Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 2, 385-398.

Paracomplex Projective Models and Harmonic Maps into Them

Sadettin Erdem

Department of Mathematics, The Middle East Technical University 06531 Ankara, Turkey e-mail: serdem@rorqual.cc.metu.edu.tr, serdem@arf.math.metu.edu.tr

Abstract. Paracomplex projective models are para-Kaehler space forms (and models of constant paraholomorphic sectional curvature of dimension greater than 2) introduced in [9]. We consider these models with their usual para-Kaehler structure and also introduce another almost para-Kaehler structure on their underlying manifolds. Next the harmonicity of maps into the two given types of manifolds is discussed and some harmonic maps are constructed from certain paraholomorphic ones. MSC 1991: 53C15, 53C50, 53C55, 58E20

Introduction

Since the first papers written on paracomplex geometry (a geometry related to the ring of para-complex numbers) by Rashevskij [23], Libermann [15] and Patterson [22] in 1948, 1952 and 1954 respectively, many other mathematicians and physicists have devoloped interests into it and used it (see [3], [4] and the references therein). The physicists 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) as developed by Kunstatter, Moffat and Malzan [14] is known to have no inconsistency problem concerning unitarity. Also Moffat [19] has interpreted the paracomplex number ε, (ε^{2} = 1) as fermion number. Consequently this interpretation has led to fundamental explanation of stability of fermionic matter. In [16], Lambert and Kibler used the paracomplex geometry to generate non-bijective canonical transformation.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

Part of this paper is devoted to describe 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 an almost para-Kaehler manifold.

The rest of the paper is devoted to describe and construct harmonic maps of manifolds with definite or neutral metrics into paracomplex projective models, examples of non-compact manifolds. One should note that harmonic maps of a manifold with neutral metric, e.g. a Lorentz surface, are the solutions of a system of wave equations while those of a Riemannian manifold are solutions of a system of elliptic ones. Harmonic maps between semi-Riemannian manifolds are connected to the hyperbolic systems. This connection is particularly significant in the areas of quantum field theory, solitons and scattering theory and general relativity, see [8]. Our work also demonstrates all along how beautifully the ring of paracomplex numbers fits to construct non-compact manifolds with neutral metrics and to attack problems of harmonic maps between them as the complex numbers do in the case of harmonic maps of Riemannian manifolds, see also [17] and [18].

I would like to thank the referee for valuable suggestions.

0. Preliminaries

Let θ denote ε with ε^{2} = 1 or i with i^{2} = −1 and then set C_{θ} = {a + θb : a, b ∈ ℝ}. Clearly C_{i} = ℂ, the field of complex numbers. But C_{ε}, which we will denote by 𝔸, forms a ring with zero divisors k(1 + ε) and k(1 − ε), k ∈ ℝ. The set 𝔸 is called paracomplex (or double or hyperbolic complex) numbers.

Notations. We frequently use one of the following notations where it is convenient.

Z = (z_{k}) = (z_{1}, …, z_{N})
= (x_{1} + θy_{1}, …, x_{N} + θy_{N}) = (x_{k}) + θ(y_{k}) = ((x_{k}), (y_{k})) = (x + θy) ∈ C_{θ}^{N}

\overline{Z} = (\overline{z}_{k}) and \overline{z} = x − θy, θ-complex conjugate of z. For Z = (z_{k}), W = (w_{k}) = (u_{k}) + θ(v_{k}), r = 0, 1, …, N − 1, let

⟨Z, W ⟩_{r}^{θ} = − ∑_{k=1}^{r} z_{k}w_{k} + ∑_{k=r+1}^{N} z_{k}w_{k}

the symmetric C_{θ}-bilinear form and

⟨Z, W ⟩\overline{θ} ; r = − ∑_{k=1}^{r} z_{k}\overline{w}_{k} + ∑_{k=r+1}^{N} z_{k}\overline{w}_{k}

the C_{θ}-Hermitian form. They are both nondegenerate on C_{θ}^{N}. For θ = ε we write ⟨, ⟩_{r}^{A} and ⟨, ⟩\overline{A} ; r for symmetric 𝔸-bilinear (or paracomplex bilinear) and para-Hermitian forms respectively. For Z, W ∈ ℝ^{N} we write

G_{r}(Z, W) = ⟨Z, W ⟩_{r} = ⟨Z, W ⟩_{r}^{R} = ⟨Z, W ⟩_{r}^{θ} = ⟨Z, W ⟩\overline{θ} ; r.

Let V be a real vector space of dimension n and J_{θ} : V → V be an endomorphism with J_{ε}^{2} = I and J_{i}^{2} = −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_{ε} corresponding to eigenvalues 1, −1 respectively. Then J_{ε} is called product structure if k_{1}, k_{2} ≥ 1. A paracomplex structure on V is a product structure with k_{1} = k_{2}. A (semi-) Riemannian metric g on V is said to be Hermitian (resp. para-Hermitian) if g(X, Y) = mg(J_{θ}X, J_{θ}Y) X, Y ∈ V, where m = 1 when J_{θ} = J_{i} (resp. m = −1 when J_{θ} = J_{ε}). 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 neutral metric, where dimV = n = 2s. (Almost) paracomplex, (almost) para-Hermitian, (almost) para-Kaehler manifolds and paracomplex, paraholomorphic, para-Hermitian vector bundles are defined in the usual way with the difference that the defining elements are replaced with the corresponding “para” ones where necessary.

Let f : (M, g, J) → (M′, g′, J′) be a differentiable map between (almost) para-Hermitian manifolds. The second fundamental form ▽df of f is given by

▽df(u, v) = D̃_{u}(df(v)) − df(D_{u}^{M}v)

where D^{M} and D̃ are the metric connection on the indicated manifold M and the pull-back connection of D^{M′} under f. Also the tension field τ(f) of f is given by

τ(f) = ∑_{i,j=1}^{n} g^{ij} ▽ df(e_{i}, e_{j}) ∈ T M′

where g^{ij} is the (i, j)-th entry of the matrix (g_{ij})^{−1}, with g_{ij} = g(e_{i}, e_{j}) and {e_{j}, …, e_{m}} is a basis for T M.

Definitions. f is said to be i) paraholomorphic if df ∘ J = J′ ∘ df, ii) harmonic if τ(f) = 0, iii) totally geodesic if ▽df = 0.

We call i) a paracomplex manifold of dimension two a para-Riemann surface, ii) an oriented, paracompact, connected surface with a neutral metric (and therefore of

signature (1,1)) a Lorentz surface, iii) a free module over the ring 𝔸 a paracomplex vector space.

