����������� ������ 54 (2002), 195–202

UDK 514.763.4

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ON HOLOMORPHICALLY PROJECTIVE MAPPINGS OF GENERALIZED KÄHLERIAN SPACES

Mića S. Stanković, Svetislav M. Minčić, Ljubica S. Velimirović

Abstract. In this work we define generalized Kählerian spaces and for them consider holomorphically projective mappings with an invariant complex structure. Also we consider equitorsion holomorphically projective mappings and for them we find some invariant geometric objects.

1. Introduction

A generalized Riemannian space GR_{N} in the sense of Eisenhart’s definition [1] is a differentiable N-dimensional manifold, equipped with nonsymmetric basic tensor g_{ij}. Connection coefficients of this space are generalized Cristoffel’s symbols of the second kind. Generally it is Γ_{jk}^{i} ≠ Γ_{kj}^{i}.

In a generalized Riemannian space one can define four kinds of covariant derivatives [3], [4]. For example, for a tensor a_{j}^{i} in GR_{N} we have

a_{j |_{1}m}^{i} = a_{j,m}^{i} + Γ_{pm}^{i}a_{j}^{p} − Γ_{jm}^{p}a_{p}^{i}, a_{j |_{2}m}^{i} = a_{j,m}^{i} + Γ_{mp}^{i}a_{j}^{p} − Γ_{mj}^{p}a_{p}^{i},

a_{j |_{3}m}^{i} = a_{j,m}^{i} + Γ_{pm}^{i}a_{j}^{p} − Γ_{mj}^{p}a_{p}^{i}, a_{j |_{4}m}^{i} = a_{j,m}^{i} + Γ_{mp}^{i}a_{j}^{p} − Γ_{jm}^{p}a_{p}^{i}.

In the case of the space GR_{N} we have five independent curvature tensors [5] (in [5] R_{5} is denoted by R̃_{2}). In this paper we consider only curvature tensors: R_{1}_{jmn}^{i} =

Γ_{jm,n}^{i} − Γ_{jn,m}^{i} + Γ_{jm}^{p}Γ_{pn}^{i} − Γ_{jn}^{p}Γ_{pm}^{i}, R_{2}_{jmn}^{i} = Γ_{mj,n}^{i} − Γ_{nj,m}^{i} + Γ_{mj}^{p}Γ_{np}^{i} − Γ_{nj}^{p}Γ_{mp}^{i}.

The Kählerian spaces and their mappings were investigated by many authors, for example K. Yano [10], [11], M. Prvanović [7], N. S. Sinyukov [8], J. Mikeš [2] and many others.

AMS Subject Classification: 53 C 25, 53 A 45, 53 B 05

Keywords and phrases: Generalized Kählerian space, holomorphically projective mappings, equitorsion holomorphically projective mappings, holomorphically projective parameter, holomorphically projective tensor.

Communicated at the 5th International Symposium on Mathematical Analysis and its Applications, Niška banja, Yugoslavia, October, 2–6, 2002.

An N-dimensional Riemannian space with basic metric tensor g_{ij}(x) is a Kählerian space if there exists an almost complex structure F_{j}^{i}(x), such that

F_{p}^{h}(x)F_{i}^{p}(x) = −δ_{i}^{h} , g_{pq}F_{i}^{p}F_{j}^{q} = g_{ij}, g^{ij} = g^{pq}F_{p}^{i}F_{q}^{j}, F_{i;j}^{h} = 0,

where ; denotes the covariant derivative with respect to the basic metric tensor g_{ij}. This paper is devoted to the generalized Kählerian spaces and their mappings.

