Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 2, 325-336.

Four-dimensional Ball-homogeneous and C-spaces

G. Calvaruso^{∗}^{1} Ph. Tondeur^{∗∗2} L. Vanhecke^{∗∗1}

1Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium

2Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA

Abstract. We prove that four-dimensional ball-homogeneous Einstein manifolds which are 2-stein or Hermitian are locally symmetric. Most of the results are extended to the class of four-dimensional C-spaces. MSC 1991: 53C25, 53C30, 53C35 Keywords and phrases: Four-dimensional Einstein manifolds, ball-homogeneous spaces, C-spaces.

1. Introduction

Locally homogeneous Riemannian manifolds have the property that the volume of “small” geodesic spheres or balls only depends on the radius. Riemannian manifolds having this property are called ball-homogeneous spaces [14]. Conversely, two-dimensional, connected ball-homogeneous spaces are clearly locally homogeneous but for all other dimensions such a possible converse property is to our knowledge and in general still an intriguing open problem.

Harmonic spaces or, more generally, Riemannian manifolds all of whose local geodesic symmetries are volume-preserving, that is, D’Atri spaces (see [13] for a survey and further references), are ball-homogeneous [10]. But also for these subclasses no examples which are not locally homogeneous are known. Three-dimensional D’Atri spaces are locally homogeneous [12] and a similar result has been proved for some special classes of four-dimensional Einstein D’Atri spaces. More specifically, taking into account Jensen’s result about locally homogeneous Einstein spaces of dimension four [11], one may state the problem whether such spaces are locally symmetric or not. A positive answer has been given for four-dimensional Kähler manifolds (without the Einstein condition) [17], Hermitian Einstein spaces [7] and 2-stein spaces [18]. In the first part of this paper (Section 3), we shall extend these results to the broader class of ball-homogeneous spaces.

∗Supported by grant decr. n. 1106 of 26/09/1995 of Università degli Studi di Lecce (Italy).

∗∗Supported by NATO grant CRG. 910034.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1997 Heldermann Verlag

In the second part (Section 4), we shall treat similar problems and prove analogous results for the class of C-spaces, that is, the Riemannian manifolds such that the eigenvalues of the Jacobi operator are constant along each geodesic. These spaces have been introduced in [3] (see also [4], [5] for a survey and more references) where it is proved that two- and three-dimensional C-spaces are D’Atri spaces, and conversely. For higher dimensions both classes seem to be related but it is yet unknown if a C-space is necessarily ball-homogeneous. Moreover, all known examples are again locally homogeneous but it remains an open problem whether this holds in general or not.

In Section 2 we start the paper with a collection of results about four-dimensional Einstein spaces which will be useful in proving our main results.

2. Four-dimensional Einstein manifolds

Let (M, g) be an Einstein manifold of dimension four which we suppose to be connected. Let R be the Riemannian curvature tensor associated to the Levi Civita connection and taken with the sign convention

R_{XY} = ∇_{[X,Y]} − [∇_{X}, ∇_{Y}] ,

for all smooth vector fields X, Y on M. ϱ denotes the Ricci tensor of type (0, 2), τ the scalar curvature and Δ the Laplacian.

As is well-known, at each point m ∈ M there exists a Singer-Thorpe basis {e_{1}, e_{2}, e_{3}, e_{4}} for the tangent space T_{m}M, that is, an orthonormal basis such that the components of R are given by

\{ \begin{matrix}R_{1212} = R_{3434} = a, R_{1313} = R_{2424} = b, R_{1414} = R_{2323} = c , \\ R_{1234} = α, R_{1342} = β, R_{1423} = γ , \\ R_{ijkl} = 0 whenever three of the indices i, j, k, l are distinct \end{matrix}  (2.1)

[20] (see also [2]). Here R_{XY ZW} = g(R_{XY} Z, W).

In what follows we shall often make use of a series of scalar curvature invariants. In particular, besides τ, we shall consider the quadratic invariant ||R||^{2} = ∑_{i,j,k,l} R_{ijkl}^{2} where {e_{i} , i = 1, 2, 3, 4} is an arbitrary orthonormal basis, and the cubic invariants

Ř = ∑_{i,j,k,l,p,q} R_{ijkl}R_{klpq}R_{pqij} ,

ˇR̄ = ∑_{i,j,k,l,p,q} R_{ikjl}R_{kplq}R_{piqj} .

