, h^c,P)$ satisfies the {\bf $(\star)$} condition. Also any para-Kaehler manifold, in particular, the $(\Bbb{P}_n(\Bbb{A}),g,J)$ satisfies the {\bf $(\star)$}. Where $h^c$ and $P$ are the structures given by {\bf (1.1)} and {\bf (1.2)} respectively. {\bf Remark}: In ${\bf [B-B]}$, the manifold $(M,h)$ in the above lemma is restricted to be Riemannian. Infact t may be allowed to be semi-Riemannian. {\bf Corollary (1.2):} The almost para-Kaehler manifold $(T {\S}^n, G^c,P) \cong (\Bbb{P}_n (\Bbb{A}),G^c,P)$ satisfies the condition ${(\bf \star)}$. {\bf 2. Construction of harmonic maps into $\Bbb{P}_n (\Bbb{A})$} Let $f: (M,h) \rightarrow (M_1,h_1)$ be a smooth map between (semi-) Riemannian manifolds of any signature. We have {\bf Lemma (2.1):} ${\bf [B-B]}$ i) If $(M,h,Q),(M_1,h_1,Q_1)$ are (almost) para-Hermitian manifolds satisfying the condition ${\bf (\star)}$ and $f$ is paraholomorphic then $f$ is harmonic. ii) The following statements are equivalent $a^\circ) \quad df=F:(TM, h^c,P') \rightarrow (TM_1,h^c_1,P_1)$ is paraholomorphic $b^\circ) \quad F$ is totally geodesic $c^\circ) \quad f$ is totally geodesic where $P',P_1$ are the respective almost paracomplex structures described by {\bf (1.2)} {\bf Remark:} In ${\bf [B-B]}$, these statements are given for the Riemannian manifolds $(M,h)$ and $(M_1,h_1)$. However they are also true when these manifolds are semi-Riemannian since the formulas and the arguments used are still valid in semi-Riemannian cases too as one can see by combining the results ${\bf ([V]}$,pgs 134-135), (${\bf [0]}$,pg 247), (${\bf [Y-I]}$, 7.1,7.3 pg 106; proposition 6.3, pg 105; 4.8,pg 100),(${\bf [Y-K]}$,proposition 7.5) {\bf Lemma (2.2):} ${\bf [V]}$, The map $f:(M,h) \rightarrow (M_1,h_1)$ between the semi-Riemannian manifolds is harmonic if and only if $df:(TM,h^c) \rightarrow (TM_1,h^c_1)$ is harmonic. {\bf Proposition (2.3)} Let $\phi_r:(M,h) \rightarrow (S^n_r,G_r)$ be a harmonic map of a (semi-) Riemannian manifold $M$ of any signature into the unit pseudo-sphere with its standart metric $G_r$ (which is of index $(r,n-r))$ and $r=0,1,\cdots,n$. Then we have i) $\phi_r$ is harmonic as a map into $(\Bbb{P}_n (\Bbb{A}),g)$ via $\ovl{\varphi}_r (S^n_r) \subseteq \Bbb{P}_n (\Bbb{A})$. ii) The map \[ d \phi_0=\Phi: (TM,h^c,P') \rightarrow (\Bbb{P}_n (\Bbb{A}),G^c,P) \] is harmonic, Moreover, $\phi$ is totally geodesic if and only if $\Phi$ is totally geodesic if and only if $\Phi$ is paraholomorphic. {\bf Proof:} By combining the lemmas (2.1) and (2.2), the result will follow. \hfill $\Box$ {\bf Corollary (2.4):} For $(M,h)$ with $\dim M=1$ and $r=0,1,\cdots,n$ we have i) $\phi_r: (M,h) \rightarrow (S^n_r, G_r)$ is harmonic (and therefore it is totally geodesic) if and only if $\ovl{\varphi}_r \circ \phi_r: (M,h) \rightarrow (\Bbb{P}_n (\Bbb{A}),g)$ is totally geodesic ii) $\phi=\phi_\circ: (M,h) \rightarrow ({\S}^n,G)$ is harmonic if and only if $\Phi$ is totally geodesic if and only if $\Phi$ is paraholomorphic. \hfill $\Box$ Note that for $M={\S}^1$, we have $(T {\S}^1, h^c, P') \cong (\Bbb{P}_1 (\Bbb{A}),h^c,P')$ which is locally diffeomorphic to $S^2_1=\{(x_1,x_2,x_3) \in \Bbb{R}^3:-x^2_1+x^2_2+x^2_3=1\} \cong {\S}^1 \times \Bbb{R}$. So any harmonic (and therefore totally geodesic) map $\phi: ({\S}^1,h) \rightarrow ({\S}^n,G)$ gives rise to a totally geodesic and therefore a paraholomorphic