DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 65–72

SOME CONNECTION CLASS ON A HYPERBOLIC KAEHLERIAN SPACE

NEVENA PUŠIĆ

Abstract. We consider a special semi-symmetric connection on a hyperbolic Kaehlerian space. We find conditions for such a connection to be of zero curvature; then, we find conditions for the curvature tensor of such a connection to be of special form.

1. Introduction

A hyperbolic Kaehlerian space is an n(= 2m)-dimensional pseudo-Riemannian space with metric tensor (g_{ij}) and non-degenerate structure tensor (F_{j} ^{i}) satisfying

F_{i}^{j}F_{j}^{k} = δ_{i}^{k} (1)

F_{ik} = g_{ka}F_{i}^{a} = −F_{ki} (2)

∇^{∘}_{k} F_{j}^{i} = 0 (3)

for the Levi-Civita connection. Formally, a hyperbolic Kaehlerian space looks very similar to a Kaehlerian space (we even call a Kaehlerian space an elliptic Kaehlerian space), but geometrically it is different.

We call a connection on a hyperbolic Kaehlerian space a semi-symmetric connection of the second kind if its torsion tensor is of the form

T_{jk}^{i} = δ_{j}^{i}p_{k} − δ_{k}^{i}p_{j} + F_{j}^{i}q_{k} − F_{k}^{i}q_{j} − 2F_{kj}q^{i} (4)

for some vector fields p, q, if it is a metric connection i. e. if there holds

∇_{k}g_{ij} = 0 (5) and if it is an F -connection, that is

1991 Mathematics Subject Classification. 53B35, 53B60.

Key words and phrases. Special semi-symmetric connection, hyperbolic Kaehlerian space.

This paper is in final form and no version of it will be submitted for publication elsewhere.

∇_{k}F_{j}^{i} = 0. (6)

It is easy to construct coefficients of such a connection with given components of torsion tensor. For such a connection,

Γ_{ki}^{a} =Γ^{∘} _{ki}^{a} + δ_{k}^{a}p_{i} − g_{ki}p^{a} + F_{k}^{a}q_{i} − F_{ki}q^{a}. (7)

The connection with coefficient (1. 7) is authomatically a metric connection and it is an F -connection if and only if

q_{i} = F_{i}^{a}p_{a}. (8)

The connection with coefficients (1. 7) is called a special semi-symmetric connection if p_{i} = \frac{∂p}{∂x^{i}} .

Yano and Imai [3] constructed a connection satisfying conditions (1. 4)-(1.

6) on a Kaehlerian space. Its coefficients had the form (1. 7), but it satisfied q_{i} = −F_{i}^{a}p_{a}. They proved that, in a Kaehlerian space, there does not exist such a connection of zero curvature. They investigated some special forms of curvature tensor of this connection and covariant derivatives of the torsion tensor.

We are going to prove here that on a hyperbolic Kaehlerian space there exists a special semi-symmetric connection of zero curvature (it is a geometric difference) and also to do the rest calculations which Yano and Imai have done for Kaehlerian spaces.

2. The form of curvature tensor

The Riemann curvature tensor of the special semi-symmetric metric connection on a hyperbolic Kaehlerian space has the form

R_{kjih} = K_{kjih} − g_{kh}p_{ji} + g_{jh}p_{ki} − g_{ji}p_{kh} + g_{ki}p_{jh} (9)

−F_{kh}q_{ji} + F_{jh}q_{ki} − F_{ji}q_{kh} + F_{ki}q_{jh} + F_{kj}β_{hi}

where

p_{ji} =∇^{∘}_{j} p_{i} − p_{j}p_{i} − q_{j}q_{i} + \frac{1}{2}g_{ji}p_{a}p^{a} (10)

q_{ji} =∇^{∘}_{j} q_{i} − p_{j}q_{i} − q_{j}p_{i} − \frac{1}{2}F_{ji}p_{a}p^{a} (11)

β_{hi} = 2(p_{h}q_{i} − q_{h}p_{i}). (12)

It is easy to notice that

q_{ji} = p_{ja}F_{i}^{a}.

It is obvious that the tensor (p_{ji}) is symmetric and the tensor (β_{hi}) is skewsymmetric.

SOME CONNECTION CLASS ON A HYPERBOLIC KAEHLERIAN SPACE 67

3. The case of vanishing curvature

If the considered connection on a hyperbolic Kaehlerian space is of zero curvature, then the Riemann-Christoffel tensor of the basic connection gets the form K_{kjih} = g_{kh}p_{ji} − g_{jh}p_{ki} + g_{ji}p_{kh} − g_{ki}p_{jh} (13)

+F_{kh}q_{ji} − F_{jh}q_{ki} + F_{ji}q_{kh} − F_{ki}q_{jh} − F_{kj}β_{hi}.