2. Generalized Kählerian spaces

A generalized N-dimensional Riemannian space with (non-symmetric) metric tensor g_{ij}, is a generalized Kählerian space GK_{N} if there exists an almost complex structure F_{j}^{i}(x) [9], such that

F_{p}^{h}(x)F_{i}^{p}(x) = −δ_{i}^{h} ,  (2.1)
g\underline{pq}F_{i}^{p}F_{j}^{q} = g\underline{ij}, g\underline{ij} = g\underline{pq}F_{p}^{i}F_{q}^{j},  (2.2)
F_{i|_{θ}j}^{h} = 0, (θ = 1, 2),  (2.3)

where |_{θ} denotes the covariant derivative of the kind θ with respect to the metric

tensor g_{ij}. From (2.2), using (2.1), we get

g\underline{ip}F_{j}^{p} + g\underline{pj}F_{i}^{p} = 0, g\underline{ip}F_{p}^{j} + g\underline{jp}F_{p}^{i} = 0.  (2.4,5)

Let us denote

F_{ji} = F_{j}^{p}g\underline{pi}, F^{ji} = F_{p}^{j}g\underline{pi}.  (2.6)

Then from (2.4) and (2.5) one obtains F_{ij} + F_{ji} = 0, F^{ij} + F^{ji} = 0. From here we prove the following theorems [9].

Theorem 2.1. For the torsion tensor of a generalized Kählerian space the relation Γi ; jm_{∨} = −Γp ; qm_{∨}F_{p}^{i}F_{j}^{q} is valid.

Theorem 2.2. The curvature tensors R_{θ}_{ijk}^{h} (θ = 1, 2) in the space GK_{N} satisfy the next relations

F_{i}^{p}R_{α}_{pjk}^{h} = F_{p}^{h}R_{α}_{ijk}^{p}, α = 1, 2.  (2.7,8)

Proof. a) From (2.3) we have F_{i|_{1}jk}^{h} − F_{i|_{1}kj}^{h} = 0, and then, using the first Ricci

identity [3], [4] we have −F_{p}^{h}R_{1}_{ijk}^{p}+F_{i}^{p}R_{1}_{pjk}^{h}−2Γp ; jk_{∨}F _{i|_{1}p}^{h} = 0, i.e. F_{i}^{p}R_{1}_{pjk}^{h}−F_{p}^{h}R_{1}_{ijk}^{p} =

0. The relation (2.7) is proved.

b) Analogously, using the Ricci identity for F_{i|_{2}jk}^{h} − F_{i|_{2}kj}^{h} and (2.3) we get

F_{i}^{p}R_{2}_{pjk}^{h} − F_{p}^{h}R_{2}_{ijk}^{p} = 0, wherefrom (2.8) follows.

Theorem 2.3. For the curvature tensors R_{θ}_{hijk} (θ = 1, 2) of the space GK_{N} the next relations are valid

F_{h}^{p}R_{α}_{pijk} = F_{i}^{p}R_{α}_{phjk}, α = 1, 2.  (2.9,10)

Proof. By composition in (2.7) with F_{h}^{q} we get F_{i}^{p}F_{q}^{h}R_{1}_{pjk}^{q} + R_{1}_{ijk}^{h} = 0. From

here we have F_{h}^{p}F_{i}^{q}R_{1}_{pqjk} − R_{1}_{hijk} = 0, and by composition with F_{r}^{i} we get

F_{h}^{p}R_{1}_{pijk} + F_{i}^{p}R_{1}_{hpjk} = 0.  (2.11)

The first kind curvature tensor satisfy the relation R_{1}_{hijk} = −R_{1}_{ihjk}. Now from

(2.11) we get the relation (2.9). The relation (2.10) we get in the same manner from (2.8) by using of anti-symmetry for the tensors R_{2}_{hijk} with respect to the two first indices.

Theorem 2.4. The curvature tensors R_{θ}_{jmn}^{i} (θ = 1, 2) of the space GK_{N} satisfy the next relations

R_{α}_{(pq)}F_{j}^{p}F_{m}^{q} = R_{α}_{(jm)} − 2Γp ; rq_{∨}Γq ; ps_{∨}F_{j}^{r}F_{m}^{s} + 2Γp ; jq_{∨}Γq ; pm_{∨}, α = 1, 2,  (2.12 a,b)

where (jm) denotes the symmetrization without division with respect to the indices j, m.