map $\Phi: (\Bbb{P}_1 (\Bbb{A}), h^c,P') \rightarrow ( \Bbb{P}_n (\Bbb{A}), G^c,P)$. {\bf Proposition (2.5):} Any paraholomorphic map $\psi: (TM, h^c,P') \rightarrow (\Bbb{P}_n (\Bbb{A}), G^c,P)$ is harmonic. Where $(M,h)$ is a (semi-) Riemannian manifold with $h$ is of any signature. {\bf Proof:} It is immediate by the lemma (2.1). \hfill $\Box$ {\bf Notation:} Set for $z=x+\varepsilon y \in \Bbb{A}$ \[ \delta '=\delta _z=\frac{1}{2} (\partial_x+ \varepsilon \partial_y), \quad \delta ''=\delta _{\ovl z}=\frac{1}{2} (\partial_x-\varepsilon \partial_y) \] and for $z=x+iy \in \Bbb{C}$ \[ \partial'=\partial_z=\frac{1}{2} (\partial_x-i \partial_y), \quad \partial''=\partial_{\ovl{z}}=\frac{1}{2} (\partial_x+i \partial_y) \] where $\partial_x=\frac{\partial}{\partial x}$ ect. Also we reserve the letters ${\L}=({\L},h,Q)$ for a Lorentz surface and ${\M}=({\M},h',Q')$ for a Riemann surface. Clearly $h$ is an indefinite metric and $Q$ is a paracomplex structure on ${\L}$ while $h'$ is positive definite and $Q'$ is a complex structure on ${\M}$. Four $\theta \in \{i,\varepsilon\}$ let $\xi: M_\theta \ra C^{2n+1}_\theta$ be a smooth map, where $M_i={\M}$ and $M_\varepsilon={\L}$. {\bf Definition (2.6):} {\em For $r=0,1,\cdots,2n;$ the map $\xi$ is said to be i) totally $\langle, \rangle^A_r$- paraisotropic if \[ \langle \delta^{'k} \xi, \delta^{'k} \xi \rangle^A_r=0 \quad \mbox{for} \quad 0 \leq k \leq n-1 \] ii) totally $\langle,\rangle^c_r$- isotropic if \[ \langle \partial^{'k} \xi, \partial^{'k} \xi \rangle>^C_r=0 \quad \mbox{for} \quad 0 \leq k \leq n-1 \]} The total $\langle, \rangle^{\ovl{A}}_r$- paraistoropy and $\langle, \rangle^{\ovl{c}}_r$ - isotropy are defined in the same way. {\em We say that a smooth map $f: M_\theta \ra {\P}_{2n} (C_\theta)$ is totally $\langle,\rangle^{C_\theta}_r$-isotropic (resp: $\langle,\rangle^{\ovl{C}_\theta}_r$-isotropic) if $f$ has a local lift $\xi: U \subseteq M_\theta \ra C^{2n+1}_\theta$ which is $\langle,\rangle^{C_\theta}_r$-isotropic (resp: $\langle,\rangle^{\ovl{C}_\theta}_r$-isotropic).} By the same argument that used for the complex case ${\bf [W]}$, one can show that smooth (resp: paraholomorphic) map $f: {\L} \ra {\P}_{2n} ({\A}$ has a smooth (resp: paraholomorphic) local lift $\xi: U \subseteq {\L} \ra {\A}^{2n+1}$ over a small enough open set $U$ in ${\L}$. It is not difficult to see that the concept of isotropy, in all cases, is well defined i.e. it does not depend on the local lift and $h$- isothermal coordinates choosen. {\bf Remark:} Four $r=0$ there is no maps $xi: M_\theta \ra C^{2n+1}_\theta$ with $\langle, \rangle^{\ovl{C}}_0$- isotropy or $\langle, \rangle^{A}_0$-paraisotropy and $f: {\L} \ra {\P}_{2n} ({\A})$ with $\langle, \rangle^{\ovl{A}}_0$-paraisotropy. Note that for a para-Riemann surface (resp: Riemann surface) $E$, a local lift $\xi: U( \subseteq E) \rightarrow \Bbb{A}^N$ (resp: $\xi: (U( \subseteq E) \rightarrow \Bbb{C}^N)$ is paraholomorphic (resp: holomorphic) if $\delta^{''} \xi=0$ (resp: $\partial^{''} \xi=0$). Now for a fixed value of $r=0,1, \cdots, 2n$ suppose $f: ({\L},h,Q) \ra ({\P}_{2n} ({\A}),g,J)$ is a totally $\langle,\rangle^{\ovl{A}}_r$- paraisotropic (resp: totally $\langle, \re^A_r$- paraisotropic ) paraholomorphic map such that for every $p \in {\L}$ there is a local paraholomorphic lift $\xi$ and $h$- isothermal coordinates $(x,y)$ on an