On the Harmonic and Killing Tensor Filed on a Compact Riemannian Manifold

Gr. Tsagas and Gr. Bitis

Abstract

Let (M, g) be an orientable and compact Riemannian manifold. The aim of the resent paper is to study the vector spaces H^{q} (M, ℜ) and K^{q} (M, ℜ) of harmonic q-forms and Killing tensor fields of order q on a compact Riemannian manifold, where q = 2, 3, ..., n − 2 and n = dim M.

Mathematics Subject Classification: 53C20 Key words: A compact Riemannian manifold, harmonic q-form, Killing tensor field of order q, Ricci tensor field, Riemannian tensor field and a special quadratic form.

1 Introduction

Let (M, g) be a compact Riemannian manifold of dimension n. We denote by K^{q} (M, ℜ) the vector space of Killing tensor fields of order q and H^{q} (M, ℜ) the vector space of harmonic q-forms on M, q = 2, ..., n − 2.

The purpose of the present paper is to study these vector spaces K^{q} (M, ℜ) and H^{q} (M, ℜ) . The whole paper contains five paragraphs. Each of them is analyzed as follows: The first paragraph is the introduction. Special tensor fields on a compact Riemannian manifolds are included in the second paragraph. The Killing tensor fields on a Riemannian manifold are studied in the third paragraph. The fourth paragraph contains harmonic q-forms on a compact Riemannian manifold. Some topological invariants of a compact manifold are studied in the last pararaph.

2 Tensor fields on a manifold

Let (M, g) be a compact Riemannian manifold of dimension n. We consider an atlas (U_{α}, ϕ_{α})_{α∈A} , where (U_{α}, ϕ_{α}) is a chart on M with local coordinate system {x_{α}^{1}, ..., x_{α}^{n}}.

Let w be a q-form on M, that is w ∈ Λ^{q} (M). Therefore w on the chart (U, ϕ) with local coordinate system {x^{1}, ..., x^{n}} can take the form

Balkan Journal of Geometry and Its Applications, Vol.6, No.2, 2001, pp. 99-108 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

w = \frac{1}{q!}w_{i_{1}i_{2}...i_{q}}dx^{i_{1}} ∧ dx^{i_{2}} ∧ ... ∧ dx^{i_{q}} (1)

where 1 ⪯ i_{1} < i_{2} < ... < i_{q} < n and w_{i_{1}...i_{q}} the components of w on the chart (U, ϕ) . The local norm of w is defined by

|w|^{2} = \frac{1}{q!}w_{i_{1}...i_{q}}w^{i_{1}...i_{q}} (2)

where

w^{i_{1}...i_{q}} = g^{i_{1}j_{1}}...g^{i_{q}j_{q}}w_{i_{1}...i_{q}} (3)

This q-form w, by means of (2), gives a function |w|^{2} on M, that means |w|^{2} ∈ D^{0} (M)

Hence we have

\frac{1}{2}Δ (|w|^{2}) =< δΔw, w > − |∇w|^{2} (4)

The formula (4) by integration implies

∫M [< δΔw, w > − |∇w|^{2}] dM (5)

where dM is the volume element of (M, g)

If w ∈ Λ^{q} (M, ℜ), then Δw ∈ Λ^{q} (M), where Δ is the Laplace operator acting on the vector space Λ^{q} (M, ℜ) as a linear operator. On the chart (U, ϕ) with local coordinate system we have

Δw = \frac{1}{q!} [g^{i j}∇_{j}∇_{i}w_{i_{1}...i_{q}} − ∑_{s=1}^{q}p_{i_{s}}^{ν}w_{i_{1}... i_{s−1} ν i_{s+1}... i_{q}}−

− ∑_{t<s}^{1,...,q} R_{i_{t} i_{s}}^{νu}w_{i_{1}... i_{t−1} ν i_{t+1}... i_{s−1} u i_{s+1}... i_{q}}] dx^{i_{1}} ∧ ... ∧ dx^{i_{q}} (6)

Hence the components of Δw on (U, ϕ) have the form

(Δw)_{i_{1}...i_{q}} = \frac{1}{q!} [−g^{i j}∇_{i}∇_{j}w_{i_{1}...i_{q}} + ∑_{s=1}^{q} p_{i_{s}}^{v}w_{i_{1}... i_{s−1} ν i_{s}... i_{q}}+

+ ∑_{t<s}^{1,...,q} R_{i_{t} i_{s}}^{uν}w_{i_{1}... i_{t−1} uν i_{t+1}... i_{s−1} ν i_{s+1}... i_{q}}] (7)