On a surface (M, g) with its metric g, local coordinates (x, y) are said to be g-isothermal if

g(∂_{x}, ∂_{x}) = kg(∂_{y}, ∂_{y}) and g(∂_{x}, ∂_{y}) = 0 where ∂_{x} = \frac{∂}{∂x} etc. 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 [7], [26]. Thus one obtains [7], that every Lorentz surface is a para-Riemann surface. The 𝔸^{N} is an obvious example of a paracomplex vector space with a canonical paracomplex structure J, J(v) = εv ∈ 𝔸^{N} = ℝ^{2N}.

1. Paracomplex projective models

We set ⟨, ⟩ = ⟨, ⟩\overline{A} ; 0, g = Re(⟨, ⟩) which is the real part of ⟨, ⟩ and 𝔸_{0}^{N} = \{Z ∈ 𝔸^{N} : ⟨Z, Z⟩ > 0\}.

Note that g is the neutral metric (of signature (N, N)) on ℝ^{2N} and ⟨Z, Z⟩ = g(Z, Z). Let ℙ_{n}(𝔸) denote the quotient of 𝔸_{0}^{n+1} under the equivalence relation: Z ≃ W if W = zZ for some z ∈ 𝔸_{+}, where 𝔸_{+} = \{z = x + εy ∈ 𝔸 : z\overline{z} > 0, x > 0\}. Then the set ℙ_{n}(𝔸) may be endowed with a differentiable manifold structure ([4], [9], [10]). We shall endow this differentiable manifold with two different metrics and paracomplex structures in order to obtain two different (almost) para-Kaehler manifolds:

A) (cf. [9], [10]) Consider the principal bundle π̃ : S → ℙ_{n}(𝔸) with its structure group R_{+} = {e^{εt} = cosh t + ε sinh t : t ∈ ℝ} ⊆ 𝔸_{+} obtained from the bundle π : 𝔸_{0}^{n+1} → ℙ_{n}(𝔸) with its structure group 𝔸_{+} by restricting π to S, where

S = S_{n+1}^{2n+1} = {Z = (x + εy) ∈ 𝔸_{0}^{n+1} : ⟨Z, Z⟩ = 1}
= {Z ∈ 𝔸_{0}^{n+1} : ⟨x, x⟩ − ⟨y, y⟩ = g(z, z) = 1}.

Then π̃ induces a metric g and an almost paracomplex structure J on ℙ_{n}(𝔸) (from that on 𝔸_{0}^{n+1}) which coincides with the ones described in [9]. Thus (ℙ_{n}(𝔸), g, J) becomes a para-Kaehler manifold of (nonzero) constant 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 ℙ_{n}(𝔸) respectively obtained in the way just described.

The para-Kaehler space form (ℙ_{n}(𝔸), g, J) has the following properties ([4], [9], [10]): It is i) complete since (S, g) is complete and π̃ is totally geodesic, ii) simply connected for n > 1 and connected for n = 1, iii) a symmetric (hence homogeneous) space, iv) Einstein, (see [4], Corollary 4.13, p.25 ).

B) We shall now put other structures on ℙ_{n}(𝔸): Let (M, β) be a (semi-) Riemannian manifold of dimension m and the signature of its metric β be (k, m − k), k = 1, 2, …, m. We have a splitting T (T M) = V (T M) ⊕ H(T M) of T (T M) into vertical and horizontal subbundles. Define, on the manifold T M:

i) A neutral metric β^{c} (i.e. a metric of signature (m, m)) by

β^{c}(X^{V} , Y^{V}) = β^{c}(X^{H}, Y^{H}) = 0 and β^{c}(X^{H}, Y^{V}) = β^{c}(X^{V} , Y^{H}) = β(X, Y), (1.1)

where X^{V} and X^{H} are the vertical and horizontal lifts of the vector fields X and Y on M respectively. The neutral metric β^{c} will be called the complete lift of β. ii) An almost paracomplex structure P by

P (X^{V}) = X^{V} , P (X^{H}) = −X^{H}.  (1.2)

Then (T M, β^{c}, P) becomes an almost para-Hermitian manifold (see [2], [20], [27] for details).

Let us now take M = 𝕊^{n}, the unit sphere in (ℝ^{n+1}, G) with its standard 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) ∈ ℝ^{n+1} × ℝ^{n+1} : G(x, x) = 1, G(x, y) = 0},

we define a map φ : T 𝕊^{n} →^{ψ} S →^{π̃} ℙ_{n}(𝔸) where

ψ(x, y) = \frac{1}{2}(2x + y, y) ≅ \frac{1}{2}((2x + y) + εy)
= \frac{1}{2}((2x_{1} + y_{1}) + εy_{1}, (2x_{2} + y_{2}) + εy_{2}, …, (2x_{n+1} + y_{n+1}) + εy_{n+1}).

Note that ψ(x, y) ∈ S = S_{n+1}^{2n+1} ⊆ ℝ^{2n+2} since g(ψ(x, y), ψ(x, y)) = 1. (Recall that g = Re(⟨, ⟩\overline{A} ; 0) which is a neutral metric on ℝ^{2n+2}.) It is not difficult to see that (cf. [9]) the map φ = ψ ∘ π̃ 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 ℙ_{n}(𝔸) the metric G^{c} and paracomplex structure P on T 𝕊^{n} described by (1.1) and (1.2) via the diffeomorphism φ so that (ℙ_{n}(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 [20]. The letters G^{c} and P will be reserved to denote the metric and almost paracomplex structure on ℙ_{n}(𝔸) respectively obtained in the way just described.

This manifold (ℙ(𝔸), G^{c}, P) has the following properties:

i) It is a locally conformally flat, locally symmetric space since 𝕊^{n} is a symmetric, connected space of constant curvature [5].

ii) Unlike (ℙ_{n}(𝔸), g, J), this is not a space of constant paraholomorphic sectional curvature since 𝕊^{n} is not locally flat. Nevertheless it has a vanishing scalar curvature [5]. Furthermore [11] it is null isotropic, that is, for every null vector u ∈ T_{p} = T_{p}(ℙ_{n}(𝔸)) the function

c_{u} = T_{p} → ℝ, 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. In fact when n ≥ 2, the null isotropy is equivalent to one of the following: a^{∘}) R(u, v, u, v) = 0 for every orthogonal pair of null vectors u, v, b^{∘}) local conformal flatness.