open set $U$ containin $p$ on which the set $V=V (\xi)=\{\xi, \delta ' \xi, \cdots, \delta^{'n-1} \xi \}$ is linearly independent and the symmetric paracomplex form $\langle, \rangle^A_r$ (resp: the para-Hermitian form $\langle, \rangle^{\ovl{A}}_r)$ is nondegenerate on $Span_{\Bbb{A}} V \subseteq \Bbb{A}^{2n+1};$. For \mb \hspace{2.0cm}$ \sigma (p)=(\xi \wedge \delta ' \xi \wedge \cdots \wedge \delta ^{' n-1} \xi) (p) \in \bigwedge^{n} \Bbb{A}^{2n+1}$ the ${\em n}$ th exterior power of ${\A}^{2n+1}$, set \[ \alpha (p)=\sigma (p) \wedge \ovl{\sigma (p)}=(\xi \wedge \cdots \wedge \delta ^{'n-1} \xi \wedge \ovl{\xi} \wedge \cdots \wedge \delta ^{''n-1} \ovl{\xi}) (p)\] and $\tilde{\alpha}=\sim \circ \alpha$ where \[ \sim (\db \sum^{2n+1}_{k=1} \lambda_k (\db \bigwedge_{j \in \Omega_k} e_j))=\db \sum^{2n+1}_{k=1} (-1)^k \lambda_k e_k \] and $\{e_j\}$ is the standart basis for ${\R}^{2n+1} \subseteq {\A}^{2n+1}$ and $\Omega_k=\{1,2,\cdots,\\ (2n+1)\} \setminus \{k\}$ {\bf Lemma (2.6)}: The map $\tilde{\alpha}: U \subseteq {\L} \ra {\A}^{2n+1}$ never vanishes i.e\\ $\tilde{\alpha} (q) \neq 0, \qquad \forall q \in U$. {\bf Proof:} For $V=V (\xi)=\{\xi, \cdots, \delta ^{'n-1} \xi\}$, note that the set $V \cup \ovl{V}$ forms a linearly independent set since $V \cap \ovl{V}=\{0\}$. Indeed, let $\;$ $v$ $ \in V \cap \ovl{V}$$\;$ then $\langle v, \ovl{w} \rangle^{\ovl{A}}_r= \langle v,w \rangle^A_r=0,\; \forall w \in \ovl{V}$ by the $\langle, \rangle^{\ovl{A}}_r$- paraisotropy. But then the nondegeneracy of $\langle, \rangle^A_r$ on $V$ gives that $V=0$. (the arguments for the case where $\langle, \rangle^{\ovl{A}}_r$- paraisotropy and $\langle, \rangle^A_r$- nondegeneracy is replaced by the $\langle, \rangle^A_r$- paraisotropy and $\langle, \rangle^{\ovl{A}}_r$- nondegenaracy is the same). So $span_{\A} (V (q) \cup \ovl{V(q)})$ is a paracomplex subspace (i.e a free submodule) of ${\A}^{2n+1}$. By the theorem 7, in ${\bf ([H-K]}$,pg 171) we deduce that $\alpha (q) \neq 0$,$\;$ and therefore $\tilde{\alpha} (q) \neq 0, \qquad \forall q \in U$. \hfill $\Box$ Observe that i) $\ovl{\tilde{\alpha}}=(-1)^n \tilde{\alpha}$, so $\tilde{\alpha}$ is pure imaginary for an odd $n$ and real for an even $n$. Thus we write $\tilde{\alpha}$ for $\varepsilon \tilde{\alpha}$ when $n$ is even so that $\tilde{\alpha}$ denotes the real section of $\unl{{\A}}^{2n+1}=U \times A^{2n+1}$ for even and odd $n$'s ii) $\langle \tilde{\alpha}, \delta ^{'k} \xi \rangle^{\ovl{A}}_0=0= \langle \tilde{\alpha}, \ovl{\delta ^{'k} \xi} \rangle^{\ovl{A}}_0, \qquad \forall k=0, \cdots, (n-1)$. Put $\;$ $F=\db \frac{\tilde{\alpha}}{\mu}$ $\;$ at points where $\;$ $\mu=| \langle \tilde{\alpha}, \tilde{\alpha} \rangle^{\ovl{A}}_r |^{1/2}\;$ $\neq 0.$ $\;$ Then {\bf Lemma (2.7):} The map $F$ does not depend on the particular choices of a local lift and $h$- isothermal coordinates. {\bf Proof:} Let $\;$ $z=x+\varepsilon y \cong (x,y) $ $\;$ and $\;$ $w=u+ \varepsilon v$ $\;$be two local $h$- isothermal coordinates on $U \subseteq {\L}$. Then \[ \delta ^k_w \xi=(\delta ^k_z \xi) (\delta _w z)^k+ \quad \mbox{terms in } \quad \delta ^s_z \xi \quad \mbox{with} \quad s0$. This gives $\;\;$ $x^2_u-x^2_v >0\;$ which proves the claim. By the claim we see that $f^w=\db \frac{\tilde{\alpha} (w)}{\mu_w}= \frac{\tilde{\alpha}(z)}{\mu_z}$. $\;$ Next, let $\xi$ and $\eta$ be two \noindent local