Proof. (a) From F_{i|_{1}j}^{h} = 0, F _{i|_{2}j}^{h} = 0 by substitution and division with 2 we get

F_{i;j}^{h} = 0, where ; denotes covariant derivative with respect to g\underline{ij}. The integrability conditions of this equation give the relation F_{p}^{h}R_{ijk}^{p} − F_{i}^{p}R_{pjk}^{h} = 0, where R_{ijk}^{h} is a curvature tensor with respect to symmetric basic tensor g\underline{ij}. Using the condition (2.1) we get F_{p}^{h}F_{i}^{q}R_{ijk}^{p} + R_{ijk}^{h} = 0, and from here F_{h}^{p}F_{i}^{q}R_{pqjk} − R_{hijk} = 0. With

respect to the condition (2.1), we get F_{h}^{p}R_{pijk} − F_{i}^{p}R_{phjk} = 0. By composition with g\underline{ij} and contraction by virtue of indices i, j, we get F_{h}^{p}R_{pk} = F_{q}^{p}R_{ph.k}^{q}. By symmetrization with respect to h, k we get

R_{hk} = F_{h}^{p}F_{k}^{q}R_{pq}.  (2.13)

We can express the tensor R_{1}_{jmn}^{i} in the form [5]:

R_{1}_{jmn}^{i} = R_{jmn}^{i} + Γ_{jm_{∨};n}^{i} − Γ_{jn_{∨};m}^{i} + Γp ; jm_{∨}Γi ; pn_{∨} − Γp ; jn_{∨}Γi ; pm_{∨}.

By contraction with respect to indices i, n, and by symmetrization with respect to j, m, we get

R_{1}_{(jm)} = R_{(jm)} − 2Γp ; jq_{∨}Γq ; pm_{∨}.  (2.14)

From (2.13) and (2.14) we have (2.12a).

(b) The tensor R_{2}_{jmn}^{i} we can express in the form [5]:

R_{2}_{jmn}^{i} = R_{jmn}^{i} − Γ_{jm_{∨};n}^{i} + Γ_{jn_{∨};m}^{i} − Γp ; jm_{∨}Γi ; pn_{∨} + Γp ; jn_{∨}Γi ; pm_{∨}.

By contraction with respect to i, n, and then by symmetrization with respect to j, m, we get R_{2}_{(jm)} = R_{(jm)} − 2Γp ; jq_{∨}Γq ; pm_{∨}, wherefrom, using (2.13), we get the relation

(2.12b).

3. Holomorphically projective mapings

Generalizing the concept of analytic planar curve in a Kählerian space [6], [8] we get an analogous notion for a generalized Kählerian space [9].

Definition 3.1. A curve l : x^{h} = x^{h}(t), (h = 1, 2,…, N) is an an analytic planar curve if the following relation is satisfied

λ^{h}_{|_{θ}p}λ^{p} = a(t)λ^{h} + b(t)F_{p}^{h}λ^{p}, (θ = 1, 2)  (3.1)

where λ^{h} = dx^{h}/dt, and a(t) and b(t) are same function of a parameter t.

In GK_{N} it is λ^{h}_{|_{1}p}λ^{p} = \frac{dλ^{h}}{dt} + Γ_{pq}^{h}λ^{p}λ^{q} = λ^{h}_{|_{2}p}λ^{p}. Then the expression on

the left-hand side in (3.1) is invariant with respect to the both kind of covariant derivative, and so we can define analytic planar curve in the space GK_{N} by one relation \frac{dλ^{h}}{dt} + Γ_{pq}^{h}λ^{p}λ^{q} = a(t)λ^{h} + b(t)F_{p}^{h}λ^{p}.

Consider two N-dimensional generalized Kählerian spaces GK_{N} and G\overline{K}_{N} with almost complex structures F_{i}^{h} and \overline{F} _{i}^{h}, respectively, where F_{i}^{h} = \overline{F} _{i}^{h}.

Definition 3.2. A diffeomorfism f: GK_{N} → G\overline{K}_{N} is holomorphically projective or analytic planar if by this mapping analytic planar curves of the space GK_{N} are mapped into analytic planar curves of the space G\overline{K}_{N}.