We obtain the inner product of the q-forms w and Δw, which takes the form

< Δw, w >=< δ∇w, w > +\frac{1}{(ϕ − 1)!}F_{q} (w) (8)

where F_{q} (w) is the following quadratic form

On the Harmonic and Killing Tensor Filed on a Compact Riemannian Manifold 101

F_{q} (w) = p_{i j}w^{i i_{2}...i_{q}}w_{i_{2}...i_{q}}^{j} + \frac{q − 1}{2}R_{i j k l}w^{i j i_{3}...i_{q}}w_{i_{3}...i_{q}}^{k l} (9)

It can be easily proved the following relation

\frac{1}{2}Δ (|w|^{2}) =< Δw, w > − |∇w|^{2} − \frac{1}{(q − 1)!}F_{q} (w, w) (10)

The integration of (10) implies

∫M [< Δw, w > − |∇w|^{2} − \frac{1}{(q − 1)!}F_{q} (w, w)] dM (11)

It can be easily obtained the relation

< Δw − (q + 1) δ∇w, w >= −q < δ∇w, w > +\frac{1}{(q − 1)!}F_{q} (w, w) (12)

The equality (12) by means of (4) becomes

\frac{1}{2}Δ (|w|^{2}) = − |∇w|^{2} + \frac{1}{q!}F_{q} (w, w) − \frac{1}{q!} < Δw − (q + 1) δ∇w, w > (13)

which by integration implies

∫M [< Δw − (q + 1) δ∇w, w > +q |∇w|^{2} − \frac{1}{(q − 1)!}F_{q} (w, w)] dM = 0 (14)

We use a new expression of the quadrate form F_{q} (w, w), which can be written

F_{q} (w, w) = B_{i_{1}i_{2}...i_{q} , j_{1}j_{2}...j_{q}}w^{i_{1}i_{2}...i_{q}}w^{j_{1}j_{2}...j_{q}} (15)

where

B_{i_{1}i_{2}...i_{q} , j_{1}j_{2}...j_{q}} = (p_{i_{1} j_{1}}g_{i_{2} j_{2}} + \frac{q − 1}{2}R_{i_{1}i_{2} j_{1}j_{2}}) g_{i_{3} j_{3}}...g_{i_{q} j_{q}} (16)

where the indices satisfy the inequalities

1 ⪯ i_{1}, i_{2}, ..., i_{q} ⪯ n 1 ⪯ j_{1}, j_{2}, ..., j_{q} ⪯ n (17)

Proposition 1 The tensor field B = (B_{i_{1}i_{2}...i_{q} , j_{1}j_{2}...j_{q}}) on a Riemannian manifold (M, g) is symmetric with respect to (i_{1}i_{2}, j_{1}j_{2}) and with respect of any of two indices (i_{ν}, j_{ν}) ν = 3, 4, ...q and as well as with respect to (i_{1}, i_{2}, ...i_{q}) and (j_{1}, j_{2}, ..., j_{q}). Therefore the quadratic form (15) is symmetric with respect to (i_{1}, i_{2}, ...i_{q}) and (j_{1}, j_{2}, ..., j_{q}).

Proof 1 It is known from the properties of the Rici tensor field p, the curvature tensor field R and the metric tensor field g, we have the following relations

p_{i_{1} j_{1}} = p_{j_{1} i_{1}} , p_{i_{2} j_{2}} = p_{j_{2} i_{2}} , R_{i_{1} i_{2} j_{1} j_{2}} = R_{j_{1} j_{2} i_{1} i_{2}} (18)

g_{i_{1} j_{1}} = g_{j_{1} i_{1}} , g_{i_{2} j_{2}} = g_{j_{2} i_{2}} , g_{i_{3} j_{3}} = g_{j_{3} i_{3}} , ..., g_{i_{q} j_{q}} = g_{j_{q} i_{q}} (19)

102 Gr. Tsagas and Gr. Bitis

From (7) and (8) we conclude the first part of the proposition. The other part, that means the symmetric property of the tensor field with respect to the pair

(i_{1}, i_{2}, ...i_{q}) , (j_{1}, j_{2}, ..., j_{q}) (20) is a consequence of the same relations (18) and (19) □

Now, we intrude a new quadratic form F^{∧}_{q} (w, w) on the vector space Λ^{q} (M, ℜ) as follow