C) Some totally geodesic submanifolds of (ℙ_{n}(𝔸), g)

For the pseudo-sphere S_{r}^{n} = {u = (u_{1}, …, u_{n+1}) ∈ ℝ^{n+1} : G_{r}(u, u) = 1} where

G_{r}(u, u) = − ∑_{i=1}^{r} u_{i}^{2} + ∑_{i=r+1}^{n+1} u_{i}^{2} consider the map

\overline{φ}_{r} : S_{r}^{n} ⟶^{\overline{ψ}_{n+1−r}} S = S_{n+1}^{2n+1} →^{π̃} ℙ_{n}(𝔸)  (1.3)

given by

\overline{ψ}_{t}(u) = (u_{1}, …, u_{t}, 0, …, 0; 0, …, 0, u_{t+1}, …, u_{n+1}) ∈ S
≅ (u_{1}, …, u_{t}, εu_{t+1}, …, εu_{n+1}) ∈ 𝔸^{n+1}.

One can show (cf. [10], Theorem 5.3) that for each r = 0, 1, …, n the map \overline{φ}_{r} is an isometric imbedding and \overline{φ}_{r}(S_{r}^{n}) is a totally geodesic submanifold of (ℙ_{n}(𝔸), g). In particular, when r = 0, we have

\overline{φ} = \overline{φ}_{0} : 𝕊^{n} = S_{0}^{n} →^{\overline{ψ}_{n+1}} S →^{π̃} ℙ_{n}(𝔸)

with \overline{φ}(u) = π̃(u, \overline{0}) ≅ span_{𝔸_{+}}\{(u, \overline{0})\} which may be identified with span_{ℝ_{+}}\{u, \overline{0}\} where (u, \overline{0}) ∈ ℝ^{n+1} × ℝ^{n+1} ≅ 𝔸^{n+1}, \overline{0} = (0, …, 0) ∈ ℝ^{n+1}. Thus \overline{φ}(𝕊^{n}) constitutes the set of all real rays in ℝ^{n+1} and it is also a totally geodesic submanifold of (ℙ_{n}(𝔸), g). This is an analogous result of the fact that the real projective space ℙ_{n}(ℝ) is a totally geodesic submanifold of the complex projective space ℙ_{n}(ℂ) with their standard metrics.

We say that an almost para-Hermitian manifold M with its almost paracomplex structure Q is (1, 2)-symplectic if

▽\overline{Z}W ∈ Γ(T^{1,0}M) ∀Z, W ∈ Γ(T^{1,0}M),

where T^{1,0}M =\{(1 + εQ)X:X ∈ Γ(T M)\} and T^{0,1}M =\overline{T^{1,0}M} =\{(1 − εQ)X:X ∈ Γ(T M)\}.

Lemma 1.1. Let (M, h) be a (semi-) Riemannian manifold. Then (T M, h^{c}, P) is (1, 2)- symplectic. In particular, (T 𝕊^{n}, G^{c}, P) ≅ (ℙ_{n}(𝔸), g, J) is (1, 2)-symplectic. Moreover any para-Kaehler manifold is (1, 2)-symplectic. Here h^{c} and P are the structures given by (1.1) and (1.2) respectively.

Proof. For any (semi-) Riemannian manifold (M, h) the manifold (T M, h^{c}, P) is (1, 2)- symplectic (cf. [2], Proposition 4.1). Then the result follows. □

2. Construction of harmonic maps into ℙ_{n}(𝔸)

Let ϕ_{r} : (M, h) → (S_{r}^{n}, G_{r}) be a smooth map of a (semi-) Riemannian manifold M of any signature into the unit pseudo-sphere with its standard metric G_{r} (which is of index (r, n−r)) and r = 0, 1, …, n. Then we have:

Proposition 2.1. If ϕ_{r} is harmonic then:

(i) The composition \overline{φ}_{r} ∘ ϕ_{r} : (M, h) → (ℙ_{n}(𝔸), g) is harmonic.

(ii) The map dϕ_{0} = Φ : (T M, h^{c}, P′) → (ℙ_{n}(𝔸), G^{c}, P) is harmonic. Moreover, ϕ_{∘} is totally geodesic if and only if Φ is totally geodesic if and only if Φ is paraholomorphic.

Proof. Since \overline{φ}_{r} a is totally geodesic map, (i) follows. For (ii), since ϕ_{∘} is harmonic by the assumption, Φ is harmonic when h is Riemannian [24]. But also Φ remains harmonic when h is semi-Riemannian as one can see by combining the results in [24], pp.134–135; [20], p.247; [27], p.106, Proposition 6.3, p.100; and [28], Proposition 7.5. The last part of (ii) follows directly from the result in [2]. □

Corollary 2.2. For (M, h) with dim M = 1 and r = 0, 1, …, n we have

i) ϕ_{r} : (M, h) → (S_{r}^{n}, G_{r}) is harmonic (and therefore it is totally geodesic) if and only if \overline{φ}_{r} ∘ ϕ_{r} : (M, h) → (ℙ_{n}(𝔸), g) is totally geodesic, ii) ϕ = ϕ_{∘} : (M, h) → (𝕊^{n}, G) is harmonic if and only if Φ is totally geodesic if and only if

Φ is paraholomorphic. □

Note that for M = 𝕊^{1}, we have (T 𝕊^{1}, h^{c}, P′) ≅ (ℙ_{1}(𝔸), h^{c}, P′) which is locally diffeomorphic to S_{1}^{2} = {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} : −x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 1} ≅ 𝕊^{1} × ℝ. So any harmonic (and therefore totally geodesic) map ϕ : (𝕊^{1}, h) → (𝕊^{n}, G) gives rise to a totally geodesic and therefore a paraholomorphic map Φ = dϕ : (ℙ_{1}(𝔸), h^{c}, P′) → (ℙ_{n}(𝔸), G^{c}, P).

Proposition 2.3. Let ψ_{1} : (T M, h^{c}, P′) → (ℙ_{n}(𝔸), G^{c}, P) and ψ_{2} : (T M, h^{c}, P′) → (ℙ_{n}(𝔸), g, J) be paraholomorphic maps, where (M, h) is a (semi-) Riemannian manifold and h is of any signature. Then ψ_{1} and ψ_{2} are harmonic.

Proof. Since the manifolds involved are all (1, 2)-symplectic and any paraholomorphic map between (1, 2)-symplectic almost para-Hermitian manifolds is harmonic [2], the harmonicity of ψ_{1} and ψ_{2} easily follows. □

Notation. For z = x + εy ∈ 𝔸 set δ′ = δ_{z} = \frac{1}{2}(∂_{x} + ε∂_{y}), δ′′ = δ\overline{z} = \frac{1}{2}(∂_{x} − ε∂_{y}) and for z = x + iy ∈ ℂ set ∂′ = ∂_{z} = \frac{1}{2}(∂_{x} − i∂_{y}), ∂′′ = ∂\overline{z} = \frac{1}{2}(∂_{x} + i∂_{y}), where ∂_{x} = \frac{∂}{∂x} etc. 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.