It is obvious that K_{kjih} is skew-symmetric in indices i, h and k, j. For K_{kjih} there should also hold K_{kjih} = K_{ihkj}. (14) According to (3. 1), we have F_{kh}(q_{ji} + q_{ij}) − F_{jh}(q_{ki} + q_{ik}) + F_{ji}(q_{kh} + q_{hk}) (15)

−F_{ki}(q_{jh} + q_{hj}) − F_{kj}β_{hi} + F_{ih}β_{jk}.

Transvecting (3. 3) by F^{hk}, we obtain

(n − 2)(q_{ji} + q_{ij}) = 0,

and, consequently, q_{ji} = −q_{ij}. (16) Besides, by (3. 3) there holds

F_{kj}β_{hi} − F_{ih}β_{jk} = 0.

Taking into account

β_{jk}F^{jk} = 4p_{a}p^{a}

and transvecting the equation above by F^{jk}, we obtain

β_{si} = \frac{4}{n}p_{a}p^{a}F_{is},

that is

p_{s}q_{i} − q_{s}p_{i} = \frac{2}{n}p_{a}p^{a}F_{is}. (17)

Transvecting (3. 5) by p^{s}, we obtain

p_{a}p^{a}q_{i} = \frac{2}{n}p_{a}p^{a}q_{i}.

as n ≥ 4, then p_{a}p^{a} = 0, (18)

p_{s}q_{i} = q_{s}p_{i} (19)

p_{s}p_{j} = q_{s}q_{j}. (20)

In such a case, the gradient vector field (p_{i}) is an isotropic one.

We have just proved

Theorem 1. If a special semi-symmetric connection (1. 7) on a hyperbolic Kaehlerian space is of zero curvature, then the vector field (p_{i}) is isotropic and the Rie mann-Christoffel tensor of the Levi-Civita connection is of the form

K_{kjih} = g_{kh}p_{ji} − g_{jh}p_{ki} + g_{ji}p_{kh} − g_{ki}p_{jh} + F_{kh}q_{ji} − F_{jh}q_{ki}
+F_{ji}q_{kh} − F_{ki}q_{jh},

where

p_{ji} =∇^{∘}_{j} p_{i} − 2p_{j}p_{i},

q_{ji} =∇^{∘}_{j} q_{i} − 2p_{j}q_{i}.

Suppose, now, that

∇_{k}T_{ji}^{h} = b_{k}T_{ji}^{h} + b̃_{k}(F_{a}^{h}T_{ji}^{a} − F_{ji}p^{h}) (21)

for some 1-forms (b_{k}) and (b̃_{k}). Then we can obtain

∇_{k}p_{j} = b_{k}p_{j} + b̃_{k}q_{j}.

But,

∇_{k}p_{j} = p_{kj} + \frac{1}{2}p_{a}p^{a}g_{kj},

and, as the vector field (p_{a}) is isotropic,

∇_{k}p_{j} = p_{kj}. (22)

Contracting the component of the curvature tensor by g^{kh}, we obtain

K_{ji} = np_{ji} + g_{ji} ∇^{∘}_{a} p^{a}

and

K = 2n ∇^{∘}_{a} p^{a},

as p_{a}^{a} =∇^{∘}_{a} p^{a}, by the fact of isotropy of (p_{a}). Then

p_{ji} = \frac{1}{n}(K_{ji}− ∇^{∘}_{a} p^{a}g_{ji}) = \frac{1}{n}(K_{ji} − (b_{a}p^{a} + b̃_{a}q^{a})g_{ji}) (23)

q_{ji} = \frac{1}{n}(K_{ja}F_{i}^{a} + (b_{a}p^{a} + b̃_{a}q^{a})F_{ji}) (24)

and the curvature tensor of the Levi-Civita connection takes the form

K_{kjih} = \frac{1}{n}[g_{kh}K_{ji} − g_{jh}K_{ki} + g_{ji}K_{kh} − g_{ki}K_{jh} (25)

+F_{kh}S_{ji} − F_{jh}S_{ki} + F_{ji}S_{kh} − F_{ki}S_{jh}
−2(b_{a}p^{a} + b̃_{a}q^{a})(g_{kh}g_{ji} − g_{jh}g_{ki} − F_{kh}F_{ji} + F_{jh}F_{ki})]

where S_{ji} stands for K_{ja}F_{i}^{a}.