F^{∧}_{q} (w, w) = B′_{(i_{1}...i_{q})( j_{1}...j_{q})}w^{i_{1}...i_{q}}w^{j_{1}...j_{q}} (21)

with the condition that the symbol (i_{1}, i_{2}, ...i_{q}) means

1 ⪯ i_{1} < i_{2} < ... < i_{q} ⪯ n

For this reason we form the tensor field

B′_{(i_{1}...i_{q})(j_{1}...j_{q})} = B_{k_{1}...k_{q}l_{1}...l_{q}}δ_{i_{1}...i_{q}}^{k_{1}...k_{q}}δ_{j_{1}...j_{q}}^{l_{1}...l_{q}} (22) where δ_{i_{1}...i_{q}}^{k_{1}...k_{q}} and δ_{j_{1}...j_{q}}^{l_{1}...l_{q}} (23) are the Kronecker’s generalized symbols which are given by the formulas

δ_{i_{1}...i_{q}}^{k_{1}...k_{q}} = | \begin{matrix}δ_{i_{1}}^{k_{1}}.....δ_{i_{1}}^{k_{q}} \\ ............ \\ ............ \\ δ_{i_{q}}^{k_{1}}.....δ_{i_{q}}^{k_{q}} \end{matrix} | , δ_{j_{1}...j_{q}}^{l_{1}...l_{q}} = | \begin{matrix}δ_{j_{1}}^{l_{1}}.....δ_{j_{1}}^{l_{q}} \\ ............ \\ ............ \\ δ_{j_{q}}^{l_{1}}.....δ_{j_{q}}^{l_{q}} \end{matrix} | (24)

Now, we can prove the following proposition

Proposition 2 The tensor field B′ = (B′_{(i_{1}...i_{q})(j_{1}...j_{q})}) is symmetric with respect to {(i_{1}...i_{q}) , (j_{1}..., j_{q})} Proof 2 According to the proposition 1 we have

B_{k_{1}...k_{q}, l_{1}...l_{q}} = B_{l_{1}...l_{q}, k_{1}...k_{q}} (25) From the properties of Kronecker’s tensor field we conclude that

δ_{j}^{i} = \{ \begin{matrix}1 if i = j \\ 0 if i ≠ j \end{matrix} (26)

δ_{j_{1}...j_{q}}^{l_{1}...l_{q}} = δ_{i_{1}...i_{q}}^{k_{1}...k_{q}} (27)

From (25), (26) and (27) we have B′_{(i_{1}...i_{q})(j_{1}...j_{q})} = B_{(j_{1}...j_{q})(i_{1}...i_{q})} □ (28)

On the Harmonic and Killing Tensor Filed on a Compact Riemannian Manifold 103

Now, under the introduction of the tensor field B the quadratic form F_{q}^{∧} (w, w) becomes

F_{q}^{∧} (w, w) = \frac{1}{(q − 1)!}B_{(j_{1}...j_{q})(i_{1}...i_{q})}w^{i_{1}...i_{q}} w^{j_{1}...j_{q}} (29)

where the indices satisfy the inequalities 1 ⪯ i_{1} < i_{2} < ... < i_{q} ⪯ n 1 ⪯ j_{1} < j_{2} < ... < j_{q} ⪯ n (30)

The relation between the quadratic forms F_{q} (w, w) and F_{q}^{∧} (w, w) is the following

F_{q}^{∧} (w, w) = \frac{1}{(q − 1)!}F_{q} (w, w) (31)

Now, we can prove the following theorem Theorem 3 Let (M, g) be a compact Riemannian manifold of dimension n, n ⪰ 3. The nullity of the quadratic form F_{q}^{∧} (w, w) is a global property. Proof 3 Let (U, ϕ) be a chart on M with local coordinate system (x^{1}, ..., x^{n}). We assume that the nullity of F_{q}^{∧} (w, w) on (U, ϕ) is κ, that is

nullity F_{q}^{∧} (w, w) = κ (32)

If (V, υ) is another chart on M with local coordinate system (y^{1}, ..., y^{n}) such that U ∩ V ≠ 0, then the components {w′ ^{l_{1}...l_{q}}} of w on V are connected with the components {w^{i_{1}...i_{q}}} of w on U with the relations

w^{i_{1}...i_{q}} = \frac{∂x^{i_{1}}}{∂y^{l_{1}}} ...\frac{∂x^{i_{q}}}{∂y^{l_{q}}} w′ ^{l_{1}...l_{q}} (33)

