DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 35–43

ON THE GEOMETRY OF SOME 4–DIMENSIONAL

EINSTEIN MANIFOLDS

Raina Ivanova

Abstract. Using a skew-symmetric curvature operator a classification of some 4dimensional Einstein manifolds has been done in [1]. In the present paper we study one of the classes of these Einstein manifolds and answer the question about the global constancy of the functions, characterizing them.

§1 Introduction.

Let (M, g) be an n-dimensional Riemannian manifold and R be the curvature tensor. In [1] we have considered a linear curvature operator κ_{X,Y}, acting in the tangent space M_{p} at a point p ∈ M, defined by

(1)  κ_{X,Y} = R(X, Y)u,

where X, Y is an arbitrary othonormal pair of vectors in M_{p} and u ∈ M_{p}. The curvature operator κ_{X,Y} is a skew-symmetric one and it leaves the 2-dimensional subspace E^{2}(p; X, Y), spaned by X, Y, invariant. That is why we have introduced the curvature operator

(2)  κ_{E^{2}} = κ_{X,Y}

with respect to any E^{2} ⊂ M_{p} at any point p ∈ M. The operator κ_{E^{2}} has a characteristical equation which in the case n = 4 has the form

λ^{4} + J_{2}.λ^{2} + J_{4} = 0,

1991 Mathematics Subject Classification. 53C25.

Key words and phrases. A skew-symmetric curvature operator, 4-dimensional Einstein manifolds, characterizing functions, global constants of the manifold.

This paper is in final form and no version of it will be submitted for publication elsewhere. Partially supported by the Ministry of Education, Science and Technology in Bulgaria under

Contract MM 413/94.

where

(3)  J_{2} = J_{2}(p; E^{2}) = ∑_{1≤i<j≤4} [R(X, Y, e_{i}, e_{j})]^{2} ,

(4) \begin{aligned}J_{4} & = J_{4}(p; E^{2}) = [ R(X, Y, e_{1}, e_{2}).R(X, Y, e_{3}, e_{4}) \\ & + R(X, Y, e_{2}, e_{3}).R(X, Y, e_{1}, e_{4}) + R(X, Y, e_{3}, e_{1}).R(X, Y, e_{2}, e_{4}) ]^{2} . \end{aligned}

If the manifold is of constant sectional curvature c, then the coefficients in characteristical equation of the operator κ_{E^{2}} are constants J_{2} = c^{2} and J_{4} = 0. We study the manifolds, for which the second characteristical coefficient of the operator κ_{E^{2}} is zero, i.e. the manifolds with the property

(5)  J_{4}(p; E^{2}) = 0.

A classification of such kind of manifolds has been done in [1]. Using the curvature operator κ_{E^{2}} we have proved ([1], [2]), that a 4-dimensional Einstein manifold (M, g) has the property (5) iff it is of one of the following two types:

1) (M, g) is of constant sectional curvature;

2) The curvature tensor R satisfies the conditions:

(Γ)  \{ \begin{matrix}R_{1213} = R_{1214} = R_{1314} = 0 \\ R_{1223} = R_{1224} = 0, R_{1323} ≠ 0 \\ R_{1234} = R_{1423} = R_{1324} = 0 \\ K_{13} + K_{23} = 0, K_{13} ≠ 0 \\ K_{12}^{2} = K_{13}^{2} + R_{1323}^{2} \\ (K_{12} is a global constant), \end{matrix}

where K_{ij} = R_{ijij} is the sectional curvature of the plane (e_{i} ∧ e_{j}).

The manifolds satisfying (Γ) are characterized by the functions K_{12}(p; E^{2}), which is a global constant of the manifold, K_{13}(p; E^{2}) and R_{1323}(p; E^{2}). There is an orthogonal group of transformations in the tangent space M_{p}, keeping the curvature conditions (Γ) invariant. We have found these transformations and also determined the changing of the functions K_{13}(p; E^{2}) and R_{1323}(p; E^{2}) by them [3].

§2. A representation of the curvature tensor.

Let (M, g) be a 4-dimensional nonflat Einstein manifold satisfying the curvature conditions (Γ) and e_{1}, e_{2}, e_{3}, e_{4} be an orthonormal base of M_{p}. Then, the following relations

(6)

R_{ijjk} + R_{illk} = 0, i, j, k, l − different,
K_{12} = K_{34}, K_{13} = K_{24}, K_{23} = K_{14}

also hold.