Further, we note that

(1/2)Δ||R||^{2} = g(ΔR, R) + ||∇R||^{2}  (2.2)

and on a four-dimensional Einstein space we always have [9]

\{ \begin{matrix}Ř = − τ^{3}/12 + (2τ/3)||R||^{2} − (1/3)g(ΔR, R) , \\ ˇR̄ = τ^{3}/48 − (τ/24)||R||^{2} − (1/6)g(ΔR, R) . \end{matrix}  (2.3)

With respect to a Singer-Thorpe basis, we then have

\begin{matrix}τ = 4(a + b + c) , \\ ||R||^{2} = 8(a^{2} + b^{2} + c^{2} + α^{2} + β^{2} + γ^{2}) , \\ Ř = 16(a^{3} + b^{3} + c^{3} + 3aα^{2} + 3bβ^{2} + 3cγ^{2}) , \\ ˇR̄ = 24(abc + aβγ + bαγ + cαβ) . \end{matrix}  (2.4)

We will also make use of the following yet unpublished result of A. Derdziński:

Proposition 2.1. Let (M, g) be a four-dimensional Einstein manifold such that its Weyl conformal tensor W ∈ C^{∞}(EndΛ^{2}M) has constant eigenvalues. Then (M, g) is locally symmetric.

Since W has constant eigenvalues if and only if a , b , c , α , β and γ are constant, and hence, if and only if (M, g) is curvature homogeneous [19], we can rewrite Proposition 2.1 in the following way:

Proposition 2.2. A four-dimensional curvature homogeneous Einstein manifold is locally symmetric.

An Einstein manifold (M, g) is called a 2-stein space if F (x) = Σ_{a,b}R_{xaxb}^{2} is independent of the unit vector x ∈ T_{m}M, for all m ∈ M. For a four-dimensional Einstein manifold we have [16]

Lemma 2.3. Let (M, g) be a four-dimensional Einstein manifold. Then (M, g) is a 2-stein space if and only if

±α = a − τ/12 , ±β = b − τ/12 , ±γ = c − τ/12  (2.5)

for each m ∈ M.

Note that if we have the + signs for one orientation of T_{m}M, then we have the – signs for the other orientation.

Now, let (M, g) be an oriented four-dimensional Riemannian manifold, m a point of M and fix an orientation on T_{m}M. The space Λ^{2}(T_{m}M) of skew-symmetric (0,2)-tensors over T_{m}M splits as Λ^{2}(T_{m}M) = Λ^{+} ⊕ Λ^{−}, Λ^{±} being the eigenspaces of the Hodge star operator ∗ ∈ EndΛ^{2}(T_{m}M). The Weyl conformal tensor W ∈ EndΛ^{2}(T_{m}M) at m leaves Λ^{±} invariant. We denote by W^{±} = \frac{1}{2}(W ± ∗W) the restriction of W to Λ^{±}. (M, g) is called self-dual (respectively, anti-self-dual) if W^{−} = 0 (respectively, W^{+} = 0) [1]. Since W^{±} = 0 if and only if (2.5) is verified with respect to a “positive” Singer-Thorpe basis (see [17]), it follows from Lemma 2.3 that a four-dimensional Einstein space is 2-stein if and only if W^{+} = 0 or W^{−} = 0.

A Riemannian manifold is called a globally Osserman space if the eigenvalues of the Jacobi operator R_{x} = R(x, ·)x are independent of m ∈ M and of the unit vector x ∈ T_{m}M. It is called a pointwise Osserman space if the eigenvalues of R_{x} only depend on m and not on x. For four-dimensional manifolds we have the following result (see [8] for more details):

Proposition 2.4. Let (M, g) be a four-dimensional manifold. Then the following statements are equivalent:

(a) (M,g) is a pointwise Osserman space;

(b) (M,g) is 2-stein;

(c) locally there is a choice of the orientation of M for which g is self-dual and Einstein.

Moreover, we have

Proposition 2.5. [6] A four-dimensional globally Osserman space is flat or locally isometric to a rank-one symmetric space.

Now we recall some facts about four-dimensional Kähler-Einstein spaces. Let (M, g, J) be such a space and m ∈ M. An orthonormal basis {e_{1}, e_{2}, e_{3}, e_{4}} of T_{m}M is called an adapted basis if e_{2} = Je_{1} and e_{4} = Je_{3} . We have

Lemma 2.6. [17] Let (M, g) be a four-dimensional Kähler-Einstein space. Then there exists an adapted Singer-Thorpe basis of T_{m}M for each m ∈ M.

With respect to an adapted Singer-Thorpe basis we obtain easily, by using the Kähler identity for the curvature tensor R,

α = b + c = τ/4 − a , β = −b , γ = −c .  (2.6)

Hence and since a Ricci-flat Riemannian manifold is obviously Einsteinian, from (2.6) and Lemma 2.3 we obtain at once the following result:

Proposition 2.7. A four-dimensional Ricci-flat Kähler manifold is a 2-stein space.