Let us denote by P_{ij}^{h} = \overline{Γ}_{ij}^{h} − Γ_{ij}^{h} the deformation tensor of connection for analytic planar mapping, where Γ_{ij}^{h} and \overline{Γ}_{ij}^{h} are the second kind Cristophell’s symbols of the spaces GK_{N} and G\overline{K}_{N}, respectively.

Analytic planar curves of the spaces GK_{N} and G\overline{K}_{N} are given by relations

\frac{dλ^{h}}{dt} + Γ_{pq}^{h}λ^{p}λ^{q} = a(t)λ^{h} + b(t)F_{p}^{h}λ^{p}, \frac{dλ^{h}}{dt} + \overline{Γ}_{pq}^{h}λ^{p}λ^{q} = \overline{a}(t)λ^{h} + \overline{b}(t)F_{p}^{h}λ^{p},

respectively. From these relations we get (\overline{Γ}_{pq}^{h} − Γ_{pq}^{h})λ^{p}λ^{q} = ψ(t)λ^{h} + σ(t)F_{p}^{h}λ^{p}, where we denote ψ(t) = \overline{a}(t) − a(t), σ(t) = \overline{b}(t) − b(t). Putting ψ(t) = ψ_{p}λ^{p}, σ(t) = σ_{q}λ^{q}, we have (\overline{Γ}_{pq}^{h} − Γ_{pq}^{h} − ψ_{p}δ_{q}^{h} − σ_{p}F_{q}^{h})λ^{p}λ^{q} = 0, wherefrom it is \overline{Γ}_{ij}^{h} = Γ_{ij}^{h} + ψ_{(i}δ_{j)}^{h} + σ_{(i}F_{j)}^{h} + ξ_{ij}^{h}, where (ij) denotes a symmetrization without division by indices i, j and ξ_{ij}^{h} is an anti-symmetric tensor. The vector σ_{i} we can select so that σ_{i} = −ψ_{p}F_{i}^{p}. Then we have

\overline{Γ}_{ij}^{h} = Γ_{ij}^{h} + ψ_{(i}δ_{j)}^{h} − ψ_{p}F_{(i}^{p}F_{j)}^{h} + ξ_{ij}^{h}.  (3.2)

Contracting by indices h, i in (3.2) and using F_{p}^{p} = 0, ξ_{pj}^{p} = 0 we have

\overline{Γ}_{pj}^{p} − Γ_{pj}^{p} = (N + 2)ψ_{j}.  (3.3)

From (3.3) we can see that ψ_{j} is a gradient vector. Substituting from (3.3) into (3.2) we get H\overline{T} _{ij}^{h} = HT_{ij}^{h}, where we denote

HT_{ij}^{h} = Γh ; \underline{ij} − \frac{1}{N + 2}(Γ_{p(i}^{p}δ_{j)}^{h} − Γ_{qp}^{q}F_{(i}^{p}F_{j)}^{h}).  (3.4)

Here H\overline{T} _{ij}^{h} denotes an object of the form (3.4) in the space G\overline{K}_{N}. The magnitude HT_{ij}^{h} is not a tensor. We shall call it holomorphically projective parameter of the type of Thomas’s projective parameter. From the facts given above, we have

Theorem 3.1. Geometric objects (3.4) of the space GK_{N} are invariants of holomorphically projective mappings.

4. Equitorsion holomorphically projective mappings

Let f: GK_{N} → G\overline{K}_{N} be a holomorphically projective mapping, and let the torsion tensors Γh ; ij_{∨} and \overline{Γ}h ; ij_{∨} of the spaces GK_{N} and G\overline{K}_{N} satisfy the condition

\overline{Γ}h ; ij_{∨} = Γh ; ij_{∨}. In this case the mapping f is called an equitorsion holomorphically

projective mapping of the spaces GK_{N} and G\overline{K}_{N}. Then (3.2) implies ξ_{ij}^{h} = 0.