Let x, y, z, u be arbitrary tangent vectors. Then,

x = ∑_{i=1}^{4} x^{i}e_{i}, y = ∑_{j=1}^{4} y^{j}e_{j}, z = ∑_{k=1}^{4} z^{k}e_{k}, u = ∑_{l=1}^{4} u^{l}e_{l},

and

(7)  R(x, y, z, u) = R (∑_{i=1}^{4} x^{i}e_{i}, ∑_{j=1}^{4} y^{j}e_{j}, ∑_{k=1}^{4} z^{k}e_{k}, ∑_{l=1}^{4} u^{l}e_{l}).

Because of (Γ), only the components K_{12}, K_{13} and R_{1323} of the sum (7) are nonzero. Let us notice

(8)  \begin{matrix}a_{ij} = x^{i}y^{j} − x^{j}y^{i} \\ b_{ij} = z^{i}u^{j} − z^{j}u^{i}. \end{matrix}

Then, taking into account (6), the coefficient of K_{12} in (7) can be written in the form

(9)  a_{12}b_{12} + a_{34}b_{34}.

Since K_{13} = K_{24}, K_{23} = K_{14} and K_{13} + K_{23} = 0, the coefficient of K_{13} is

(10)  a_{13}b_{13} + a_{24}b_{24} − a_{23}b_{23} − a_{14}b_{14}.

Analogously, for the coefficients of R_{1323} we obtain

(11)  a_{13}b_{23} + a_{23}b_{13} − a_{14}b_{24} − a_{24}b_{14}.

Now, using that

x^{i} = g(x, e_{i}), y^{j} = g(y, e_{j}), z^{k} = g(z, e_{k}), u^{l} = g(u, e_{l}),

we can present the coefficients (9),(10),(11) in the following way:

α_{12}(x, y, z, u)
= [g(x, e_{1})g(y, e_{2}) − g(x, e_{2})g(y, e_{1})][g(z, e_{1})g(u, e_{2}) − g(z, e_{2})g(u, e_{1})]
+ [g(x, e_{3})g(y, e_{4}) − g(x, e_{4})g(y, e_{3})][g(z, e_{3})g(u, e_{4}) − g(z, e_{4})g(u, e_{3})],
α_{13}(x, y, z, u)
= [g(x, e_{1})g(y, e_{3}) − g(x, e_{3})g(y, e_{1})][g(z, e_{1})g(u, e_{3}) − g(z, e_{3})g(u, e_{1})]
+ [g(x, e_{2})g(y, e_{4}) − g(x, e_{4})g(y, e_{2})][g(z, e_{2})g(u, e_{4}) − g(z, e_{4})g(u, e_{3})]
− [g(x, e_{2})g(y, e_{3}) − g(x, e_{3})g(y, e_{2})][g(z, e_{2})g(u, e_{3}) − g(z, e_{3})g(u, e_{2})]
− [g(x, e_{1})g(y, e_{4}) − g(x, e_{4})g(y, e_{1})][g(z, e_{1})g(u, e_{4}) − g(z, e_{4})g(u, e_{1})],

α_{1323}(x, y, z, u)
= [g(x, e_{1})g(y, e_{3}) − g(x, e_{3})g(y, e_{1})][g(z, e_{2})g(u, e_{3}) − g(z, e_{3})g(u, e_{2})]
+ [g(x, e_{2})g(y, e_{3}) − g(x, e_{3})g(y, e_{2})][g(z, e_{1})g(u, e_{3}) − g(z, e_{3})g(u, e_{1})]
− [g(x, e_{1})g(y, e_{4}) − g(x, e_{4})g(y, e_{1})][g(z, e_{2})g(u, e_{4}) − g(z, e_{4})g(u, e_{2})]
− [g(x, e_{2})g(y, e_{4}) − g(x, e_{4})g(y, e_{2})][g(z, e_{1})g(u, e_{4}) − g(z, e_{4})g(u, e_{1})]

and the expression (7) takes the form: (12)

R(x, y, z, u) = α_{12}(x, y, z, u).K_{12} + α_{13}(x, y, z, u).K_{13} + α_{1323}(x, y, z, u).R_{1323}.