There arises the question of existence of such a construction. In the hyperbolic Kaehlerian space, for an isotropic vector field there holds

p_{i} = u_{i} + v_{i}, (26)

in the adapted coordinate system, where u = u_{i}l^{i}, l^{i} being m = \frac{n}{2} eigenvectors for the structure and eigenvalue 1 and v = v_{j}λ^{j}, λ^{j} being m = \frac{n}{2} eigenvectors and eigenvalue −1. l^{i}, λ^{j} form an adapted basis of the hyperbolic Kaehlerian space. Then p_{α} = u_{α} (α = 1, …, m) and pα̂ = vα̂ (α̂ = α + m = m + 1, …, n) as uα̂ = v_{α} = 0. Then

q_{i} = u_{i} − v_{i} (27)

by the obvious reason. u and v are mutually orthogonal because p and q are mutually orthogonal. Then

p_{αβ} =∇^{∘}_{α} u_{β} − 2u_{α}u_{β} = b_{α}u_{β} + b̃_{α}u_{β}

q_{αβ} =∇^{∘}_{α} v_{β} − 2u_{α}v_{β} = b_{α}v_{β} + b̃_{α}v_{β}

and

b_{α}p_{β} + b̃_{α}q_{β} = 0
p_{α̂β̂} = bα̂vβ̂ − b̃α̂vβ̂

q_{α̂β̂} = −bα̂vβ̂ + b̃α̂vβ̂

bα̂pβ̂ + b̃α̂qβ̂ = 0.

In the similar way, we can show that

b_{α}pβ̂ + b̃_{α}qβ̂ = 0

and, consequntly, there holds

b_{i}p_{j} + b̃_{i}q_{j} = 0 (28)

in any coordinate system. We proved

Theorem 2. A hyperbolic Kaehlerian space admits a special semi-symmetric connection of zero curvature satisfying (3. 9) for an isotropic vector field p_{i} if and only if (3. 16) holds. Then, by (3. 10), K_{kjih} = 0 and such a space is flat.

70 NEVENA PUŠIĆ

4. The case R_{kjih} = α_{kj}F_{ih}

If the curvature tensor of a special semi-symmetric metric connection is of the form α_{kj}F_{ih} for some skew-symmetric tensor α_{kj}, then the Riemann-Christoffel tensor of the Levi-Civita connection is of the form K_{kjih} = g_{kh}p_{ji} − g_{jh}p_{ki} + g_{ji}p_{kh} − g_{ki}p_{jh} (29)

+F_{kh}q_{ji} − F_{jh}q_{ki} + F_{ji}q_{kh} − F_{ki}q_{jh} − F_{kj}β_{hi} − α_{kj}F_{ki}

Then, as there holds

K_{kjih} = K_{ihkj},

we obtain F_{kh}(q_{ij} + q_{ji}) − F_{jh}(q_{ki} + q_{ik}) + F_{ji}(q_{kh} + q_{hk}) (30)

−F_{ki}(q_{jh} + q_{hj}) − F_{kj}(β_{hi} − α_{hi}) + F_{ih}(β_{jk} − α_{jk}) = 0.

Transvecting (4. 2) by F^{hk}, we obtain

(n − 2)(q_{ij} + q_{ji}) = 0,

and, consequently, q_{ij} + q_{ji} = 0. (31) By (4. 3),we can see from (4. 2) that

F_{kj}(α_{hi} − β_{hi}) − F_{ih}(α_{jk} − β_{jk}) = 0.

Transvecting the equation above by F^{jk}, we obtain

α_{hi} = β_{hi} − \frac{1}{n}(α − 4p_{a}p^{a})F_{hi}, (32)

where α stands for α_{jk}F^{jk}. By the first Bianchi identity, there holds

F_{kh}(2q_{ji} + α_{ji}) − F_{jh}(2q_{ki} + α_{ki}) + F_{ih}(2q_{kj} + α_{kj})

+F_{ji}(2q_{kh} + β_{kh}) − F_{ki}(2q_{jh} + β_{jh}) + F_{kj}(2q_{ih} + β_{ih}) = 0.

If we transvect the equation above by F^{kh}, then

(n − 4)(2q_{ji} + α_{ji}) = 2(p_{a}^{a} + 2p_{a}p^{a})F_{ji}.

As

p_{a}^{a} = p_{ai}g^{ia} =∇^{∘}_{a} p^{a} + \frac{n}{2}p_{a}p^{a},

then

α_{ji} = −2q_{ji} + \frac{2}{n − 4}(∇^{∘}_{a} p^{a} + \frac{n + 4}{2}p_{a}p^{a})F_{ji}. (33)

After transvection by F^{ji}, we obtain

α = α_{ji}F^{ji} = −\frac{2}{n − 4}[2(n − 2) ∇^{∘}_{a} p^{a} + n^{2}p_{a}p^{a}]

SOME CONNECTION CLASS ON A HYPERBOLIC KAEHLERIAN SPACE 71

and, consequently

α − 4p_{a}p^{a} = −\frac{4(n − 2)}{n − 4}(∇^{∘}_{a} p^{a} + \frac{n + 4}{2}p_{a}p^{a}).

