����������� ������ 50 (1998), 57–64

UDK 514.763.4

���������� ������ ��� research paper

ON A CURVATURE TENSOR OF KÄHLER TYPE IN AN ALMOST HERMITIAN AND ALMOST PARA-HERMITIAN MANIFOLD

Mileva Prvanović

Abstract. We find a tensor of Kähler type for an almost Hermitian and almost para-Hermitian manifold. We show that this tenosr is closely related with the problem of almost Hermitian and almost para-Hermitian manifold with pointwise constant holomorphic sectional curvature.

1. Preliminaries

Let (M, J, g) be an n (= 2m)-dimensional Riemannian manifold endowed with endomorphism J satisfying

J^{2} = εI,  g(Jx, Jy) = −εg(x, y),  (1.1)

where g is the metric of the manifold, ε = ±1, I indicates the identity mapping, x, y ∈ T_{p} and T_{p} is the tangent space to M at p ∈ M. If ε = −1, (M, J, g) is an almost Hermitian manifold. If ε = +1, (M, J, g) is an almost para-Hermitian manifold [1]. In both cases

F (x, y) = g(Jx, y)  (1.2)

satisfies

F (x, y) = −F (y, x).  (1.3)

It is worth to mention that an almost para-Hermitian manifold is a semi-Riemannian manifold of signature (m, m).

In both cases, the vectors x and Jx are mutually orthogonal. This, in the case ε = −1 implies that x and Jx are linearly independent. In the case of almost para-Hermitian manifold, there exist vectors satisfying Jx = x or Jx = −x. Such vector is a null vector and each null vector x satisfies Jx = x or Jx = −x. For non-null vector x, x and Jx are linearly independent, too.

AMS Subject Classification: 53 C 25, 53 C 55

Keywords and phrases: almost Hermitian manifold, almost para-Hermitian manifold, curvature tensor

2. Curvature tensor of Kähler type

We denote by ∇ and R the Riemannian connection and the curvature tensor of (M, J, g), respectively. An almost Hermitian (almost para-Hermitian) manifold satisfying ∇J = 0 is a Kähler (para-Kähler) manifold. It is well known that the Riemannian curvature tensor of a Kähler (para-Kähler) manifold satisfies the condition

R(x, y, z, w) = −εR(x, y, Jz, Jw).  (2.1)

Thus, we say that, in an almost Hermitian (almost para-Hemritian) manifold, the tensor Q(x, y, z, w) is a curvature tensor of Kähler type, if it satisfies the conditions

Q(x, y, z, w) = −Q(x, y, w, z),  (2.2)
Q(x, y, z, w) = −Q(y, x, z, w),  (2.3)
Q(x, y, z, w) = Q(z, w, x, y),  (2.4)
Q(x, y, z, w) + Q(x, z, w, y) + Q(x, w, y, z) = 0,  (2.5)
Q(x, y, z, w) = −εQ(x, y, Jz, Jw).  (2.6)

For a general almost Hermitian (almost para-Hemritian) manifold, the Riemannian curvature tensor R(x, y, z, w) does not satisfy (2.6). Thus, we shall try to construct a new tensor satisfying (2.2)–(2.6). The construction that follows is inspired by the paper [4].

We start with the tensor

H(x, y, z, w) = = 4[R(x, y, z, w) − εR(x, y, Jz, Jw) − εR(Jx, Jy, z, w) + R(Jx, Jy, Jz, Jw)]. (2.7)

It is easy to see that this tensor satisfies identities (2.2), (2.3), (2.4) and (2.6). If it satisfies (2.5), too, it must be

R(x, y, Jz, Jw) + R(Jx, Jy, z, w) + R(x, z, Jw, Jy)+ + R(Jx, Jz, w, y) + R(x, w, Jy, Jz) + R(Jx, Jw, y, z) = 0. (2.8)

Putting into (2.8) Jx and Jy instead of x and y, respectively, and taking into account (1.1), we get

R(Jx, Jy, Jz, Jw) + R(x, y, z, w) + εR(Jx, z, Jw, y)+ + εR(x, Jz, w, Jy) + εR(Jx, w, y, Jz) + εR(x, Jw, Jy, z) = 0. (2.9)

Subtracting (2.8) (multiplied by ε) from (2.9), we find

R(x, y, z, w) − εR(x, y, Jz, Jw) − εR(Jx, Jy, z, w) + R(Jx, Jy, Jz, Jw) =

= ε[R(Jx, Jz, w, y) + R(x, w, Jy, Jz) + R(x, z, Jw, Jy) + R(Jx, Jw, y, z) − R(Jx, z, Jw, y) − R(x, Jz, w, Jy) − R(Jx, w, y, Jz) − R(x, Jw, Jy, z)].