The tensors R, α_{12}, α_{13} and α_{1323} are of type (0,4). For the corresponding tensors of type (1,3) we get

(13) R(z, u, y) = α_{12}(z, u, y).K_{12} + α_{13}(z, u, y).K_{13} + α_{1323}(z, u, y).R_{1323},

where

α_{12}(z, u, y)
= [g(z, e_{1})g(u, e_{2}) − g(z, e_{2})g(u, e_{1})][g(y, e_{2})e_{1} − g(y, e_{1})e_{2}]
+ [g(z, e_{3})g(u, e_{4}) − g(z, e_{4})g(u, e_{3})][g(y, e_{4})e_{3} − g(y, e_{3})e_{4}],
α_{13}(z, u, y)
= [g(z, e_{1})g(u, e_{3}) − g(z, e_{3})g(u, e_{1})][g(y, e_{3})e_{1} − g(y, e_{1})e_{3}]
+ [g(z, e_{2})g(u, e_{4}) − g(z, e_{4})g(u, e_{2})][g(y, e_{4})e_{2} − g(z, e_{2})e_{4}]
− [g(z, e_{2})g(u, e_{3}) − g(z, e_{3})g(u, e_{2})][g(y, e_{3})e_{2} − g(y, e_{2})e_{3}]
− [g(z, e_{1})g(u, e_{4}) − g(z, e_{4})g(u, e_{1})][g(y, e_{4})e_{1} − g(y, e_{1})e_{4}],
α_{1323}(z, u, y)
= [g(z, e_{2})g(u, e_{3}) − g(z, e_{3})g(u, e_{2})][g(y, e_{3})e_{1} − g(y, e_{1})e_{3}]
+ [g(z, e_{1})g(u, e_{3}) − g(z, e_{3})g(u, e_{1})][g(y, e_{3})e_{2} − g(y, e_{2})e_{3}]
− [g(z, e_{2})g(u, e_{4}) − g(z, e_{4})g(u, e_{2})][g(y, e_{4})e_{1} − g(y, e_{1})e_{4}]
− [g(z, e_{1})g(u, e_{4}) − g(z, e_{4})g(u, e_{1})][g(y, e_{4})e_{2} − g(y, e_{2})e_{4}]

So, we have represented the curvature tensor only by the metric tensor of the manifold. Moreover, this representation of R is with respect to an arbitrary orthonormal base of M_{p}.

§3. Global constancy of the characterizing functions.

According to the curvature conditions (Γ) the functions K_{12}(p; E^{2}), K_{13}(p; E^{2}) and R_{1323}(p; E^{2}) have a special role for the characterizing the manifold. From (6) and (Γ) we have K_{13} + K_{14} = 0 and then for the Ricci curvature with respect to e_{1}, we obtain:

(14)  ρ(e_{1}, e_{1}) = ∑_{i=2}^{4} K_{1i} = K_{12}.

(M, g) is an Einstein manifold and it follows from (14) and the well known theorem of Herglotz that K_{12} is a global constant of the manifold. It is interesting to answer the question whether the other two characteizing functions K_{13}(p; E^{2}) and K_{1323}(p; E^{2}) are global constants too? For this purpose, let us differentiate the representation (13), considering now x, y, z, u and e_{1}, e_{2}, e_{3}, e_{4} as smooth vector fields around a point p ∈ M.

(15) \begin{matrix}(∇_{x}R)(z, u, y) \\ = ∇_{x}[α_{12}(z, u, y).K_{12} + α_{13}(z, u, y).K_{13} + α_{1323}(z, u, y).R_{1323}] \\ − [α_{12}(∇_{x}z, u, y).K_{12} + α_{13}(∇_{x}z, u, y).K_{13} + α_{1323}(∇_{x}z, u, y).R_{1323}] \\ − [α_{12}(z, ∇_{x}u, y).K_{12} + α_{13}(z, ∇_{x}u, y).K_{13} + α_{1323}(z, ∇_{x}u, y).R_{1323}] \\ − [α_{12}(z, u, ∇_{x}y).K_{12} + α_{13}(z, u, ∇_{x}y).K_{13} + α_{1323}(z, u, ∇_{x}y).R_{1323}]. \end{matrix}

From the well known second Bianchi identity

∑_{(x,z,u)}(∇_{x}R)(z, u, y) = 0

and (15), we get: (16)

∑_{(x,z,u)} { x(K_{12}).α_{12}(z, u, y) + K_{12}.∇_{x}(α_{12}(z, u, y))
+ x(K_{13}).α_{13}(z, u, y) + K_{13}.∇_{x}(α_{13}(z, u, y))
+ x(R_{1323}).α_{1323}(z, u, y) + R_{1323}.∇_{x}(α_{1323}(z, u, y))
− K_{12}[α_{12}(∇_{x}z, u, y) + α_{12}(z, ∇_{x}u, y) + α_{12}(z, u, ∇_{x}y)]
− K_{13}[α_{13}(∇_{x}z, u, y) + α_{13}(z, ∇_{x}u, y) + α_{13}(z, u, ∇_{x}y)]
− R_{1323}[α_{1323}(∇_{x}z, u, y) + α_{1323}(z, ∇_{x}u, y) + α_{1323}(z, u, ∇_{x}y)] } = 0.