The relations (33) are valid on U ∩ V and the local coordinates (x^{1}, ..., x^{n}) and (y^{1}, ..., y^{n}) are connected by the relations x^{1} = x^{1} (y^{1}, ..., y^{n}) ...x^{n} = x^{n} (y^{1}, ..., y^{n}) (34)

The quadratic form F_{q}^{∧} (w, w) on the local coordinate system (y^{1}, ..., y^{n}) and by meaning of (33) and (34) takes the expression

F_{q}^{∧} (w′, w′) = P · Q (35)

where

P = w′ ^{l_{1}...l_{q}} \frac{∂x^{i_{1}}}{∂y^{l_{1}}} ...\frac{∂x^{i_{q}}}{∂y^{l_{q}}} , Q = \frac{∂x^{j_{1}}}{∂y^{m_{1}}} ...\frac{∂x^{j_{q}}}{∂y^{m_{q}}} w′ ^{m_{1}...m_{q}} (36)

The change of the local coordinate system (x^{1}, ..., x^{n}) on the chart (U, ϕ) to the (y^{1}, ..., y^{n}) on (V, υ) brings a change into the base {dy^{i_{1}} ∧ dy^{i_{2}} ∧ ... ∧ dy^{i_{q}}} (37)

of the vector space Λ^{q} (U ∩ V, ℜ), which is determined by the relation

dx^{i_{1}} ∧ dx^{i_{2}} ∧ ... ∧ dx^{i_{q}} = \frac{∂x^{i_{1}}}{∂y^{l_{1}}} \frac{∂x^{i_{2}}}{∂y^{l_{2}}} ...\frac{∂x^{i_{q}}}{∂y^{l_{q}}} dy^{l_{1}} ∧ dy^{l_{2}} ∧ ... ∧ dy^{l_{q}} (38)

This expression implies

rank (\frac{∂x^{i_{1}}}{∂y^{l_{1}}} \frac{∂x^{i_{2}}}{∂y^{l_{2}}} ...\frac{∂x^{i_{q}}}{∂y^{l_{q}}} ) = ( \begin{matrix}n \\ q \end{matrix} ) (39)

From (35) by means of (39) implies that the nullity of the quadratic form F_{q}^{∧} (w, w) is constant and equal κ on the whole manifold.

3 Killing tensor fields

Let (M,g) be a compact Riemannian manifold of dimension n. Let T be an antisymmetric tensor field of type (0,q). It is known that for T we can associate an exterior q-form w. This exterior q-form w is called Killing if it satisfies the relation

(q + 1) ∇w = dw (40)

The relation (40) implies that the tensor field ∇w is antisymmetric and the same time

δw = 0 (41)

The tensor field T is called Killing if the associated exterior q-form w is Killing. Let (U, ϕ) be a chart on (M, g) with local coordinate system (x^{1}, ..., x^{n}). Let

{w_{i_{1}...i_{q}}} be the components of w on (U, ϕ). If w is a Killing, then {w_{i_{1}...i_{q}}} satisfy the relations

g^{ji}∇_{j}∇_{i}w_{i_{1}...i_{q}}+\frac{1}{q} ∑_{s=1}^{n} p_{i_{s}}^{v}w_{i_{1}...i_{s−1} v i_{s+1}...i_{q}}+\frac{1}{q} ∑_{t<s}^{1...q} R_{i_{t} i_{s}}^{vu}w_{i_{1}...i_{t−1} v i_{t+1}...i_{s−1} u i_{s+1}...i_{q}} = 0

(42) and

g^{i j}∇_{j}w_{i_{1}i_{2}...i_{q}} = 0 (43)

on the chart (U, ϕ).

If w is a Killing exterior q-form, the formula (11) takes the form

∫M [q |∇w|^{2} − \frac{1}{(q − 1)!}F_{q} (w, w)] dM (44)

Now, we can prove the following theorem Theorem 4 Let (M, g) be a compact orientable Riemannian manifold of dimension

n. We assume that the quadrate form F_{q}^{∧} (w, w) is semi-negative on the whole manifold

M. We also assume that on one chart of M the nullity of F_{q}^{∧} (w, w) is equal to the number of linearly independent parallel exterior forms of order q, then the dimesion of K_{q} (M, ℜ) of Killing exterior q-form on M is given by

(45) dim K_{q} (M, ℜ) = nulity F_{q}^{∧} (w, w)

Proof 4 Let w be a Killing exterior form of order q. Under the assumption, we have

that the quadratic form F_{q}^{∧} (w, w) is semi-definite on the whole manifold. From the formula (44) we obtain

∇w = 0 F_{q}^{∧} (w, w) = 0 (46)

It is known that if f ∈ D^{o} (M) and Δf ⪰ 0 or Δf ⪯ 0, then f = c.