paraholomorphic lifts of $f$, then $\xi=\lambda \eta$ for some $\lambda: {\L} \ra {\A}$ with $\;$ $\lambda \ovl{\lambda}>0$. $\;$ So $\;$ $\alpha_\xi=(\lambda \ovl{\lambda})^n \alpha_\eta$ $\;$ and consequently $F_\xi=F_\eta$ \hfill $\Box$ Now observe that the paracomplex subspace $W=Span_{\A} (V \cup \ovl{V})$ has a basis $\{w_1, \cdots, w_{2n}\}$ whose elements are all real since $\ovl{W}=W$. Set\\ $W^R=span_{\R} \{w_1, \cdots, w_{2n}\}$ and let $\langle, \rangle_r$ denote the metric which is the restriction of $\langle, \rangle^A_r$ (or equivalently $\langle, \rangle^{\ovl{A}}_r)$ on ${\R}^{2n+1} \subseteq {\A}^{2n+1})$. Further suppose that $f$ is as such so that $\langle, \rangle_r$ is of signature $(r,2n-r)$ on $W^R$. As this signature does not depend on the particular choices of charts and local lifts, one can see that $\; \mu \neq 0$ $\;$ on the whole of ${\L}$ and therefore $\;$ $F=\frac{\tilde{\alpha}}{\mu}$ defines a map \[F: {\L} \ra S^{2n}_r \subseteq {\R}^{2n+1}.\] from the whole of the Lorentz surface into a pseudo-sphere \[ S^{2n}_r=\{(x_k) \in {\R}^{2n+1}: -\db \sum^r_{k=1} x^2_k+ \db \sum^{2n+1}_{k=r+1} x^2_k=1\} \] Define an endomorphism $\rho_r: {\A}^N \ra {\A}^N$ by: \[ \rho(z)=\rho_r (z)=(-z_1, \cdots, -z_r,z_{r+1}, \cdots,z_N)\] for $Z=(z_k) \in {\A}^N$. Then we have, for $r=0,1, \cdots,(N-1)$ i) $\langle Z,W \rangle^A_r=\langle \rho_r (Z), w \rangle^A_0= \langle Z,\rho_r (W)\rangle^A_0$ ii) $\langle \rho_r (Z), \rho_r (W) \rangle^A_r=\langle Z, W \rangle^A_r$ iii) For a smooth map $f: M_\theta \ra {\C}_\theta^N, \quad \theta=i,\varepsilon$ \[ \rho_r (\partial^{'k} f)= \partial^{'k} \rho_r (f) \quad \mbox{and} \quad \rho_r (\delta ^{'k} f)= \delta ^{'k} (\rho_r (f))\] where $M_i={\M} \quad \mbox{and} \quad M_\varepsilon={\L}$ These relations are valid also for the operators $\delta ^{''k}$ and $\partial^{''k}$ Note that $\;$ $\rho_r (F) \not\in W=span_{{\A}} (V \cup \ovl{V})$ $\;$ so one can write $\;$ $\delta ^{'n} \xi=v+w+s \rho_r (F)$ $\;$ where $\;$ $v \in V$ $\;$ and $\;$ $w \in \ovl{V}$. $\;$ We have \[ \langle \delta ^{'n} \xi, \delta ^{'k} \xi \rangle^{\ovl{A}}_r= \langle v, \delta ^{'k} \xi \rangle^{\ovl{A}}_r+ \langle w, \delta ^{'k} \xi \rangle^{\ovl{A}}_r+s \langle \rho_r(F), \delta ^{'k} \xi \rangle^{\ovl{A}}_r \] Since $\;$ $\langle v, \delta ^{'k} \xi \rangle^{\ovl{A}}_r= \langle \delta ^{'n} \xi, \delta ^{'k} \xi \rangle^{\ovl{A}}_r=0; \; \quad \forall k=0, \cdots, (n-1)\;$ by the $\;\langle, \rangle^{\ovl{A}}_r$- paraisotropy and $\;$ $\langle \rho_r (F), \delta ^{'k} \xi \rangle^{\ovl{A}}_r=\langle F,\delta ^{'k} \xi \rangle^{\ovl{A}}_0=0$ we have $\langle w, \delta ^{'k} \xi \rangle^{\ovl{A}}_r= \langle w, \delta ^{''k} \ovl{\xi} \rangle^A_r=0$. But then from the nondegeneracy of $\langle, \rangle^A_r$ on $V$ and therefore on $\ovl{V}$, this gives that $\;$ $w=0$ $\;$ so that $$ \delta ^{'n} \xi=v+ \langle \delta ^{'n} \xi, \rho_r (F) \rangle^A_r \rho_r (F) \quad \cdots \cdots \eqno{(2.1)} $$ {\bf Remark:} For the case where $\langle, \rangle^A_r$- paraisotropy and $\langle, \rangle^{\ovl{A}}_r$ nondegenarcy involves instead, we still get the same formula ${\bf (2.1)}$ As we sum up: Let $f:({\L},h,Q) \ra ({\P}_{2n} ({\A}), g,J)$ be a paraholomorphic totally $\langle, \rangle^{\ovl{A}}_r$- paraisotropic (resp: totally $\langle, \rangle^A_r$- paraisotropic) map such that for every $p \in {\L}$ there is a local paraholomorphic lift $\xi$ and $h$- isothermal coordinates $(x,y)$ on an open set $U$ containing $p$ on which the set $V=V (\xi)=\{\xi, \cdots,\delta ^{'n-1} \xi\}$ is linearly independent and the symmetric paracomplex form $\langle, \rangle^A_r$ (resp: para-Hermitian form $\langle, \rangle^{\ovl{A}}_r)$ is nondegenerate on $span_{\A} V \subseteq {\A}^{2n+1}$; and is of signature $(r,2n-r)$ on $W^R \subseteq W=span_{\Bbb{A}}(V \cup \ovl{V})$. {\bf Remark:} The symmetric paracomplex form $\langle, \rangle^A_0$ is always nondegenarate on any subspace of $\Bbb{A}^{2n+1}$ and it is always of singnature $(0,2n)$ on $W^R$. Therefore we drop the assumptions of nondegenerancy of $\langle, \rangle^A_0$ on $span_{{\A}} V$ and the signature of being $(0,2n)$. The map $f$ determines the maps. \[ F:({\L},h,Q) \ra S^{2n}_r \subseteq {\R}^{2n+1} \] into the pseudo-sphere with its standart semi-Riemannian metric of signature $(r,2n-r)$ and \[ K:U \subseteq ({\L}, h,Q) \ra {\A} \] given by \[ K=\langle \delta^{'n} \xi, F \rangle^A_0= \langle \delta^{'n} \xi, \rho_r (F) \rangle^A_r \] \[ =det [\xi, \delta' \xi,\cdots,\delta^{'n} \xi,\ovl{\xi}, \cdots, \ovl {\delta^{'(n-1)} \xi}] \] the determinant of a matrix whose columns formed by the indicated elements $\delta^{'k} \xi$ and $\ovl{\delta^{'k} \xi}$ of ${\A}^{2n+1}$. Note that unlike $F$, the map $K$ is defined locally and depends on the local lift $\xi$ and $h$-isothermal coordinates choosen. Nevertheless its paraholomorphicity is global i.e. if it is paraholomorphic with respect to particular local lift and $h$- isothermal coordinates then it is paraholomorphic with respect to any other choices too. {\bf Proposition (2.8):} Let $f:({\L},h,Q) \ra ({\P}_{2n} ({\A}), g,J),$ $F$ and $K$ be the maps described as above. Then $F$ is harmonic if $K$ is paraholomorphic. {\bf Proof:} One can show that $F$ is harmonic if and only if $\delta' \delta{''} F=\lambda F$ for some smooth map $\lambda: {\L} \ra {\R}. (c.f: {\bf [E]_1,[E]_2})$. In doing that it is enough show \[ \langle \delta' \delta'' F, \delta^{'k} \xi \rangle^A_0=0; \quad \forall k=0,1,\cdots, (n-1) \] Note that \[ \langle \delta' \delta'' F, \delta^{'k} \xi \rangle^A_0 =\delta^{''} (\delta'\langle F, \delta^{'k} \xi \rangle^A_0- \langle F, \delta^{'k+1} \xi \rangle^A_0)=0; \; \forall k=0, \cdots, (n-2) \] Thus \[ \langle \delta' \delta'' F, \delta^{'n-1} \xi \rangle^A_0=0; \quad =-\delta^{''} \langle F, \delta^{'n} \xi \rangle^A_0=-\delta^{''} K \] which completes the proof. {\bf Corollary (2.9)} When $K$ is paraholomorphic then the map 1$^\circ$) $\ovl{\varphi}_r \circ F: ({\L}, h) \stackrel{F}{\ra} S^{2n}_r \stackrel{\ovl{\varphi}_r}{\rightarrow} ({\P}_{2n} ({\A}), g)$ is harmonic. In particular $\ovl{\varphi}_0 \circ F: ({\L},h) \ra \ovl{({\P}_{2n} ({\R})},g) \subseteq ({\P}_{2n} ({\A}),g)$ is harmonic. 2$^\circ$) $dF: (T {\L},h^c) \ra (T S^{2n}_0, G^c) \simeq ({\P}_{2n} ({\A}),G^c)$ is harmonic. {\bf Lemma (2.10):} Let $f$ be as above with $\langle, \rangle^A_r$ paraisotropy and for $K = a+\varepsilon b$, $\;$ set $\;$ $u = \db \frac{\partial a}{\partial x}+ \frac{\partial b}{\partial y},\;\; v = \db \frac{\partial a}{\partial y}+ \frac{\partial b}{\partial x}$ $\;$ so that $\;$ $\delta^{''} K = u+ \varepsilon v$. $\;$ Then either $\;$ $u = v$ or $u = -v$. $\;$ Further if $\;$ $u (q) \neq 0$ $\;$ for