Substituting this into (2.7) we obtain H(x, y, z, w) = = 3[R(x, y, z, w) − εR(x, y, Jz, Jw) − εR(Jx, Jy, z, w) + R(Jx, Jy, Jz, Jw)] + ε[R(x, z, Jw, Jy) + R(Jx, Jz, w, y) + R(x, w, Jy, Jz) + R(Jx, Jw, y, z)

− R(Jx, z, Jw, y) − R(x, Jz, w, Jy) − R(Jx, w, y, Jz) − R(x, Jw, Jy, z)] (2.10)

We can verify, by direct computation, that the tensor (2.10) satisfies all identities (2.2)–(2.6), that is, it is a curvature tensor of Kähler type.

If (M, J, g) is a Kähler (para-Kähler) manifold, then (2.1) holds good, because of which we have also

\begin{aligned}R(x, y, z, w) & = −εR(Jx, Jy, z, w), \\ R(x, y, z, w) & = R(Jx, Jy, Jz, Jw), \\ R(x, Jy, Jz, w) & = −R(x, Jy, z, Jw), \\ R(x, Jy, Jz, w) & = −R(Jx, y, Jz, w). \end{aligned}  (2.11)

Therefore, (2.10) reduces to

H(x, y, z, w) = 14R(x, y, z, w) − 2ε[R(x, Jz, w, Jy) + R(x, Jw, Jy, z)]. (2.12)

But, in view of Bianchi identity and (2.11),

R(x, Jz, w, Jy) + R(x, Jw, Jy, z) =

= −R(x, w, Jy, Jz) − R(x, Jy, Jz, w) − R(x, Jy, z, Jw) − R(x, z, Jw, Jy)

= ε[R(x, w, y, z) + R(x, z, w, y)] − R(x, Jy, Jz, w) + R(x, Jy, Jz, w)
= −εR(x, y, z, w).

This means that (2.12) reduces to H(x, y, z, w) = 16R(x, y, z, w). Thus, we can state

Theorem 1. For an almost Hermitian (almost para-Hermitian) manifold (M, J, g), the tensor

A(x, y, z, w) =

= \frac{1}{16}\{3[R(x, y, z, w) − εR(x, y, Jz, Jw) − εR(Jx, Jy, z, w) + R(Jx, Jy, Jz, Jw)]

+ ε[R(x, z, Jw, Jy) + R(Jx, Jz, w, y) + R(x, w, Jy, Jz) + R(Jx, Jw, y, z) − R(Jx, z, Jw, y) − R(x, Jz, w, Jy) − R(Jx, w, y, Jz) − R(x, Jw, Jy, z)]} (2.13)

is a curvature tensor of Kähler type. If (M, J, g) is a Kähler (para-Kähler) manifold, (2.13) reduces to the Riemannian curvature tensor.

We note that, beside (2.2)–(2.6) the tensor (2.13) satisfies the identities of type (2.11), too, i.e.

\begin{aligned}A(x, y, z, w) & = −εA(Jx, Jy, z, w), \\ A(x, y, z, w) & = A(Jx, Jy, Jz, Jw), \\ A(x, Jy, Jz, w) & = −A(x, Jy, z, Jw), \\ A(x, Jy, Jz, w) & = −A(Jx, y, Jz, w). \end{aligned}  (2.14)

3. The Ricci tensor with respect to the tensor (2.13)

Since we have R(x, y, z, w) = g(R(z, w)y, x) = −εg(JR(z, w)y, Jx), we have also

R(Jx, Jy, z, w) = −g(JR(z, w)Jy, x),
R(Jx, Jy, Jz, Jw) = −g(JR(Jz, Jw)Jy, x),
R(Jx, z, Jw, y) = −g(JR(Jw, y)z, x),
R(Jx, w, y, Jz) = −g(JR(y, Jz)w, x).

Therefore, if the tensor A(z, w)y is defined by A(x, y, z, w) = g(A(z, w)y, x), we can rewrite (2.13) in the form

A(z, w)y = \frac{1}{16}\{3[R(z, w)y − εR(Jz, Jw)y + εJ(R(z, w)Jy) − J(R(Jz, Jw)Jy)]

+ ε[R(Jw, Jy)z − J(R(w, y)Jz) + R(Jy, Jz)w − J(R(y, z)Jw)

+ J(R(Jw, y)z) − R(w, Jy)Jz + J(R(y, Jz)w) − R(Jy, z)Jw]}. (3.1)

We recall that the Ricci tensor ρ(x, y) and the Hermitian Ricci tensor ρ^{∗}(x, y) of (M, J, g) are defined by

ρ(y, w) = trace: z → R(z, w)y = trace: z → εJ(R(Jz, w)y),
ρ^{∗}(y, w) = trace: z → R(Jz, w)y = trace: z → J(R(z, w)y),

and are symmetric and skew-symmetric respectively, i.e.