4.1.Holomorphically projective parameters of the first kind Curvature tensors of the first kind R_{1} and \overline{R}_{1} of the spaces GK_{N} and G\overline{K}_{N},

respectively, are connected by the relation [5]

\overline{R}_{1}_{jmn}^{i} = R_{1}_{jmn}^{i} + P_{jm|_{1}n}^{i} − P_{jn|_{1}m}^{i} + P_{jm}^{p}P_{pn}^{i} − P_{jn}^{p}P_{pm}^{i} + 2Γp ; mn_{∨}P_{jp}^{i},  (4.1)

where P_{ij}^{h} = \overline{Γ}_{ij}^{h} − Γ_{ij}^{h} is a deformation tensor. Substituting (3.3) and ξ_{ij}^{h} = 0 into (4.1) we get

\overline{R}_{1}_{jmn}^{i} = R_{1}_{jmn}^{i} + δ_{m}^{i} ψ_{1}_{jn} + δ_{j}^{i} ψ_{1}_{[ mn]} − δ_{j}^{i} ψ_{1}_{jm}

+ F_{j}^{p}(F_{n}^{i}ψ_{1}_{pm} − F_{m}^{i}ψ_{1}_{pn}) + F_{j}^{i}(F_{n}^{p}ψ_{1}_{pm} − F_{m}^{p}ψ_{1}_{pn})

+ 2Γi ; mn_{∨}ψ_{j} + 2Γp ; mn_{∨}ψ_{p}δ_{j}^{i} − 2Γp ; mn_{∨}ψ_{q}F_{j}^{q}F_{p}^{i} − 2Γp ; mn_{∨}ψ_{q}F_{p}^{q}F_{j}^{i}, (4.2)

where we denote ψ_{1}_{ij} = ψ_{i|_{1}j} − ψ_{i}ψ_{j} + ψ_{p}F_{i}^{p}ψ_{q}F_{j}^{q}. Contracting with respect to

indices i, n in (4.2) we obtain

\overline{R}_{1}_{jm} = R_{1}_{jm} + ψ_{1}_{[ mj ]} − Nψ_{1}_{jm} − F_{j}^{p}F_{m}^{q}ψ_{1}_{(pq)}
+ 2Γp ; mj_{∨}ψ_{p} − 2Γp ; mr_{∨}ψ_{q}F_{j}^{q}F_{p}^{r} − 2Γp ; mr_{∨}ψ_{q}F_{p}^{q}F_{j}^{r}.  (4.3)

Anti-symmetrization without division in (4.3) with respect to indices j, m yields

(N + 2)ψ_{1}_{[ jm]} = R_{1}_{[ jm]} − \overline{R}_{1}_{[ jm]} + 4Γp ; mj_{∨}ψ_{p} − 2Γp ; mr_{∨}ψ_{q}F_{j}^{q}F_{p}^{r}
+ 2Γp ; jr_{∨}ψ_{q}F_{m}^{q}F_{p}^{r} − 2Γp ; mr_{∨}ψ_{q}F_{p}^{q}F_{j}^{r} + 2Γp ; jr_{∨}ψ_{q}F_{p}^{q}F_{m}^{r}. (4.4)

Symmetrizing without division with respect to j, m in (4.3) we arrive at

\overline{R}_{1}_{(jm)} = R_{1}_{(jm)} − Nψ_{1}_{(jm)} − 2F_{j}^{p}F_{m}^{q}ψ_{1}_{(pq)} − 2Γp ; mr_{∨}ψ_{q}F_{j}^{q}F_{p}^{r}
− 2Γp ; jr_{∨}ψ_{q}F_{m}^{q}F_{p}^{r} − 2Γp ; mr_{∨}ψ_{q}F_{p}^{q}F_{j}^{r} − 2Γp ; jr_{∨}ψ_{q}F_{p}^{q}F_{m}^{r}.  (4.5)

By composition with F_{p}^{j}F_{q}^{m}, contraction with respect to j, m, and using the conditions (2.12a), we get from (4.5)

\overline{R}_{1}_{(jm)} = R_{1}_{(jm)} − Nψ_{1}_{(pq)}F_{j}^{p}F_{m}^{q} − 2ψ_{1}_{(jm)} + 2Γp ; qr_{∨}ψ_{j}F_{p}^{r}F_{m}^{q}
+ 2Γp ; qr_{∨}ψ_{m}F_{p}^{r}F_{j}^{q} + 2Γp ; rj_{∨}ψ_{q}F_{p}^{q}F_{m}^{r} + 2Γp ; rm_{∨}ψ_{m}F_{p}^{q}F_{j}^{r}.  (4.6)