For θ ∈ {i, ε} let ξ_{θ} : M_{θ} → C_{θ}^{2n+1} be a smooth map, where M_{i} = 𝕄 and M_{ε} = 𝕃.

Definition 2.4. For r = 0, 1, …, 2n

i) the map ξ = ξ_{ε} is said to be totally ⟨, ⟩_{r}^{A}-paraisotropic if ⟨δ^{′k}ξ, δ^{′k}ξ⟩_{r}^{A} = 0 for 0 ≤ k ≤ n − 1,

ii) the map ξ = ξ_{i} is said to be totally ⟨, ⟩_{r}^{C}-isotropic if ⟨∂^{′k}ξ, ∂^{′k}ξ⟩_{r}^{C} = 0 for 0 ≤ k ≤ n − 1.

The total ⟨, ⟩\overline{A} ; r-paraisotropy and ⟨, ⟩\overline{c} ; r-isotropy are defined in the same way.

We say that a smooth map f : M_{θ} → ℙ_{2n}(C_{θ}) is totally ⟨, ⟩_{r}^{C_{θ}}-isotropic (resp. ⟨, ⟩_{r}^{\overline{C}_{θ}}-isotropic) if f has a local lift ξ : U ⊆ M_{θ} → C_{θ}^{2n+1} which is ⟨, ⟩_{r}^{C_{θ}}-isotropic (resp. ⟨, ⟩_{r}^{\overline{C}_{θ}}-isotropic).

By the same argument that was used for the complex case [25], one can show that a smooth (resp. a paraholomorphic) map f : 𝕃 → ℙ_{2n}(𝔸) has a smooth (resp. paraholomorphic) local lift ξ : U ⊆ 𝕃 → 𝔸^{2n+1} over a small enough open set U in 𝕃. It is not difficult to see that the concept of paraisotropy is well-defined in all cases, i.e. it does not depend on the local lift and h-isothermal coordinates choosen.

Remark. For r = 0, there are no nonconstant maps ξ : M_{θ} → C_{θ}^{2n+1} with ⟨, ⟩\overline{C} ; 0-isotropy or ⟨, ⟩_{0}^{A}-paraisotropy and f : 𝕃 → ℙ_{2n}(𝔸) with ⟨, ⟩\overline{A} ; 0-paraisotropy, since ⟨ξ, ξ⟩\overline{C} ; 0 = 0 = ⟨ξ, ξ⟩_{0}^{A} implies that ξ = 0, and f(p) ∈ ℙ_{2n}(𝔸) implies that ⟨f(p), f(p)⟩\overline{A} ; 0 > 0. Also note that for a para-Riemann surface (resp. Riemann surface) E, a local lift ξ : U(⊆ E) → 𝔸^{N} (resp. ξ : U(⊆ E) → ℂ^{N}) is paraholomorphic (resp. holomorphic) if and only if δ′′ξ = 0 (resp. ∂′′ξ = 0).

Now for a fixed value of r = 0, 1, …, 2n suppose f : (𝕃, h, Q) → (ℙ_{2n}(𝔸), g, J) is a totally ⟨, ⟩\overline{A} ; r-paraisotropic (resp. totally ⟨, ⟩_{r}^{A}-paraisotropic) paraholomorphic map such that for every p ∈ 𝕃 there is a local paraholomorphic lift ξ and h-isothermal coordinates (x, y) on an open set U containing p on which the set V = V (ξ) = {ξ, δ′ξ, …, δ^{′n−1}ξ} is linearly independent and the symmetric paracomplex form ⟨, ⟩_{r}^{A} (resp. the para-Hermitian form ⟨, ⟩\overline{A} ; r) is nondegenerate on span_{𝔸}V ⊆ 𝔸^{2n+1}.

For σ(p) = (ξ ∧ δ′ξ ∧ ⋯ ∧ δ^{′n−1}ξ)(p) ∈ ⋀^{n} 𝔸^{2n+1}, the n-th exterior power of 𝔸^{2n+1}, set α(p) = σ(p) ∧ \overline{σ(p)} = (ξ ∧ ⋯ ∧ δ^{′n−1}ξ ∧ \overline{ξ} ∧ ⋯ ∧ δ^{′′n−1}\overline{ξ})(p) and α̃ =∼ ∘α, where

∼ ( ∑_{k=1}^{2n+1} λ_{k}( ⋀_{j∈Ω_{k}} e_{j})) = ∑_{k=1}^{2n+1}(−1)^{k+1}λ_{k}e_{k}

and {e_{j}} is the standard basis for ℝ^{2n+1} ⊆ 𝔸^{2n+1} and Ω_{k} = {1, 2, …, 2n + 1} \ {k}.

Lemma 2.5. The map α̃ : U ⊆ 𝕃 → 𝔸^{2n+1} never vanishes, i.e. α̃(q) ≠ 0, ∀q ∈ U.

Proof. For V = V (ξ) = \{ξ, …, δ^{′n−1}ξ\}, note that the set V ∪ \overline{V} forms a linearly independent set since V ∩ \overline{V} is empty. Indeed, let v ∈ V ∩ \overline{V}, then ⟨v, \overline{w}⟩\overline{A} ; r = ⟨v, w⟩_{r}^{A} = 0, ∀w ∈ V by the ⟨, ⟩\overline{A} ; r-paraisotropy. But then the nondegeneracy of ⟨, ⟩_{r}^{A} on V gives that V = 0. (The arguments for the case where ⟨, ⟩\overline{A} ; r-paraisotropy and ⟨, ⟩_{r}^{A}-nondegeneracy are involved are still valid when the ⟨, ⟩_{r}^{A}-paraisotropy and ⟨, ⟩\overline{A} ; r-nondegeneracy involve instead.) So span_{𝔸}(V (q) ∪ \overline{V (q)}) is a paracomplex subspace (i.e. a free submodule) of 𝔸^{2n+1}. By Theorem 7 in [13], p.171, we deduce that α(q) ≠ 0, and therefore α̃(q) ≠ 0, ∀q ∈ U. □

Observe that

i) \overline{α̃} = (−1)^{n}α̃, so α̃ is pure imaginary for an odd n and real for an even n. Thus we write α̃ for εα̃ when n is even so that α̃ denotes the real section of \underline{𝔸}^{2n+1} = U × A^{2n+1} for even and odd n.

ii) ⟨α̃, δ^{′k}ξ⟩\overline{A} ; 0 = 0 = ⟨α̃, \overline{δ^{′k}ξ}⟩\overline{A} ; 0, ∀k = 0, …, n − 1.