ρ(y, w) = ρ(w, y), ρ^{∗}(y, w) = −ρ^{∗}(w, y). (3.2)

Now, if we define A(y, w) by A(y, w) = trace: z → A(z, w)y, then we get from (3.1):

A(y, w) = \frac{1}{16}\{3[ρ(y, w) − ερ^{∗}(y, Jw) − ερ^{∗}(w, Jy) − ερ(Jy, Jw)]

+ ε[2ρ(Jw, Jy) − ρ(Jy, Jw) − 2ερ(w, y) + ερ(y, w)

− 2ρ^{∗}(w, Jy) + 2ρ^{∗}(Jw, y) − ρ^{∗}(y, Jw) + ρ^{∗}(Jy, w)]},

i.e.

A(y, w) = \frac{1}{8}[ρ(y, w) − 3ερ^{∗}(y, Jw) − 3ερ^{∗}(w, Jy) − ερ(Jy, Jw)].  (3.3)

If (M, J, g) is a Kähler (para-Kähler) manifold, then ρ(y, w) = −ερ(Jy, Jw) = −ερ^{∗}(y, Jw) = −ερ^{∗}(w, Jy), and (3.3) reduces to A(y, w) = ρ(y, w). Thus, we have

Theorem 2. The Ricci tensor with respect to the tensor A(x, y, z, w) is given by (3.3), and if (M, J, g) is a Kähler (para-Kähler) manifold, it reduces to the Ricci tensor ρ(y, w).

It is easy to see that the tensor A(y, w) has the properties like the Ricci tensor ρ(y, w) of a Kähler (para-Kähler) manifold, namely

A(x, y) = A(y, x), A(x, y) = −εA(Jx, Jy), A(x, Jy) = −A(Jx, y),

A(x, y) = −εA^{∗}(x, Jy), A^{∗}(x, y) = −A^{∗}(y, x),

where A^{∗}(x, y) = trace: z → A(Jz, y)x.

We define the scalar curvature with respect to the curvature tensor A(x, y, z, w) by

A = trace: w → A(w),  (3.4)

where

A(y, w) = g(y, A(w)).  (3.5)

To obtain the expression for this scalar curvature, we consider the vectors ρ(w) and ρ^{∗}(w) defined by

\begin{aligned}ρ(y, w) & = g(y, ρ(w)) = −εg(Jy, Jρ(w)), \\ ρ^{∗}(y, w) & = g(y, ρ^{∗}(w)) = −εg(Jy, Jρ^{∗}(w)), \end{aligned}  (3.6)

and note that the scalar curvature τ and the Hermitian scalar curvature τ^{∗} can be defined by

τ = trace: w → ρ(w) = trace: w → εJρ(Jw),
τ^{∗} = trace: w → ρ^{∗}(w) = trace: w → Jρ^{∗}(w).

(3.7)

Using (3.5) and (3.6), we can rewrite (3.3) in the form

A(w) = \frac{1}{8}[ρ(w) − 3ερ^{∗}(Jw) − 3εJρ^{∗}(w) + εJρ(Jw)],

from which, in view of (3.4) and (3.7), we get

A = \frac{1}{4}(τ − 3ετ^{∗}).  (3.8)

If (M, J, g) is a Kähler (para-Kähler) manifold, then τ^{∗} = −ετ and (3.8) reduces to A = τ. Thus, we can state

Theorem 3. The scalar curvature with respect to the tensor A(x, y, z, w) is given by (3.8). If (M, J, g) is a Kähler (para-Kähler) manifold, it reduces to the scalar curvature τ with respect to the Riemannian curvature tensor.

4. Almost Hermitian and almost para-Hermitian manifold with pointwise constant holomorphic sectional curvature

As is well known, the holomprphic sectional curvature of (M, J, g) at p ∈ M, determined by non-null vector x ∈ T_{p}, is given by

\frac{R(x, Jx, x, Jx)}{g(x, x)g(Jx, Jx) − g(x, Jx)g(x, Jx)} = −c(p),

i.e. by

\frac{R(x, Jx, x, Jx)}{−εg(x, x)g(x, x)} = −c(p)  (4.1)

because of (1.1) and the orthogonality of vectors x and Jx.