Using (4.4,5,6) and (2.12a) we get

(N + 2)ψ_{1}_{jm} = R_{1}_{jm} − \overline{R}_{1}_{jm} + 2Γp ; mj_{∨}ψ_{p} − \frac{2N − 2}{N − 2}Γp ; mr_{∨}ψ_{q}F_{j}^{q}F_{p}^{r}

− \frac{2}{N − 2}Γp ; jr_{∨}ψ_{q}F_{m}^{q}F_{p}^{r} − \frac{2}{N − 2}Γp ; qr_{∨}ψ_{j}F_{p}^{r}F_{m}^{q}

− \frac{2}{N − 2}Γp ; qr_{∨}ψ_{m}F_{p}^{r}F_{j}^{q} − 2Γp ; mr_{∨}ψ_{q}F_{p}^{q}F_{j}^{r}.  (4.7)

Eliminating ψ_{i} by using the condition (3.3) we reduce the equation (4.7) to the form (N + 2)ψ_{1}_{jm} = R_{1}_{jm} − \overline{R}_{1}_{jm} + \overline{P}_{1}_{jm} − P_{1}_{jm}, where we denote

P_{1}_{jm} = \frac{2}{N + 2}(Γp ; mj_{∨}Γ_{qp}^{q} − \frac{N − 1}{N − 2}Γp ; mr_{∨}Γ_{sq}^{s}F_{j}^{q}F_{p}^{r} − \frac{1}{N − 2}Γp ; jr_{∨}Γ_{sq}^{s}F_{m}^{q}F_{p}^{r}

− \frac{1}{N − 2}Γp ; qr_{∨}Γ_{sj}^{s}F_{p}^{r}F_{m}^{q} − \frac{1}{N − 2}Γp ; qr_{∨}Γ_{sm}^{s}F_{p}^{r}F_{j}^{q} − Γp ; mr_{∨}Γ_{sq}^{s}F_{p}^{q}F_{j}^{r}).  (4.8)

In the same manner the geometric objects \overline{P}_{1}_{jm} of the space G\overline{K}_{N} is defined. Elim-

On holomorphically projective mappings … 201

inating ψ_{1}_{jm} from (4.2) we obtain HP \overline{W}_{1}^{i} _{jmn} = HP W_{1}^{i} _{jmn}, where the magnitude

HP W_{1}^{i} _{jmn} = R_{1}_{jmn}^{i} + \frac{1}{N + 2}[δ_{m}^{i} (R_{1}_{jn} − P_{1}_{jn}) + δ_{j}^{i} (R_{1}_{[ mn]} − P_{1}_{[ mn]})

− δ_{n}^{i} (R_{1}_{jm} − P_{1}_{jm}) + F_{j}^{p}F_{n}^{i}(R_{1}_{pm} − P_{1}_{pm}) − F_{j}^{p}F_{m}^{i}(R_{1}_{pn} − P_{1}_{pn})

+ F_{j}^{i}F_{n}^{p}(R_{1}_{pm} − P_{1}_{pm}) − F_{j}^{i}F_{m}^{p}(R_{1}_{pn} − P_{1}_{pn}) − 2Γi ; mn_{∨}Γ_{qj}^{q} − 2δ_{j}^{i} Γp ; mn_{∨}Γ_{qp}^{q}

+ 2Γp ; mn_{∨}Γ_{sq}^{s}F_{j}^{q}F_{p}^{i} + 2Γp ; mn_{∨}Γ_{sq}^{s}F_{p}^{q}F_{j}^{i}]  (4.9)

is expressed by geometric objects of the space GK_{N}. In the same manner the magnitude HP \overline{W}_{1}^{i} _{jmn} is expressed by geometric objects of the space G\overline{K}_{N}. The magnitude HP W_{1}^{i} _{jmn} is not a tensor, and we call it the equitorsion holomorphically projective parameter of the first kind of the space GK_{N}. From the facts given above, we have

Theorem 4.1. The equitorsion holomorphically projective parameter (4.9) is an invariant of equitorsion holomorphically projective mapping f: GK_{N} → G\overline{K}_{N}.