We may not write (16) in detail (which takes more than 20 pages) because we can choose x, y, z, u to be some of e_{1}, e_{2}, e_{3}, e_{4} in a proper way, so that many of the addends in (16) are zero. Let us choose x = e_{4}, y = e_{1}, z = e_{1}, u = e_{3}. Using

(17)  \begin{matrix}g(∇_{e_{i}}e_{j}, e_{j}) = 0 \\ g(∇_{e_{i}}e_{j}, e_{k}) = −g(∇_{e_{i}}e_{k}, e_{j}), \end{matrix}

we get

e_{4}(K_{13}).e_{3} + K_{13}.∇_{e_{4}}e_{3}
+ K_{12} { [g(∇_{e_{4}}e_{2}, e_{3}).e_{2} − 2g(∇_{e_{1}}e_{1}, e_{4})].e_{3} + g(∇_{e_{1}}e_{1}, e_{3}).e_{4} }
+ K_{13} { [2g(∇_{e_{1}}e_{4}, e_{3}) + g(∇_{e_{4}}e_{1}, e_{3})].e_{1} + 2g(∇_{e_{1}}e_{1}, e_{4}).e_{3}
+ [g(∇_{e_{1}}e_{1}, e_{3}) − g(∇_{e_{4}}e_{4}, e_{3})].e_{4} }
+ R_{1323} { [2g(∇_{e_{1}}e_{4}, e_{3}) + g(∇_{e_{4}}e_{1}, e_{3})].e_{2} − 2[g(∇_{e_{1}}e_{4}, e_{2})
+ g(∇_{e_{4}}e_{1}, e_{2})].e_{3} − g(∇_{e_{1}}e_{3}, e_{2}).e_{4} } = 0.

The vectors e_{1}, e_{2}, e_{3}, e_{4} are linearly independent. Having in mind

(19)  ∇_{e_{4}}e_{3} = g(∇_{e_{4}}e_{3}, e_{1}).e_{1} + g(∇_{e_{4}}e_{3}, e_{2}).e_{2} + g(∇_{e_{4}}e_{3}, e_{4}).e_{4}

and substituting it into (18), it is easily seen that

(20)  e_{4}(K_{13}) = λ_{1}.K_{12} + λ_{2}.K_{13} + λ_{3}.R_{1323},

where

(21)  \begin{matrix}λ_{1} = 2g(∇_{e_{1}}e_{1}, e_{4}) \\ λ_{2} = −2g(∇_{e_{1}}e_{1}, e_{4}) \\ λ_{3} = 2[g(∇_{e_{1}}e_{4}, e_{2}) + g(∇_{e_{4}}e_{1}, e_{2})]. \end{matrix}

The coefficients λ_{i}, i = 1, 2, 3 in (21) do not depend on the appropriate choice of the vectors (x, y, z, u). If for (x, y, z, u), inspite of (e_{4}, e_{1}, e_{1}, e_{3}) we take (e_{4}, e_{2}, e_{2}, e_{3}), i.e. changing 1 → 2, then the new coefficients are obtained from (21) by changing 1 → 2, too.

In order to express e_{4}(R_{1323}) we replace (x, y, z, u) by (e_{1}, e_{4}, e_{2}, e_{3}). Then from (16) and (17) we get