some $\;$ $q$ $\;$ then $\;$ $K(q)$ $\;$ is either zero or a zero divisor. {\bf Proof} By the remark before proposition (2.8) we have $\delta^{'n} \xi=v+K \rho_r (F)$ so \begin{eqnarray*} \langle \delta^{'n} \xi, \delta^{'n} \xi \rangle^A_r&=& \langle v, v \rangle^A+ 2 K \langle v, \rho_r (F) \rangle^A+ K^2 \langle \rho_r (F), \rho_r (F) \rangle^A_r\\ &=& K^2 \end{eqnarray*} Since $\langle v,v \rangle^A=0$ by the $\langle, \rangle^A$- paraisotropy and $\langle v, \rho_r (F) \rangle^A_r= \langle v, F \rangle^A=0$ by the construction and $\langle \rho_r (F), \rho_r (F) \rangle^A_r= \langle F, F \rangle^A_r=1$ Thus $\delta^{''} \langle \delta^{'n} \xi, \delta^{'n} \xi \rangle^A_r= \delta^{''} K^2=2K \delta^{''} K=0$. Hence the result follows. {\bf Corollary (2.11):} Let $f,K, u$ be as in Lemma (2.10). If $K(q)$ is not a zero divisor for all $q$ or $u (q)=0$ whenever $K (q)$ is a zero divisor then $\delta^{''} K=0$, that is $K$ is paraholomorphic. {\bf Proposition (2.12):} Let $\; f,\; \ovl{\varphi_r} \circ F,\; K,\; u\; $ be the maps described earlier with $f$ being totally $\langle, \rangle^A_r$- paraisotropic and $K(q)$ is not a zero divisor for all $q$ or $u (q)=0$ whenever $K (q)$ is a zero divisor for $r=1,2,\cdots,2n$. Then the map \[ \ovl{\varphi}_r \circ F: ({\L},h) \ra ({\P}_{2n} ({\A}), g) \] is harmonic. {\bf Proof:} It follows from the proposition (2.8) and corally (2.11) \hfill $\Box$ {\bf Remark:} For the map $f: {\L} \ra {\P}_{2n} ({\A})$, as there is no local lift $\xi_f: U \subseteq {\L} \ra {\A}^{2n+1}$ of $f$ with $\langle, \rangle^{\ovl{A}}_0$- paraisotropy, we start with any paraholomorphic map $\xi: {\L} \ra {\A}^{2n+1}$ with $\langle, \rangle^{\ovl{A}}_0$- paraisotropy in order to construct $F: {\L} \ra ({\P}_{2n} ({\A}),g)$ a candidate for a harmonic one. Let $f: {\M} \ra ({\P}_{2n} ({\C}), g)$ be a map of a Riemann surface into complex projective spaces with the properties that $f$ is totally $\langle, \rangle^c_r$ isotropic, holomorphic and the Hermitian form $\langle, \rangle^{\ovl{c}}_r$ is nondegenerate on $span_{{\C}} V$ and is of singnature $(r, 2n-r)$ on $W^R \equiv W=span_{{\C}} (V \cup \ovl{V})$. We have {\bf Lemma (2.13)}: $[{\bf E}]_1$ The map $K: \cup (\subseteq M) \ra {\C}$ is holomorphic and therefore $F: {\M} \ra S^{2n}_r$ is harmonic, where $S^{2n}_r$ is with its standart semi-Riemannian metric and $r=0,1,\cdots,2n$ {\bf Proposition (2.14):} Let $f$ be just as above. Then the maps. \[ \ovl{\varphi}_r \circ F : ({\M},h) \ra ({\P}_{2n} ({\A}),g) \] with $r=0,1, \cdots, 2n$; and \[ dF: (T{\M},h^c) \ra (T S_0^{2n}, G^c) \cong ({\P}_{2n}, G^c) \] are harmonic. \hfill $\Box$ {\bf Examples:} {\bf I)} For the symmetric bilinear form $\langle Z,W \rangle^A_1=-z_1 w_1+z_2 w_2+z_3 w_3=\ell (z,w), \; z=(z_i), \; w=(w_i) \in {\A}^3$; let $\{u_1,u_2,u\} \subseteq {\R}^3 \subseteq {\A}^3$ be an $\ell$- orthonormal basis for ${\R}^3$ with $\ell (u_1,u_1)=-1$ and $\ell (u_2,u_2)=\ell (u,u)=1$. Then set $E=k(u_2+\varepsilon u_1)$ so that ${\ovl E}=k(u_2-\varepsilon u_1), \; k=\frac{1}{\sqrt{2}}.$ $\; $ We see that $\ell (E,E)=\ell(\ovl{E}, \ovl{E})=0$ and $\ell (E, \ovl{E})=1$. Now define a map $\xi: {\L} \ra {\A}^3$ by \[ \xi (z)=- \frac{1}{2} E+e^{mz} u+e^{2mz} \ovl{E}, \qquad m \in {\R} \] Note that 1 $^\circ$) $e^z=e^{x+\varepsilon y}=e^x (\cosh y + \varepsilon \sinh y)$ 2 $^\circ$) $\ell (\xi, \ovl{\xi})=\frac{1}{4}+ e^{2mx}+e^{4mx}>0$ so $\xi (z) \in {\A}^3_0$ and $\delta'' \xi=0$ i.e. $\xi$ is paraholomorphic. So $f=\pi \circ \xi: (\Bbb{L}, h, Q) \ra ({\P}_2 ({\A}),g,J)$ is paraholomorphic and therefore harmonic. 