4.2. Holomorphically projective parameters of the second kind For the curvature tensors R_{2} and \overline{R}_{2} of the spaces GK_{N} and G\overline{K}_{N} the relation

\overline{R}_{2}_{jmn}^{i} = R_{2}_{jmn}^{i} + P_{mj |_{2}n}^{i} − P_{nj |_{2}m}^{i} + P_{mj}^{p}P_{np}^{i} − P_{nj}^{p}P_{mp}^{i} + 2Γp ; nm_{∨}P_{pj}^{i}  (4.10)

is valid [5], where P_{jm}^{i} is a deformation tensor.

Analogously to the previous case we obtain HP \overline{W}_{2}^{i} _{jmn} = HP W_{2}^{i} _{jmn}, where we denote

HP W_{2}^{i} _{jmn} = R_{2}_{jmn}^{i} + \frac{1}{N + 2}[δ_{m}^{i} (R_{2}_{jn} − P_{2}_{jn}) + δ_{j}^{i} (R_{2}_{[ mn]} − P_{2}_{[ mn]})

− δ_{n}^{i} (R_{2}_{jm} − P_{2}_{jm}) + F_{j}^{p}F_{n}^{i}(R_{2}_{pm} − P_{2}_{pm}) − F_{j}^{p}F_{m}^{i}(R_{2}_{pn} − P_{2}_{pn})

+ F_{j}^{i}F_{n}^{p}(R_{2}_{pm} − P_{2}_{pm}) − F_{j}^{i}F_{m}^{p}(R_{2}_{pn} − P_{2}_{pn}) − 2δ_{j}^{i} Γp ; nm_{∨}Γ_{qp}^{q} − 2Γi ; nm_{∨}Γ_{qj}^{q}

+ 2Γp ; nm_{∨}Γ_{sq}^{s}F_{p}^{q}F_{j}^{i} + 2Γp ; nm_{∨}Γ_{sq}^{s}F_{j}^{q}F_{p}^{i}],  (4.11)

P_{2}_{jm} = \frac{2}{N + 2}(Γp ; jm_{∨}Γ_{qp}^{q} − Γp ; rm_{∨}Γ_{sq}^{s}F_{p}^{q}F_{j}^{r} − \frac{N − 1}{N − 2}Γp ; rm_{∨}Γ_{sq}^{s}F_{j}^{q}F_{p}^{r}

− \frac{1}{N − 2}Γp ; rj_{∨}Γ_{sq}^{s}F_{m}^{q}F_{p}^{r} − \frac{1}{N − 2}Γp ; rq_{∨}Γ_{sj}^{s}F_{p}^{r}F_{m}^{q} − \frac{1}{N − 2}Γp ; rq_{∨}Γ_{sm}^{s}qF_{p}^{r}F_{j}^{q}).(4.12)

The magnitude HP W_{2}^{i} _{jmn} is not a tensor, and we call it the equitorsion holomorphically projective parameter of the second kind of the space GK_{N}. From the facts given above, we have

Theorem 4.2. The equitorsion holomorphically projective parameter of the second kind is an invariant of equitorsion holomorphically projective mapping of the spaces GK_{N} and G\overline{K}_{N}.

4.3. The case of Kählerian spaces In the case of holomorphically projective mappings of Kählerian spaces the

magnitudes HP W_{θ}^{i} _{jmn}, (θ = 1, 2), given by (4.9,11) reduce to the holomorphically

projective curvature tensor [8]

HP W^{i} _{jmn} = R_{jmn}^{i} + \frac{1}{N + 2}(R_{j[n}δ_{m]}^{i} + F_{j}^{p}R_{p[m}F_{n]}^{i} + 2F_{j}^{i}F_{n}^{p}R_{pm}).

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(received 23.12.2002)

Prirodno matematički fakultet, Višegradska 33, 18000 Niš, Yugoslavia E-mail: stmica@ptt.yu, vljubica@pmf.pmf.ni.ac.yu