Put F = \frac{α̃}{μ} at points where μ = |⟨α̃, α̃⟩\overline{A} ; r|^{1/2} ≠ 0. Then

Lemma 2.6. The map F does not depend on the particular choices of a local lift and hisothermal coordinates.

Proof. Let z = x+εy ≅ (x, y) and w = u+εv be two local h-isothermal coordinates on U ⊆ 𝕃. Then δ_{w}^{k}ξ = (δ_{z}^{k}ξ)(δ_{w}z)^{k}+ terms in δ_{z}^{s}ξ with s < k, so α(w) = α(z)||δ_{w}z||^{2(1+2+···+(n−1))}, where ||δ_{w}z|| = (δ_{w}z)\overline{(δ_{w}z)}.

Claim. ||δ_{w}z|| ≠ 0 on U.

Indeed, note that z = z(w) is paraholomorphic, so for z = x(u, v) + εy(u, v) with w = u + εv

we have x_{u} = \frac{∂x}{∂u} = y_{v} and x_{v} = y_{u}. But then δ_{w}z = (x_{u} + y_{v}) + ε(x_{v} + y_{u}) = 2(x_{u} + εx_{v}),

so ||δ_{w}z||^{2} = 4(x_{u}^{2} − x_{v}^{2}). On the other hand \frac{∂}{∂u} = x_{u} \frac{∂}{∂x} + y_{u} \frac{∂}{∂y} = x_{u} \frac{∂}{∂x} + x_{v} \frac{∂}{∂y} and then

h(\frac{∂}{∂u}, \frac{∂}{∂u}) = (x_{u}^{2} − x_{v}^{2})h(\frac{∂}{∂x}, \frac{∂}{∂x}) ≠ 0. This gives x_{u}^{2} − x_{v}^{2} ≠ 0 which proves the claim.

By the claim we see that F_{w} = \frac{α̃(w)}{μ_{w}} = \frac{α̃(z)}{μ_{z}} = F_{z}. Next, let ξ and η be two local

paraholomorphic lifts of f, then ξ = λη for some λ : 𝕃 → 𝔸 with λ\overline{λ} > 0. So α_{ξ} = (λ\overline{λ})^{n}α_{η} and consequently F_{ξ} = F_{η}. □

Now observe that for V = V (ξ)(q) = {ξ(q), …, δ^{′n−1}ξ(q)} the paracomplex subspace W (q) = span_{𝔸}(V (q)∪\overline{V (q)}) has a basis \{w_{1}, …, w_{2n}\} with w_{t} ∈ ℝ^{2n+1}, t = 1, 2, …, 2n since \overline{W} = W. Set W^{R}(q) = span_{ℝ}{w_{1}, …, w_{2n}} and let ⟨, ⟩_{r} denote the metric which is the restriction of

⟨, ⟩_{r}^{A} (or equivalently ⟨, ⟩\overline{A} ; r) on ℝ^{2n+1} ⊆ 𝔸^{2n+1}.

Further suppose that f is such that ⟨, ⟩_{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 μ ≠ 0 on the whole of 𝕃 and therefore F = \frac{α̃}{μ} defines a map

F : 𝕃 → S_{r}^{2n} ⊆ ℝ^{2n+1}.

from the whole of the Lorentz surface into a pseudo-sphere. Define an endomorphism ρ_{r} : 𝔸^{N} → 𝔸^{N} by ρ(Z) = ρ_{r}(Z) = (−z_{1}, …, −z_{r}, z_{r+1}, …, z_{N}) for Z = (z_{k}) ∈ 𝔸^{N}. Then we have for r = 0, 1, …, N − 1:

i) ⟨Z, W ⟩_{r}^{A} = ⟨ρ_{r}(Z), W ⟩_{0}^{A} = ⟨Z, ρ_{r}(W)⟩_{0}^{A}.

ii) ⟨ρ_{r}(Z), ρ_{r}(W)⟩_{r}^{A} = ⟨Z, W ⟩_{r}^{A}.

iii) For a smooth map f : M_{θ} → ℂ_{θ}^{N}, θ = i, ε

ρ_{r}(∂^{′k}f) = ∂^{′k}ρ_{r}(f) and ρ_{r}(δ^{′k}f) = δ^{′k}(ρ_{r}(f))

where M_{i} =𝕄 and M_{ε} =𝕃. These relations are valid also for the operators δ^{′′k} and ∂^{′′k}.

Note that ρ_{r}(F) ∉ W = span_{𝔸}(V ∪ \overline{V}), so one can write δ^{′n}ξ = v + w + s(ρ_{r}(F)) where v ∈ span_{𝔸}(V), w ∈ span_{𝔸}(\overline{V}) and s(q) ∈ 𝔸. We have

⟨δ^{′n}ξ, δ^{′k}ξ⟩\overline{A} ; r = ⟨v, δ^{′k}ξ⟩\overline{A} ; r + ⟨w, δ^{′k}ξ⟩\overline{A} ; r + s⟨ρ_{r}(F), δ^{′k}ξ⟩\overline{A} ; r.

Since ⟨v, δ^{′k}ξ⟩\overline{A} ; r = ⟨δ^{′n}ξ, δ^{′k}ξ⟩\overline{A} ; r = 0, ∀k = 0, …, n − 1 by the ⟨, ⟩\overline{A} ; r-paraisotropy and

⟨ρ_{r}(F), δ^{′k}ξ⟩\overline{A} ; r = ⟨F, δ^{′k}ξ⟩\overline{A} ; 0 = 0 we have ⟨w, δ^{′k}ξ⟩\overline{A} ; r = ⟨w, δ^{′′k}\overline{ξ}⟩_{r}^{A} = 0. But then by the

nondegeneracy of ⟨, ⟩_{r}^{A} on V and therefore on \overline{V}, this gives that w = 0 so that

δ^{′n}ξ = v + ⟨δ^{′n}ξ, ρ_{r}(F)⟩_{r}^{A}ρ_{r}(F).  (2.1)

Remark. For the case where ⟨, ⟩_{r}^{A}-paraisotropy and ⟨, ⟩\overline{A} ; r-nondegeneracy involve instead, we still get the same formula (2.1).