3 $^\circ$) $\ell (\xi,\xi)=0$ so $\xi$ is totally $\ell$-paraisotropic. Further $V=\{\xi\}$ is linearly independent and the para-Hermitian form $\ovl{\ell}=\langle,\rangle^{\ovl{A}}_1$ is nondegenerate on $span_{\A} V.$ 4 $^\circ$) Set $w_1=\xi+\ovl{\xi}, \qquad w_2=\varepsilon(\xi-\ovl{\xi})$ and observe that $W^R=span_{\R} \{w_1,w_2\} \subseteq {\R}^3$. Further, since \[ \ovl{\ell} (w_1,w_1)=\ell (\xi+\ovl{\xi}, \xi+\ovl{\xi})=2\ell (\xi,\ovl{\xi})= 2(\frac{1}{4}+e^{2mx}+e^{4mx})>0 \] \[ \ovl{\ell}(w_2,w_2)=-2 \ovl{\ell} (\xi,\ovl{\xi})<0 \quad \mbox{and} \quad \ovl{\ell} (w_1,w_2)=0,\] we see that $\ovl{\ell}$ is of signature $(1,1)$ on $W^R$. 5 $^\circ$) Set $\tilde{\alpha}=\sim (\xi \wedge \ovl{\xi}) \in {\A}^3$ and $F=\db \frac{\tilde{\alpha}}{\mu}$ with $\mu= [\ell (\tilde{\alpha}, \ovl{\tilde{\alpha}})]^{1/2}$. Then $F: ({\L},h) \ra S^2_1$ is defined on the whole of ${\L}$. 6 $^\circ$) $K=det [\xi,\ovl{\xi},\xi']= \left[ \begin{array}{ccc} -1/2 & e^{2mz} & e^{mz}\\ e^{2m \ovl{z}} & -1/2 & e^{m \ovl{z}}\\ 0 & me^{mz} & 2me^{2mz}\\ \end{array} \right]$ $=m e^{mz}[ (e^{mx} \cosh my +e^{3mx} \cosh my -2 e^{5mx} \cosh my)$ $- \varepsilon (e^{3mx} \sinh my + 2 e^{5mx} \sinh my)]= m e^{mz} (u-\varepsilon v)$ $K(z) $ is a zero divisor if and only if $u+v=0$ or $u-v=0$. But \[ u+v=e^{mx} \cosh my+e^{m(3x+y)}+2e^{m(5x-y)}\rangle 0 \] and \[ u-v=e^{mx} \cosh my+e^{m(3x-y)}+2e^{m(5x-y)} > 0 \] for all $z \in {\L}$. Thus $K(z)$ is not a zero divisor for all $z$ and hence by proposition (2.8) and corollary (2.11), $F$ is harmonic. By proposition (2.12) the map $\ovl{\varphi}_1 \circ F: ({\L},h) \ra ({\P}_2 ({\A}),g)$ is harmonic. {\bf II)} Let $\xi=\xi(z)=((1+\varepsilon) e^{m(x+y)}, e^{mz},\varepsilon e^{mz})$, with $z=x+\varepsilon y$, where $m$ is a nonzero real number. Then note that 1 $^\circ$) $\xi: ({\L},h,Q) \ra {\A}^3 $ is a paraholomorphic map. 2 $^\circ$) Denoting $\vartheta = \langle , \rangle^{\ovl{A}}_0$, the para-Hermitian form, we have $$\vartheta (\xi, \xi) = \langle \xi, \xi \rangle^{\ovl{A}}_0 = \langle \xi, \ovl{\xi} \rangle^{A}_0 = (1 \pm \varepsilon) (1 \mp \varepsilon)e^{2k (x+y)} + e^{2mx} - e^{2mx} = 0$$ so $f$ is totally $\vartheta$-paraisotropic. Further, $\{\xi, \ovl{\xi}\}$ is a linearly independent set. By the remark before proposition (2.8) it is enough to check the paraholomorphicity of $K = \xi \wedge \ovl{\xi} \wedge \delta' \xi$. But, $K \equiv 0$ since $\xi$ and $\delta' \xi$ are linearly dependent. So the maps $F = \frac{\tilde{\alpha}}{\mu}$: $(\Bbb{L}, h) \ra \Bbb{S}^2$ and $\ovl{\varphi}_0 \circ F: (\Bbb{L}, h) \ra \ovl{\Bbb{P}_2 (\Bbb{R})}$, (the real projective model), are harmonic. Thus the map $dF: (T \Bbb{L}, h^c) \ra (\Bbb{P}_2 (\Bbb{A}), G^c)$ is harmonic too. {\bf III)} See ([{\bf B}], pg 101) for the examples of holomorphic maps $f: (\Bbb{M},h) \ra (\Bbb{P}_{2n} (\Bbb{C}), g,J)$ satisfying the required properties stated just before the lemma (2.13), for $r = 0$. This $f$ gives rise the following harmonic maps: $$F: (\Bbb{M}, h) \ra \ovl{(\Bbb{P}_{2n} (\Bbb{R}), g)}$$ and $$dF: (T \Bbb{M}, h^c) \ra (\Bbb{P}_{2n} (\Bbb{A}), G^c)$$ {\bf IV)} Consider $Z = (z_i) \in \Bbb{A}^{m+1}$ as $(m+1) \times 1$ matrix and put $Z^{\#} = \ovl{Z}^t$, the tranpose of $\ovl{Z} = (\ovl{z}_i)$. We then set $p (Z) = ZZ^{\#} / ||Z||^2$, where $||Z||^2 = \langle Z, Z \rangle^{\ovl{A}}_0 = \displaystyle \sum_{i=1}^{m+1} z_i \ovl{z}_i$ for nonisotropic vector $z$, i.e. $||Z|| \neq 0$. Note that $p (Z)$ is a projector i.e. $p (Z)$ satisfies that $p (Z)^2 = p (Z)^{\#} = p (Z)$ with $1 = Tr (p (Z))$, the trace of $p (Z)$. {\bf Lemma:} (c.f. {\bf [L-T]}) The set of all projectors $p (Z)$ can be identified with the nonisotropic part of a straight line $span_{\Bbb{A}} \{ Z \}$ which is diffeomorphic to $T (\Bbb{P}_m (\Bbb{R}))$, the tangent bundle of the real projective space. For a nonisotropic paraholomorphic map $W : \Bbb{A} \ra \Bbb{A}^m$, define a map $$B (z) = \frac{1}{1 + ||W (z)||^2} \left[ \begin{array}{ll} 1 & W (z)^{\#}\\ W (z) & W (z) W (z)^{\#}\\ \end{array} \right].$$ Then $B (z)$ is a projector and therefore, by the above lemma, it may be viewed as a map into paracomplex projective model $(\Bbb{P}_m (\Bbb{A}), g)$. Then the map $B: (\Bbb{A}, \langle , \rangle^{\ovl{A}}) \ra (\Bbb{P}_m (\Bbb{A}), g)$ is infact harmonic (see [{\bf L - T}]). {\bf References:} \begin{itemize} {\bf \item[[B]]-} J.1.M.Barbosa: On minimal immersions of ${\S}^2$ into ${\S}^{2m}$. Trans. Am. Math. Soc. Vol. 210, (1975), 75-106 {\bf \item[[B-B]]-} C.L. Bejan and M.Benyounes: {\em Harmonic maps between almost para-Hermitian manifolds}. (to appear), Kluwer Acad.Publ., Proc. Conf., 1997. {\bf \item[[C-F-G]]-} V. Cruceanu, P.Fortuny and P.M. Gadea: {\em A survey on paracomplex geometry.} Rocky Mountain J.Math. 26(1996), 83-115. {\bf \item[[C-G-M]]-} V. Cruceanu, P.M. 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Moffat and J.Malzan: {\em Geometrical interpretation of a generalized theory of gravitation.} J.Math.Phys. 24 (1986), 886. {\bf \item[[L]]-} P.Libermann: {\em Sur les structures presque paracomplexes} C.R. Acad. Sci. Paris Ser. I Mat. 234 (1952), 2517-2519. {\bf \item[[L-T]]-} D.Lambert and Ph. Tombal: {\em Hyperbolic complex numbers and nonlinear sigma models.} Internat.J.Theoret. Phys. Vol. 26, No.10, 1987. {\bf \item[[M]]-} J. Malzan {\bf \item[[0]]-} V. Oproiu: {\em A pseudo-Riemannian structure in Lagrange geometry} Analele stiintifice ale Universitatii 'Al.I. Cuza' din Iai. Tomul XXXIII, s.I.a. Matematica, 1987, f3. {\bf \item[[P]]-} V.J.Parmar: {\bf \item[[Pa]]-} E.M.Paterson: {\bf Riemann extensions which have Kaehler metrics.} Proc. Roy. Soc. edinburgh (A) 64 (1954), 113-126. {\bf \item[[R]]-} P.K. Rashkevskij: {\em The scalar field in a stratified space.} Trudy Sem. Vekt. Tenz. Anal. 6(1948), 225-248. {\bf \item[[V]]}- M.E. Vazquez-Abal: {\em Harmonicity on the tangent bundle of order} $r$. C.R. Acad. Sci. Paris, t. 312, serie I, p. 131-136, 1991. {\bf \item[[W]]}- H.Wu: {\em The equdistribution theory of homorphic curves.} Anals of mathematics studies 64. (Princeton University press, 1970) {\bf \item[[We]]}- T.Weinstein: {\em An Introduction to Lorentz surfaces.} Walter de Gruyter; Berlin, New York (1996) {\bf \item[[Y-I]]}-K.Yano and S.Ishhara: {\em Tangent and cotangent bundles.} M.Dekker, New York (1973). {\bf \item[[Y-K]]-} K.Yano and S.Kobayashi: {\em Prolongations of tensor fields and connections to tangent bundles I-General theory}. J.Math.Soc. Japan Vol. 18, No.2, 1996. \end{itemize} \end{document}