Then

α_{ji} = β_{ji} + \frac{4(n − 2)}{n(n − 4)}(∇^{∘}_{a} p^{a} + \frac{n + 4}{2}p_{a}p^{a})F_{ji}. (34)

β_{ji} = −2q_{ji} − \frac{2}{n}(∇^{∘}_{a} p^{a} + \frac{n + 4}{2}p_{a}p^{a})F_{ji}. (35)

We can substitute (4. 7) and (4. 5) into the first Bianchi identity. Then

0 = (\frac{2}{n − 4} − \frac{2}{n})(∇^{∘}_{a} p^{a} + \frac{n + 4}{2}p_{a}p^{a})(F_{kh}F_{ji} − F_{jh}F_{ki} + F_{ih}F_{kj})

= \frac{8}{n(n − 4)}(∇^{∘}_{a} p^{a} + \frac{n + 4}{2}p_{a}p^{a})(F_{kh}F_{ji} − F_{jh}F_{ki} + F_{ih}F_{kj}).

As

F_{kh}F_{ji} − F_{jh}F_{ki} + F_{ih}F_{kj} ≠ 0,

then ∇^{∘}_{a} p^{a} + \frac{n+4}{2}p_{a}p^{a} = 0, and

α_{ji} = β_{ji} = −2q_{ji}. (36) Then, according to (4. 1),

K_{kjih} = g_{kh}p_{ji} − g_{jh}p_{ki} + g_{ji}p_{kh} − g_{ki}p_{jh} + F_{kh}q_{ji} − F_{jh}q_{ki} +F_{ji}q_{kh} − F_{ki}q_{jh} + 2F_{kj}q_{hi} + 2F_{hi}q_{kj}.

After transvection by g^{kh}, we get K_{ji} = (n + 4)p_{ji} + g_{ji}p_{a}^{a}, (37) As

∇^{∘}_{a} p^{a} = −\frac{n + 4}{2}p_{a}p^{a},

then

p_{a}^{a} = −2p_{a}p^{a}, K = −4(n + 2)p_{a}p^{a},

p_{a}p^{a} = −\frac{K}{4(n + 2)},

p_{a}^{a} = \frac{K}{2(n + 2)},

p_{ji} = \frac{1}{n + 4}[K_{ji} − \frac{K}{2(n + 2)}g_{ji}], (38)

q_{ji} = \frac{1}{n + 4}[K_{ja}F_{i}^{a} + \frac{K}{2(n + 2)}F_{ji}]. (39)

If we substitute (4. 10) and (4. 11) into (4. 1), we obtain that there holds:

Theorem 3. If the curvature tensor of the special semi-symmetric connection on a hyperbolic Kaehlerian space admits the form R_{kjih} = α_{kj}F_{ih}, then the HBtensor [2] of the hyperbolic Kaehlerian space vanishes.

Suppose that the covariant derivative of the torsion tensor vanishes

∇_{k}T_{ij}^{h} = 0. (40)

Then ∇_{k}T_{hj}^{h} = 0. As T_{hj}^{h} = np_{j}, then there holds

∇_{k}p_{j} = 0. (41)

According to (1. 7)

∇_{k}p_{j} =∇^{∘}_{k} p_{j} + g_{kj}p_{a}p^{a} − p_{k}p_{j} − q_{k}q_{j} = 0. (42)

Then

p_{kj} = −\frac{1}{2}p_{a}p^{a}g_{kj}. (43)

Comparing (4. 15) to (4. 10), we obtain

K_{ji} = K \frac{n + 8}{8(n + 2)}g_{ji}

and

K \frac{(n + 4)(n − 4)}{8(n + 2)} = 0,

and, consequently,

K = 0.

We have proved

Corollary 1. If the curvature tensor of the special semi-symmetric connection on a hyperbolic Kaehlerian space admits the form R_{kjih} = α_{kj}F_{ih}, then the condition (4. 12) is satisfied if and only if the curvature scalar of the space vanishes and the vector (p_{i}) is isotropic.

References

1. M. Prvanović, N. Pušić: On manifolds admitting some semi–symmetric connections, in print (Indian Jounal of Mathematics)

2. N. Pušić: On an invariant tensor of a conformal transformation of a hyperbolic Kaehlerian space, Zbornik radova Filozofskog fakulteta u Nišu, Serija Matematika, 4 (1990.) 55-64

3. K.Yano, T. Imai: On semi-symmetric F -connection, Tensor, N. S, vol 29 (1975.) 135-138

Institute of Mathematics, Faculty of Science, University of Novi Sad Trg Dositeja Obradovi’ca 3, 21000 Novi Sad, Yugoslavia