e_{4}(R_{1323}).e_{3} + R_{1323}.∇_{e_{4}}e_{3}

+ K_{12} { g(∇_{e_{4}}e_{3}, e_{2}).e_{1} − [g(∇_{e_{1}}e_{2}, e_{4}) + g(∇_{e_{2}}e_{1}, e_{4})].e_{3} + g(∇_{e_{2}}e_{1}, e_{3}).e_{4} } + K_{13} { g(∇_{e_{4}}e_{2}, e_{3}).e_{1} + [g(∇_{e_{1}}e_{3}, e_{4}) − g(∇_{e_{1}}e_{4}, e_{3})].e_{2} + [g(∇_{e_{1}}e_{4}, e_{2}) − g(∇_{e_{4}}e_{2}, e_{1}) + g(∇_{e_{4}}e_{1}, e_{2}) − g(∇_{e_{2}}e_{4}, e_{1})].e_{3} − g(∇_{e_{2}}e_{3}, e_{1}).e_{4} } + R_{1323} { [g(∇_{e_{1}}e_{4}, e_{3}) − g(∇_{e_{1}}e_{3}, e_{4})].e_{1} + [g(∇_{e_{4}}e_{2}, e_{3}).e_{2} − [g(∇_{e_{1}}e_{4}, e_{1}) + g(∇_{e_{2}}e_{4}, e_{2})].e_{3} − [g(∇_{e_{2}}e_{3}, e_{2}) + g(∇_{e_{4}}e_{3}, e_{4})].e_{4} } = 0.

From (19),(20) and due to the linear independence of the vectors e_{1}, e_{2}, e_{3}, e_{4} one can easily obtain

(23)  e_{4}(R_{1323}) = μ_{1}.K_{12} + μ_{2}.K_{13} + μ_{3}.R_{1323},

where

(24)  \begin{matrix}μ_{1} = g(∇_{e_{1}}e_{2}, e_{4}) + g(∇_{e_{2}}e_{1}, e_{4}) \\ μ_{2} = g(∇_{e_{1}}e_{2}, e_{4}) + g(∇_{e_{2}}e_{4}, e_{1}) − 2g(∇_{e_{4}}e_{1}, e_{2}) \\ μ_{3} = −g(∇_{e_{1}}e_{1}, e_{4}) − g(∇_{e_{2}}e_{2}, e_{4}). \end{matrix}

The coefficients μ_{i}, i = 1, 2, 3 in (24) do not depend on the appropriate choice of (x, y, z, u). Indeed, if inspite of (e_{1}, e_{4}, e_{2}, e_{3}), we take (e_{4}, e_{2}, e_{1}, e_{3}) or (e_{4}, e_{1}, e_{2}, e_{3}), the new coefficients coincide with these in (24). In this way, we obtained the system for the characterizing functions:

(25)  \{ \begin{matrix}e_{4}(K_{12}) = 0 \\ e_{4}(K_{13}) = λ_{1}.K_{12} + λ_{2}.K_{13} + λ_{3}.R_{1323} \\ e_{4}(R_{1323}) = μ_{1}.K_{12} + μ_{2}.K_{13} + μ_{3}.R_{1323}, \end{matrix}

where λ_{i}, i = 1, 2, 3 and μ_{j}, j = 1, 2, 3 are given by (21) and (24), respectively. Now let us differentiate the last equation of the curvature conditions (Γ):

(26)  K_{12}^{2} = K_{13}^{2} + R_{1323}^{2}.

We have

(27)  K_{12}.e_{4}(K_{12}) = K_{13}.e_{4}(K_{13}) + R_{1323}.e_{4}(R_{1323}).

Substituting (25) into (27), we obtain

(28) K_{12}(λ_{1}.K_{13} + μ_{1}.R_{1323}) = −[λ_{2}.K_{13}^{2} + μ_{3}.R_{1323}^{2} + (λ_{3} + μ_{2}).K_{13}.R_{1323}].

From (26) we can express K_{12}, replace it into (28) and square both sides of that equality. The result is a polynomial of the functions K_{13}(p; E^{2}) and R_{1323}(p; E^{2}), namely:

(29) A_{1}.K_{13}^{4} + A_{2}.K_{13}^{3}.R_{1323} + A_{3}.K_{13}^{2}.R_{1323}^{2} + A_{4}.K_{13}.R_{1323}^{3} + A_{5}.R_{1323}^{4} = 0,

where

(30)  \begin{matrix}A_{1} = λ_{1}^{2} − λ_{2}^{2} \\ A_{2} = 2λ_{1}μ_{1} − 2λ_{2}(λ_{3} + μ_{2}) \\ A_{3} = λ_{1}^{2} + μ_{1}^{2} − (λ_{3} + μ_{2})^{2} − 2λ_{2}μ_{3} \\ A_{4} = 2λ_{1}μ_{1} − 2μ_{3}(λ_{3} + μ_{2}) \\ A_{5} = μ_{1}^{2} − μ_{3}^{2}. \end{matrix}

From (21) and (24) we have

(31)  \{ \begin{matrix}λ_{1} = −λ_{2} \\ λ_{3} + μ_{2} = −μ_{1}, \end{matrix}

and then A_{1} = A_{2} = 0. The equation (29) takes the form

R_{1323}^{2}(A_{3}.K_{13}^{2} + A_{4}.K_{13}.R_{1323} + A_{5}.R_{1323}^{2}) = 0.