As we sum up: Let f : (𝕃, h, Q) → (ℙ_{2n}(𝔸), g, J) be a paraholomorphic totally ⟨, ⟩\overline{A} ; r-paraiso-tropic (resp. totally ⟨, ⟩_{r}^{A}-paraisotropic) map such that for every p ∈ 𝕃 there is a local paraholomorphic lift ξ and h-isothermal coordinates (x, y) on an open set U containing p on which the set V = V (ξ) = {ξ, …, δ^{′n−1}ξ} is linearly independent and the symmetric paracomplex form ⟨, ⟩_{r}^{A} (resp. para-Hermitian form ⟨, ⟩\overline{A} ; r) is nondegenerate on span_{𝔸}V ⊆ 𝔸^{2n+1}; and is of signature (r, 2n − r) on W^{R} ⊆ W = span_{𝔸}(V ∪ \overline{V}).

Remark. The symmetric paracomplex form ⟨, ⟩_{0}^{A} is always nondegenerate on any subspace of 𝔸^{2n+1} and it is always of signature (0, 2n) on W^{R}. Therefore we drop the assumptions of nondegeneracy of ⟨, ⟩_{0}^{A} on span_{𝔸}V and the signature of being (0, 2n).

The map f determines the maps

F : (𝕃, h, Q) → S_{r}^{2n} ⊆ ℝ^{2n+1}

into the pseudo-sphere with its standard semi-Riemannian metric of signature (r, 2n−r) and

K : U ⊆ (𝕃, h, Q) → 𝔸

given by

K = ⟨δ^{′n}ξ, F ⟩_{0}^{A} = ⟨δ^{′n}ξ, ρ_{r}(F)⟩_{r}^{A} = det[ξ, δ′ξ, …, δ^{′n}ξ, \overline{ξ}, …, \overline{δ^{′(n−1)}ξ}],

the determinant of a matrix whose columns formed by the indicated sections δ^{′k}ξ and \overline{δ^{′k}ξ} of \underline{𝔸}^{2n+1}. Note that unlike F, the map K is defined locally and depends on the local lift ξ and h-isothermal coordinates choosen. Nevertheless its paraholomorphicity does not depend on the particular choice of ξ and h.

Proposition 2.7. Let f : (𝕃, h, Q) → (ℙ_{2n}(𝔸), g, J), F and K be the maps described as above. Then F is harmonic if K is paraholomorphic.

Proof. One can show that F is harmonic if and only if δ′δ′′F = λF for some smooth map λ : 𝕃 → ℝ (cf. [6], [7]). In doing that it is enough to show

⟨δ′δ′′F, δ^{′k}ξ⟩_{0}^{A} = 0, ∀k = 0, 1, …, n − 1.

Note that

⟨δ′δ′′F, δ^{′k}ξ⟩_{0}^{A} = δ′′(δ′⟨F, δ^{′k}ξ⟩_{0}^{A} − ⟨F, δ^{′k+1}ξ⟩_{0}^{A}) = 0, ∀k = 0, …, n − 2.

Thus 0 = ⟨δ′δ′′F, δ^{′n−1}ξ⟩_{0}^{A} = −δ′′⟨F, δ^{′n}ξ⟩_{0}^{A} = −δ′′K which completes the proof. □

Corollary 2.8. When K is paraholomorphic then the maps

1^{∘}) \overline{φ}_{r} ∘ F : (𝕃, h) →^{F} S_{r}^{2n} →^{\overline{φ}_{r}} (ℙ_{2n}(𝔸), g),

2^{∘}) dF : (T 𝕃, h^{c}) → (T S_{0}^{2n}, G^{c}) ≃ (ℙ_{2n}(𝔸), G^{c}) are harmonic.

Lemma 2.9. Let f be as above with ⟨, ⟩_{r}^{A} paraisotropy and for K = a + εb, set u = \frac{∂a}{∂x} +

\frac{∂b}{∂y} , v = \frac{∂a}{∂y} + \frac{∂b}{∂x} so that δ′′K = u + εv. Then either u = v or u = −v. Further if u(q) ≠ 0

for some q then K(q) is either zero or a zero divisor.

Proof. By the remark before Proposition 2.7 we have δ^{′n}ξ = v + Kρ_{r}(F). So

⟨δ^{′n}ξ, δ^{′n}ξ⟩_{r}^{A} = ⟨v, v⟩_{r}^{A} + 2K⟨v, ρ_{r}(F)⟩_{r}^{A} + K^{2}⟨ρ_{r}(F), ρ_{r}(F)⟩_{r}^{A} = K^{2}

since ⟨v, v⟩_{r}^{A} = 0 by the ⟨, ⟩_{r}^{A}-paraisotropy and ⟨v, ρ_{r}(F)⟩_{r}^{A} = ⟨v, F ⟩_{r}^{A} = 0 by the construction and ⟨ρ_{r}(F), ρ_{r}(F)⟩_{r}^{A} = ⟨F, F ⟩_{r}^{A} = 1. Thus δ′′⟨δ^{′n}ξ, δ^{′n}ξ⟩_{r}^{A} = δ′′K^{2} = 2K(δ′′K) = 0. Hence the result follows. □

Corollary 2.10. Let f, K, u be as in Lemma 2.9. If K(q) is not a zero divisor for all q or u(q) = 0 whenever K(q) is a zero divisor then δ′′K = 0, that is, K is paraholomorphic.

Proposition 2.11. Let f, \overline{φ_{r}} ∘ F, K, u be the maps described earlier with f being totally ⟨, ⟩_{r}^{A}-paraisotropic and K(q) not be a zero divisor for all q or u(q) = 0 whenever K(q) is a zero divisor for r = 1, 2, …, 2n. Then the map

\overline{φ}_{r} ∘ F : (𝕃, h) → (ℙ_{2n}(𝔸), g)

is harmonic.

Proof. It follows from Proposition 2.7 and Corollary 2.10. □

Remark. For a map f : 𝕃 → ℙ_{2n}(𝔸), as there is no local lift ξ_{f} : U ⊆ 𝕃 → 𝔸^{2n+1} of f with ⟨, ⟩\overline{A} ; 0-paraisotropy, we start with any paraholomorphic map ξ : 𝕃 → 𝔸^{2n+1} with ⟨, ⟩\overline{A} ; 0-paraisotropy in order to construct F : 𝕃 → S_{r}^{2n}, a candidate for a harmonic one.