If the scalar c(p) is independent of the choice of vector x ∈ T_{p}, then (M, J, g) is said to be a manifold of pointwise constant holomorphic sectional curvature at p ∈ M. The almost Hermitian manifolds of pointwise constant holomorphic sectional curvature have been studied by many authors ([2], [5], [6], [7], [8]). The purpose of this section is to show that some of these questions (parallely for almost Hermitian and almost para-Hermitian manifolds) can be considered with the help of the tensor (2.13).

We define the sectional curvature at the point p ∈ M, with respect to the tensor A(x, y, z, w), determined by non-null vectors x, y ∈ T_{p}, as follows

\frac{A(x, y, x, y)}{g(x, x)g(y, y) − g(x, y)g(x, y)} = −c(p).

If y = Jx, we have the holomorphic sectional curvature with respect to A(x, y, z, w):

\frac{A(x, Jx, x, Jx)}{−εg(x, x)g(x, x)} = −c(p).  (4.2)

On the other hand, putting in (2.13) y = Jx, z = x, w = Jx, we find A(x, Jx, x, Jx) = R(x, Jx, x, Jx). Thus, (M, J, g) is a manifold of pointwise constant holomorphic sectional curvature if and only if it has this property with respect to the tensor A(x, y, z, w), too. But, A(x, y, z, w) is a curvature tensor of Kähler type and we can proceed with it in the same manner as with the Riemannian curvature tensor in the case of Kähler space, e.g. [9], p. 75 or [3], pp. 165–168. Namely, we rewrite (4.2) in the form

A(x, Jx, x, Jx) − εc(p)g(x, x)g(x, x) = 0,  (4.3)

and consider the quadrilinear mapping which sends (x, y, z, w) ∈ T_{p} ⊗ T_{p} ⊗ T_{p} ⊗ T_{p} into

A(Jx, y, Jz, w) + A(Jy, z, Jx, w) + A(Jz, x, Jy, w)

− εc(p)[g(x, y)g(z, w) + g(y, z)g(x, w) + g(x, z)g(y, w)]. (4.4)

This mapping is symmetric in x, y, z and w because A(x, y, z, w) satisfies the identities (2.2), (2.3), (2.4) and (2.6). For x = y = z = w, (4.4) reduces to 3A(Jx, x, Jx, x) − 3εc(p)g(x, x)g(x, x). But this vanishes by the assumption (4.3). Thus, the mapping (4.4) vanishes identically, i.e.

A(Jx, y, Jz, w) + A(Jy, z, Jx, w) + A(Jz, x, Jy, w) = = εc(p)[g(x, y)g(z, w) + g(y, z)g(x, w) + g(x, z)g(y, w)]. (4.5)

Putting into (4.5) Jx and Jy instead of x and y respectively, and taking into account (1.1) and (2.14), we have

A(x, y, z, w) − A(y, z, x, w) + εA(z, Jx, Jy, w) = = εc(p)[g(Jx, y)g(Jz, w) + g(y, Jz)g(Jx, w) − εg(x, z)g(y, w)].

Taking the skew-symmetric part of this equation with respect to x and y, we obtain

2A(x, y, z, w)−A(y, z, x, w)+A(x, z, y, w)+ε[A(z, Jx, Jy, w)−A(z, Jy, Jx, w)] =

= εc(p){2g(Jx, y)g(Jz, w) + g(y, Jz)g(Jx, w) − g(x, Jz)g(Jy, w) − ε[g(x, z)g(y, w) − g(x, w)g(y, z)]}. (4.6)

In view of (2.2)–(2.6), we have −A(y, z, x, w) + A(x, z, y, w) = A(x, y, z, w) and

ε[A(z, Jx, Jy, w) − A(z, Jy, Jx, w)] = −εA(z, w, Jx, Jy) = A(x, y, z, w).

Therefore, and taking into account (1.2), we can rewrite (4.6) as follows:

A(x, y, z, w) = \frac{c(p)}{4}\{g(x, w)g(y, z) − g(x, z)g(y, w)

+ ε[−F (x, w)F (y, z) + F (x, z)F (y, w) + 2F (x, y)F (z, w)]}. (4.7)

Thus, we can state

Theorem 4. An almost Hermitian (almost para-Hermitian) manifold is a manifold with a pointwise constant holomorhic sectional curvature if and only if the tensor A(x, y, z, w) has the form (4.7).