Because of (Γ), the polynomial of K_{13} and R_{1323} in the brackets of the last equation is identically equal to zero, which implicates that all of its coefficients are zero, i.e.

(32)  A_{3} = A_{4} = A_{5} = 0.

The equations (30),(31) and (32) give us the system

(33)  \begin{matrix}33.1. λ_{1}^{2} + 2λ_{1}μ_{3} = 0 \\ 33.2. λ_{1}μ_{1} + μ_{1}μ_{3} = 0 \\ 33.3 μ_{1}^{2} − μ_{3}^{2} = 0 \end{matrix}

From (33.2) it follows that there are 3 possible cases:

1) μ_{1} = 0, λ_{1} ≠ −μ_{3}. Then from (33.3) we get μ_{3} = 0 and from (33.1): λ_{1} = 0, i.e. in this case we have λ_{1} = −λ_{2} = 0, μ_{1} = μ_{3} = 0. But λ_{1} ≠ −μ_{3} and thus, 1) is impossible;

2) μ_{1} ≠ 0, λ_{1} = −μ_{3}. Then, from (33.1) we obtain μ_{3} = 0 and from (33.3) it follows μ_{1} = 0, which is impossible;

3) μ_{1} = 0, λ_{1} = −μ_{3}. Then from (33.3) and (33.1) it follows μ_{3} = 0 and λ_{1} = 0, respectively. So, in this case we have

(34)  λ_{1} = −λ_{2} = 0, μ_{1} = μ_{3} = 0, λ_{3} + μ_{2} = 0.

Then the system (25) can be written in the form

(35)  \begin{matrix}e_{4}(K_{12}) = 0 \\ e_{4}(K_{13}) = λ_{3}R_{1323} \\ e_{4}(R_{1323}) = −λ_{3}.K_{13}. \end{matrix}

This means that in the general case K_{12} is the only global constant of the manifold. The functions K_{13}(p; E^{2}) and R_{1323}(p; E^{2}) are not global constants.

Let us now suppose that K_{13} and R_{1323} are global constants too, i.e.

e_{4}(K_{13}) = e_{4}(R_{1323}) = 0.

Then, from (35) we get λ_{3} = 0 which is equivalent to the equation

(36)  g(∇_{e_{1}}e_{4}, e_{2}) = −g(∇_{e_{4}}e_{1}, e_{2}).

As we have already noticed, the coefficients λ_{i}, i = 1, 2, 3 do not depend on the appropriate choice of (x, y, z, u) = (e_{i}, e_{j}, e_{k}, e_{l}) and from (36) we have

(37)  g(∇_{e_{i}}e_{j}, e_{k}) = −g(∇_{e_{j}}e_{i}, e_{k}) , i, j, k = 1, 2, 3, 4.

According to (17) and (37) we can write

(38)  g(∇_{e_{i}}e_{j}, e_{k}) = g(∇_{e_{j}}e_{k}, e_{i}) = g(∇_{e_{k}}e_{i}, e_{j}).

Thus, we have proved the following

Theorem. Let (M, g) be a 4-dimensional nonflat Einstein manifold satisfying the curvature conditions (Γ). Then, (M, g) is one of the following types:

1) Only one of the characterizing functions, namely K_{12}, is a global constant of the manifold and the other two functions K_{13}(p; E^{2}), R_{1323}(p; E^{2}) are not global constants.

2) Three of the functions K_{12}, K_{13} and R_{1323} are global constants of the manifold and the metric tensor has the property (38).

REFERENCES

[1] R. Ivanova, G. Stanilov. A skew-symmetric curvature operator in the Riemannian geometry, Symposia Gaussiana. Proc. 2nd Gauss Symposium, Münich (Germany), 1993 (391 - 395).

[2] R. Ivanova. About a classification of some Einstein manifolds. Proc. XXIII Spring Conference of Union of Mathematicians in Bulgaria, St. Zagora, 1994 (175 - 182).

[3] R. Ivanova. An orthogonal tangential group of transformations of a 4-dimensional Riemannian manifold. Math. Balkanica 10, 1996 (to appear).

Dept. of Descriptive Geometry University of Architecture, Civil Engineering and Geodesy 1, Christo Smirnenski Bul. 1421 Sofia, Bulgaria