Let f : 𝕄 → (ℙ_{2n}(ℂ), g) be a map of a Riemann surface into complex projective spaces with the properties that f is totally ⟨, ⟩_{r}^{C}-isotropic, holomorphic and the Hermitian form ⟨, ⟩\overline{c} ; r is nondegenerate on span_{ℂ}V and is of singnature (r, 2n − r) on W^{R} ≡ W = span_{ℂ}(V ∪ \overline{V}). We see that the map K : U(⊆ M) → ℂ is holomorphic and therefore F : 𝕄 → S_{r}^{2n} is harmonic [6].

Proposition 2.12. Let f be just as above. Then the maps

\overline{φ}_{r} ∘ F : (𝕄, h) → (ℙ_{2n}(𝔸), g), r = 0, 1, …, 2n; and
dF : (T 𝕄, h^{c}) → (T S_{0}^{2n}, G^{c}) ≅ (ℙ_{2n}, G^{c})

are harmonic. □

Examples. I) For the symmetric bilinear form ⟨Z, W ⟩_{1}^{A} = −z_{1}w_{1} +z_{2}w_{2} +z_{3}w_{3} = ℓ(Z, W), Z = (z_{i}), W = (w_{i}) ∈ 𝔸^{3}, let {u_{1}, u_{2}, u} ⊆ ℝ^{3} ⊆ 𝔸^{3} be an ℓ-orthonormal basis for ℝ^{3} with ℓ(u_{1}, u_{1}) = −1 and ℓ(u_{2}, u_{2}) = ℓ(u, u) = 1. Then set E = k(u_{2} + εu_{1}) so that \overline{E} = k(u_{2} − εu_{1}), k = \frac{1}{\sqrt{2}}. We see that ℓ(E, E) = ℓ(\overline{E}, \overline{E}) = 0 and ℓ(E, \overline{E}) = 1. Now define a map ξ : 𝕃 → 𝔸^{3} by

ξ(z) = −\frac{1}{2}E + e^{mz}u + e^{2mz}\overline{E},

where m is a nonzero real number. Note that 1^{∘}) e^{z} = e^{x+εy} = e^{x}(cosh y + ε sinh y).

2^{∘}) ℓ(ξ, \overline{ξ}) = \frac{1}{4} + e^{2mx} + e^{4mx} > 0, so ξ(z) ∈ 𝔸_{0}^{3} and δ′′ξ = 0, i.e. ξ is paraholomorphic. So f = π ∘ ξ : (𝕃, h, Q) → (ℙ_{2}(𝔸), g, J) is paraholomorphic and therefore harmonic.

3^{∘}) ℓ(ξ, ξ) = 0, so ξ is totally ℓ-paraisotropic. Further V = {ξ} is linearly independent and the para-Hermitian form \overline{ℓ} = ⟨, ⟩\overline{A} ; 1 is nondegenerate on span_{𝔸}V.

4^{∘}) Set w_{1} = ξ + \overline{ξ}, w_{2} = ε(ξ − \overline{ξ}) and observe that W^{R} = span_{ℝ}\{w_{1}, w_{2}\} ⊆ ℝ^{3}. Further, since

\overline{ℓ}(w_{1}, w_{1}) = ℓ(ξ + \overline{ξ}, ξ + \overline{ξ}) = 2ℓ(ξ, \overline{ξ}) = 2(\frac{1}{4} + e^{2mx} + e^{4mx}) > 0,

\overline{ℓ}(w_{2}, w_{2}) = −2\overline{ℓ}(ξ, \overline{ξ}) < 0 and \overline{ℓ}(w_{1}, w_{2}) = 0,

we see that \overline{ℓ} is of signature (1, 1) on W^{R}.

5^{∘}) Set α̃ =∼ (ξ ∧ \overline{ξ}) ∈ 𝔸^{3} and F = \frac{α̃}{μ} with μ = [ℓ(α̃, \overline{α̃})]^{1/2}. Then F : (𝕃, h) → S_{1}^{2} is defined

on the whole of 𝕃.

6^{∘}) K = det[ξ, \overline{ξ}, ξ′] = det [ \begin{matrix}−1/2 & e^{2mz} & e^{mz} \\ e^{2m\overline{z}} & −1/2 & e^{m\overline{z}} \\ 0 & 2me^{2mz} & me^{mz} \end{matrix} ] = me^{mz}(e^{2mx} + e^{4mx} + 1/4).

So

K\overline{K} = m^{2}(e^{2mx} + e^{4mx} + 1/4)^{2}e^{2mx} ≠ 0.

Thus K(z) is not a zero divisor for all z and hence by Proposition 2.7 and Corollary 2.10, F is harmonic. By Proposition 2.11 the map \overline{φ}_{1} ∘ F : (𝕃, h) → (ℙ_{2}(𝔸), g) is harmonic.

II) Let ξ = ξ(z) = ((1 + ε)e^{m(x+y)}, e^{mz}, εe^{mz}), with z = x + εy, where m is a nonzero real number. Then note that 1^{∘}) ξ : (𝕃, h, Q) → 𝔸^{3} is a paraholomorphic map. 2^{∘}) Denoting ϑ = ⟨, ⟩\overline{A} ; 0, the para-Hermitian form, we have

ϑ(ξ, ξ) = ⟨ξ, ξ⟩\overline{A} ; 0 = ⟨ξ, \overline{ξ}⟩_{0}^{A} = (1 ± ε)(1 ∓ ε)e^{2k(x+y)} + e^{2mx} − e^{2mx} = 0,

so ξ is totally ϑ-paraisotropic. Further, \{ξ, \overline{ξ}\} is a linearly independent set.

By the remark before Proposition 2.7 it is enough to check the paraholomorphicity of K = ξ ∧ \overline{ξ} ∧ δ′ξ. But, K ≡ 0 since ξ and δ′ξ are linearly dependent. So the maps F = \frac{α̃}{μ}: (𝕃, h) → 𝕊^{2} = S_{0}^{2} and \overline{φ}_{0} ∘ F : (𝕃, h) → ℙ_{2}(𝔸), (the real projective model), are harmonic. Thus the map dF : (T 𝕃, h^{c}) → (ℙ_{2}(𝔸), G^{c}) is harmonic too.

III) See [1], p.101, for the examples of holomorphic maps f : (𝕄, h) → (ℙ_{2n}(ℂ), g, J) satisfying the properties stated just before Lemma 2.12, for r = 0. This f gives rise to following harmonic maps:

F : (𝕄, h) → (ℙ_{2n}(ℝ), g) and dF : (T 𝕄, h^{c}) → (ℙ_{2n}(𝔸), G^{c}).