From the relation (8), the equation Δw = (q + 1) δ ∇w and the above remark we conclude that

(47) |w|^{2} = c, c = constant

We assume that on the chart (U, ϕ) of the manifold M with local coordinate system

(x^{1}...x^{n}) the nullity of the quadratic form F_{q}^{∧} (w, w) is equal to κ and there are κ linearly independent different than zero exterior q-forms, w_{(i_{1}i_{2}...i_{q})}^{(β)} β = 1, 2, ..., κ, which satisfy the relations (46). These exterior q-forms are parallel and the same time Killing

Let w^{(κ+1)} be another parallel exterior q-form different than zero, which satisfies the relation

F_{q}^{∧} (w^{(κ+1)}, w^{(κ+1)}) = 0 (48)

We shall prove that the exterior q-form w^{(κ+1)} can be written with a unique manner as a linear combination of w_{(i_{1}i_{2}...i_{q})}^{(β)} β = 1, 2, ..., κ with constant coefficients on the chart (U, ϕ).

We assume that

w_{(i_{1}i_{2}...i_{q})}^{(κ+1)} = ∑_{β=1}^{κ} ϕ_{β} w_{(i_{1}i_{2}...i_{q})}^{(β)} (49)

where ϕ_{β} are functions of (x^{1}, x^{2}, ..., x^{n}). If we apply the operator of covariant differentiation ∇_{j} on the relation (49), we obtain for every j the system

∑_{β=1}^{κ} ∂_{j}ϕ_{β (i_{1}i_{2}...i_{q})}^{(β)} = 0 (50)

which has a number ( \begin{matrix}n \\ q \end{matrix} ) equations with κ unknown ∂_{j}ϕ_{1}, ∂_{j}ϕ_{2}, ..., ∂_{j}ϕ_{κ} (51)

From our assumption we know that the degree of the matrix

{w_{(i_{1}i_{2}...i_{q})}^{(β)}} (52)

of the coefficient of the unknown is equal to κ

Therefore for every point of U, there exists at least one determinant of order κ different than zero, the referred above matrix. The corresponding homogeneous system, which has as matrix of the coefficients the determinant which is different than zero. Hence, we obtain

(53) ∂_{j}ϕ_{β} = 0 (β = 1, 2, ...κ) for every j = 1, 2, ...n

which imply

ϕ_{β} = c_{β} β = 1, 2, ...κ (54)

Hence, we have

w_{(i_{1}i_{2}...i_{q})}^{(κ+1)} = ∑_{β=1}^{κ} c_{β} w_{(i_{1}i_{2}...i_{q})}^{(β)} (55)

On the other hand, by means of the formulas of changing the local coordinate system we easily conclude that the linear conection (55) with constant coefficients is valid on the whole manifold.

Finally, since every exterior q-form w^{(β)} satisfies the relation

|w^{(β)}| = c′_{β} (c′_{β} ≠ 0) (56)

and taking under consideration the theorem 3, we conclude that the exterior q-form w^{(β)} β = 1, 2, ...κ are the only non-zero linearly independent Killing exterior q-form on the whole manifold □

4 Harmonic q-forms

Let w be an exterior q-form. This is called harmonic if it satisfies the relations

dw = 0, δw = 0 (57)

or equivalently

Δw = 0 (58)

Let (U, ϕ) be a chart on the manifold with local coordinate system (x^{1}...x^{n}). Let {w_{i_{1}...i_{q}}} be the components of w on U. If w is harmonic, then its components satisfy the conditions

g^{ji}∇_{j}∇_{i}w_{i_{1}...i_{q}} − ∑_{s=1}^{q} R_{i_{s}}^{v}w_{i_{1}...i_{s−1} v i_{s+1}...i_{q}} − ∑_{t<s}^{1...q} R_{i_{t} i_{s}}^{vu}w_{i_{1}...i_{t−1} v i_{t+1}...i_{s−1} u i_{s+1}...i_{q}} = 0

(59) The integral formula (14), if w is a harmonic form and by means of (59), takes the form

∫M [|∇w|^{2} + \frac{1}{(q − 1)!}F_{q} (w, w)] dM = 0 (60)

The following results are known. If F_{q} (w, w) is semi-definite on the whole manifold, then every harmonic form is parallel. If F_{q} (w, w) > 0, then there exists no harmonic form and the q Betti number is zero, then

b_{q} (M) = 0 (61)

Now, we can prove the following theorem

Theorem 5 Let (M, g) be an orientable and compact Riemannian manifold of dimen-

sion n. We assume that the quadratic form F_{q}^{∧} (w, w) is semi-positive on the whole manifold M and the same time the nullity of F_{q} (w, w) is equall with the number of linearity independent parallel exterior q-forms, then q Betti number of M is equall to

the nullity of F_{q}^{∧} (w, w), that is b_{q} (M) = nullity (F_{q}^{∧} (w, w)) .