Remark. We obtain the same result dealing directly with (4.1). Namely, we rewrite (4.1) in the form

R(x, Jx, x, Jx) − εc(p)g(x, x)g(x, x) = 0  (4.8)

and consider the quadrilinear mapping which sends (x, y, z, w) ∈ T_{p} ⊗ T_{p} ⊗ T_{p} ⊗ T_{p} into

\frac{1}{16}\{R(Jx, y, Jw, z) + R(Jx, z, Jw, y) + R(Jx, w, Jz, y) + R(Jx, y, Jz, w)

+ R(Jx, z, Jy, w) + R(Jx, w, Jy, z) + R(Jy, w, Jz, x) + R(Jy, x, Jz, w) + R(Jy, x, Jw, z) + R(Jy, z, Jw, x) + R(Jw, y, Jz, x) + R(Jw, x, Jz, y)

− 4εc(p)[g(x, y)g(w, z) + g(x, w)g(z, y) + g(x, z)g(y, w)]}.

This mapping is symmetric in x, y, z and w and since it vanishes for x = y = z = w by the assumption (4.8), it must vanish identically, i.e.

\frac{1}{16}[R(Jx, y, Jw, z) + R(Jx, z, Jw, y) + R(Jx, w, Jz, y) + R(Jx, y, Jz, w)

+ R(Jx, z, Jy, w) + R(Jx, w, Jy, z) + R(Jy, w, Jz, x) + R(Jy, x, Jz, w) + R(Jy, x, Jw, z) + R(Jy, z, Jw, x) + R(Jw, y, Jz, x) + R(Jw, x, Jz, y)] =

= ε\frac{c(p)}{4}[g(x, y)g(w, z) + g(x, w)g(z, y) + g(x, z)g(y, w)]. (4.9)

Now, proceeding with (4.9) in a similar manner as with (4.5), we get (4.7).

5. Sato’s form of the tensor A(x, y, z, w)

Sato proved ([5], Theorem 4.2) that the curvature tensor of an almost Hermitian manifold of pointwise constant holomorphic sectional curvature c(p) is given by

R(x, y, z, w) = \frac{c(p)}{4}[g(x, w)g(y, z) − g(x, z)g(y, w)

+ F (x, w)F (y, z) − F (x, z)F (y, w) − 2F (x, y)F (z, w)]

= \frac{1}{96}\{26[G(x, y, z, w) − G(z, w, x, y)] − 6[G(Jx, Jy, Jz, Jw) + G(Jz, Jw, Jx, Jy)]

+ 13[G(x, z, y, w) + G(y, w, x, z) − G(x, w, y, z) − G(y, z, x, w)]

−3[G(Jx, Jz, Jy, Jw)+G(Jy, Jw, Jx, Jz)−G(Jx, Jw, Jy, Jz)−G(Jy, Jz, Jx, Jw)]

+ 4[G(x, Jy, z, Jw) + G(Jx, y, Jz, w)] + 2[G(x, Jz, y, Jw) + G(Jx, z, Jy, w) − G(x, Jw, y, Jz) − G(Jx, w, Jy, z)]}, where G(x, y, z, w) = R(x, y, z, w) − R(x, y, Jz, Jw).

To comprise the almost para-Hermitian manifold, too, we shall consider the tensor

R(x, y, z, w) = \frac{1}{96}\{26[G(x, y, z, w) − G(z, w, x, y)] − 6[G(Jx, Jy, Jz, Jw)

+ G(Jz, Jw, Jx, Jy)] + 13[G(x, z, y, w) + G(y, w, x, z) − G(x, w, y, z) − G(y, z, x, w)] −3[G(Jx, Jz, Jy, Jw)+G(Jy, Jw, Jx, Jz)−G(Jx, Jw, Jy, Jz)−G(Jy, Jz, Jx, Jw)]

− 4ε[G(x, Jy, z, Jw) + G(Jx, y, Jz, w)] − 2ε[G(x, Jz, y, Jw)

+ G(Jx, z, Jy, w) − G(x, Jw, y, Jz) − G(Jx, w, Jy, z)]}, (5.1)

where

G(x, y, z, w) = R(x, y, z, w) + εR(x, y, Jz, Jw).  (5.2)

Substituting (5.2) into (5.1), we obtain, after some simple but long computation, the expression on right-hand side of the relation (2.13). Thus, we can say that (5.1) is Sato’s form of the tensor A(x, y, z, w). Also, we see that, for an almost Hermitian manifold, Theorem 4 coincides with mentioned Sato’s theorem.

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(received 16.04.1998.)

Ljube Stojanovića 14, 11000 Beograd, Yugoslavia