IV) Consider Z = (z_{i}) ∈ 𝔸^{m+1} as (m + 1) × 1 matrix and put Z^{#} = \overline{Z}^{t}, the transpose of

\overline{Z} = (\overline{z}_{i}). We then set p(Z) = ZZ^{#}/||Z||^{2}, where ||Z||^{2} = ⟨Z, Z⟩\overline{A} ; 0 = ∑_{i=1}^{m+1} z_{i}\overline{z}_{i} for a non-

isotropic vector Z, i.e. ||Z|| ≠ 0. Note that p(Z) is a projector, i.e. p(Z)^{2} = p(Z)^{#} = p(Z) with 1 = Tr (p(Z)), the trace of p(Z).

Lemma. (cf. [18]) The set of all projectors p(Z) can be identified with the nonisotropic part of the straight line span_{𝔸}{Z} which is diffeomorphic to T (ℙ_{m}(ℝ)), the tangent bundle of the real projective space.

For a nonisotropic paraholomorphic map W : 𝔸 → 𝔸^{m}, define a map

B(z) = \frac{1}{1 + ||W (z)||^{2}} [ \begin{matrix}1 & W (z)^{#} \\ W (z) & W (z)W (z)^{#} \end{matrix} ] .

Then B(z) is a projector and therefore, by the above Lemma, it may be viewed as a map into the paracomplex projective model (ℙ_{m}(𝔸), g). This map B : (𝔸, ⟨, ⟩\overline{A}) → (ℙ_{m}(𝔸), g) is in fact harmonic (see [18]).

References

[1] Barbosa, J. L. M.: On minimal immersions of 𝕊^{2} into 𝕊^{2m}. Trans. Amer. Math. Soc. 210 (1975), 75–106.

[2] Bejan, C. L.; Benyounes, M.: Harmonic maps between almost para-Hermitian manifolds. To appear, Kluwer Acad. Publ., Proc. Conf. 1997.

[3] Cruceanu, V.; Fortuny, P; Gadea, P. M.: A survey on paracomplex geometry. Rocky Mountain J. Math. 26 (1996), 83–115.

[4] Cruceanu, V.; Gadea, P. M.; Masque, J. M.: Para-Hermitian and para-Kaehler manifolds. Quaderni dell’Istituto Matematico (Messina) 1 (1995), 1–70.

[5] Cendan-Verdes, J.; Garcia-Rio, E.; Vazquez-Abal, M. E.: On the semi-Riemannian structure of the tangent bundle to a two-point homogeneous space. Riv. Mat. Univ. Parma (5) 3 (1994)(2), 253–270.

[6] Erdem, S.: Harmonic maps from Riemann surfaces into pseudo-Riemannian spheres and hyperbolic spaces. Math. Proc. Cambridge Philos. Soc. 94 (1983), 483.

[7] Erdem, S.: Harmonic maps of Lorentz surfaces, quadratic differentials and paraholomorphicity. Beiträge Algebra Geom. 38 (1997)(1), 19–32.

[8] Glazebrook, J. F.: Strings, harmonic maps and hyperbolic systems. Comput. Math. Appl. 19 (1990)(8/9), 117–125.

[9] Gadea, P. M.; Amilibia, A. M.: Spaces of constant paraholomorphic sectional curvature. Pacific J. Math. 136 (1989), 85–101.

[10] Gadea, P. M.; Amilibia, A. M.: The paracomplex projective model and the para-Grassmannian manifolds. In: Contribuciones matematicas. Estudios en honor del Profesor Jose Javier Etayo Miqueo. Madrid 1994, 315–334.

[11] Garcia-Rio, E.; Kupeli, D.: Null and infinitestimal isotropy in semi-Riemannian geometry. J. Geom. Phys. 13 (1994), 207–222.

[12] Gadea, P. M.; Montesinos, A.: The paracomplex projective spaces as symmetric and natural spaces. Indian J. Pure Appl. Math. 23 (1992)(4), 261–275.

[13] Hoffman, K.; Kunze, R.: Linear Algebra. Prentice-Hall Inc., Englewood Cliffs, New Jersey 1971 (second edition).

[14] Kunstatter, G.; Moffat, J. W.; Malzan, J.: Geometrical interpretation of a generalized theory of gravitation. J. Math. Phys. 24 (1983), 886–889.

[15] Libermann, P.: Sur les structures presque paracomplexes. C. R. Acad. Sci. Paris Ser. I Math. 234 (1952), 2517–2519.

[16] Lambert, D.; Kibler, M.: An algebraic and geometric approach to nonbijective quadratic transformations. Lycen (Lyon) (1986), 8642.

[17] Lambert, D.; Piette, B.: Solutions of Minkowskian σ-models on hyperbolic complex Grassmannian manifolds. Classical Quantum Gravity 5 (1988), 307–319.

[18] Lambert, D.; Tombal, Ph.: Hyperbolic complex numbers and nonlinear sigma models. Internat. J. Theoret. Phys. 26 (1987)(10), 943–950.

[19] Moffat, J. W.: Lectures given at the VIIth international school of gravitation and cosmology, Erice 1981. Proceedings to be published by World Scientific Publ. Co., Singapore 1982.

[20] Oproiu, V.: A pseudo-Riemannian structure in Lagrange geometry. An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. XXXIII (1987)(3), 239–254.

[21] Parmar, V. J.: Harmonic morphism between semi-Riemannian Manifolds. Ph.D. Thesis, The Univ. of Leeds, U.K. (1991).

[22] Patterson, E. M.: Riemann extensions which have Kaehler metrics. Proc. Roy. Soc. Edinburgh Sect. A 64 (1954), 113–126.

[23] Rashevskij, P. K.: The scalar field in a stratified space. Trudy Sem. Vektor. Tenzor. Anal. 6 (1948), 225–248.

[24] Vazquez-Abal, M. E.: Harmonicity on the tangent bundle of order r. C. R. Acad. Sci. Paris Ser. I Math. 312 (1991), 131–136.

[25] Wu, H.: The equidistribution theory of holomorphic curves. Ann. of Math. Stud. 64, Princeton University Press 1970.

[26] Weinstein, T.: An Introduction to Lorentz surfaces. Walter de Gruyter, Berlin – New York 1996.

[27] Yano, K.; Ishihara, S.: Tangent and cotangent bundles. M. Dekker, New York 1973.

[28] Yano, K.; Kobayashi, S.: Prolongations of tensor fields and connections to tangent bundles I, General theory. J. Math. Soc. Japan 18 (1966)(2), 194–211.

Received January 27, 1998