Proof 5 From the assumption the F_{q}^{∧} (w, w) is semi-positive on the whole manifold M, we conclude, by the meaning of the formula of (60), the relations

F_{q}^{∧} (w, w) = 0 , ∇w = 0 (62)

From the equation (10) using the known result if f ∈ D^{o} (M) and Δf ⪰ 0 implies f = constant we conclude that |w|^{2} = c.

If we use the same method as in the theorem 3 we conclude that there are κ linearly independent paraller exterior q-forns, which are the only harmonic q-forms on the manifold. Therefore we have the equality dim H^{q} (M, ℜ) = b_{q} = κ. □

5 Determination of the Betti numbers

Let M be an orientable and compact manifold. It is known that the q Betti number b_{q} is a topological invariant.

One problem of algebraic topology is to determine b_{q} (M), q = 1, 2, ..., n − 1, where n = dim M. In some cases we use Riemannian metrics on M to determine b_{q} q = 1, 2, ..., n − 1.

This method connects Algebraic Topology and Differential Geometry.

Let H (M) be the set of all Riemannian metrics on M. It is known that H (M) is a Bancach space of infinite dimension.

Let g ∈ H (M) be a Riemannian metric on M. If w ∈ Λ^{q} (M) then we can form

F_{q}^{∧} (w, w) (63)

If

F_{q}^{∧} (w, w) ⪰ 0 and ∇w = 0 (64)

then w is parallel and the same time harmonic. If there are κ exterior q-forms w_{1}, w_{2}, ..., w_{κ} such that they satisfy

F_{q}^{∧} (w_{1}, w_{1}) ⪰ 0 ∇w_{1} = 0, F_{q}^{∧} (w_{2}, w_{2}) ⪰ 0 ∇w_{2} = 0, ....., F_{q}^{∧} (w_{κ}, w_{κ}) ⪰ 0 ∇w_{κ} = 0

(65) then b_{q} = κ. From the above we have the proposition.

Proposition 6 Let M be an orientable and compact manifold of dimension n. We assume that there are κ exterior q-forms w_{1}, w_{2}, ..., w_{κ} and Riemannian metrics g such that thes forms are parallel with respect to the Luvi-Civita connection ∇ and the

same time F_{q}^{∧} (w_{j}, w_{j}) ⪰ 0, j = 1, ..., κ, then b_{q} = κ

The above proposition permits to determine the Betti numbers on an orientable and compact Riemannian manifold.

References

[1] P. H. Berard, Spectral Geometry. Direct and Inverse Problems, Lecture Notes in Mathematics, 1207 Springer-Verlag, Berlin, 1986.

[2] R. Brooks, A Construction of Metrics of Negative Ricci Curvature, Journal of Differential Geometry, 29, 1989, 85-94

[3] J. Lohcamp, Negatively Ricci Curved Manifold, Bulletin of A.M.S., 27, 288-291.

[4] Gr. Tsagas, A Relation Between Killing Tensor Fields and Negative Pinched Riemannian Manifolds, Proc. A.M.S., 22, 1969, 476-478.

[5] Gr. Tsagas, Some Properties of Negative Pinched Riemannian Manifolds of Dimension 5 and 7, Journal of Differential Geometry, 3, 1969, 523-530.

[6] Gr. Tsagas and Ch. Christophoridou, Relation Between Topology and the Killing Vector Fields, Proceedings of the International workshop on Global analysis, Differential Geometry and Lie algebras, 1996, 115-119.

[7] K. Yano and Bochner, Curvature and Betti Numbers, Princeton University Press, 1953, Princeton.

[8] K. Yano, On Harmonic and Killing Vector Fields, Ann. Math., 55, 1952, 38-45.

[9] Gr. Tsagas and Gr. Bitis, On the Harmonic and Killing Vector Fields on Compact Decomposable Riemannian Manifolds, Harmonic Journal.

Departement of Mathematics and Physics

Aristotle University of Thessaloniki

Thessaloniki 54006, Greece