Finally, we recall the following useful formula for ||R||^{2} when (M, g, J) is a Hermitian Einstein manifold [7, (2.50)]:

||R||^{2}g(x, x)g(x, x) = 16 F (x) + 2 (τ − 3τ^{∗}) H(x) +  (2.7)
+ (1/4)(3τ^{∗^{2}} − τ^{2})g(x, x)g(x, x) ,

for all m ∈ M and x ∈ T_{m}M. Here, τ^{∗} is the ∗-scalar curvature of (M, g, J), that is, τ^{∗} = trQ^{∗} where g(Q^{∗}x, y) = ϱ^{∗}(x, y) = −(1/2)tr(z ↦ R(x, Jy)Jz, and H(x) is the holomorphic sectional curvature of the holomorphic plane determined by x. In particular, when (M, g, J) is Kählerian, we have τ = τ^{∗} and hence (2.7) becomes

||R||^{2}g(x, x)g(x, x) = 16F (x) + (τ^{2}/2)g(x, x)g(x, x) − 4τH(x) .  (2.8)

This last formula implies at once that a four-dimensional 2-stein Kähler manifold with nonvanishing scalar curvature has constant holomorphic sectional curvature (see also [17]).

3. Four-dimensional ball-homogeneous spaces

Following [14], a Riemannian manifold (M, g) is called ball-homogeneous if the volume V_{m}(r) of any geodesic ball B_{m}(r) of sufficiently small radius r and center m does not depend on m.

For the volume of geodesic balls we have the following power series expansion:

Proposition 3.1. [9] For any point m ∈ M and any small radius r>0 we have

V_{m}(r) = V_{0}^{n}(r)(1 + A(m)r^{2} + B(m)r^{4} + C(m)r^{6} + O(r^{8}))  (3.1)

where V_{0}^{n}(r) denotes the volume of a Euclidean ball of dimension n=dimM and radius r. The coefficients A, B and C are given, in function of linear, quadratic and cubic scalar curvature invariants, respectively by:

A(m) = −\frac{τ(m)}{6(n + 2)} ,  (3.2)
B(m) = \frac{(−3||R||^{2} + 8||ϱ||^{2} + 5τ^{2} − 18τΔτ)(m)}{360(n + 2)(n + 4)} ,  (3.3)
C(m) = \frac{1}{720(n + 2)(n + 4)(n + 6)} \{ − \frac{5}{9}τ^{3} − \frac{8}{3}τ||ϱ||^{2} +  (3.4)

+τ||R||^{2} + \frac{64}{63}ϱ̌ − \frac{64}{21}g(ϱ ⊗ ϱ, R̄) + \frac{32}{7}g(ϱ, Ṙ) −

− \frac{110}{63}Ř − \frac{200}{63} ˇR̄ + \frac{45}{7}||∇τ||^{2} + \frac{45}{14}||∇ϱ||^{2} +

+\frac{45}{7}α(ϱ) − \frac{45}{14}||∇R||^{2} + 6τ Δτ + \frac{48}{7} g(Δϱ, ϱ) +

+\frac{54}{7}g(∇^{2}τ, ϱ) − \frac{30}{7}g(ΔR, R) − \frac{45}{7}Δ^{2}τ\} (m)

where

ϱ̌ = ∑_{i,j,k} ϱ_{ij}ϱ_{jk}ϱ_{ki} ,

g(ϱ ⊗ ϱ, R̄) = ∑_{i,j,k,l} ϱ_{ij}ϱ_{kl}R_{ikjl} ,

g(ϱ, Ṙ) = ∑_{i,j,p,q,r} ϱ_{ij}R_{ipqr}R_{jpqr} ,

α(ϱ) = ∑_{i,j,k} ∇_{i}ϱ_{jk}∇_{k}ϱ_{ij} .

So, if (M, g) is ball-homogeneous, then A, B and C must be constant functions on M. Now, we first prove Lemma 3.2. Let (M,g) be a four-dimensional ball-homogeneous Einstein space. Then ||R||^{2} and ||∇R||^{2} are constant on M.

Proof. Let (M, g) be a ball-homogeneous Einstein space. Since τ and B are constant, from (3.3) it follows that 8||ϱ||^{2} − 3||R||^{2} is constant. Moreover, the Einstein condition ϱ = \frac{τ}{n}g is equivalent to ||ϱ||^{2} = \frac{τ^{2}}{n} and so ||ϱ||^{2} is constant. Hence ||R||^{2} is also constant on M. From (3.4) we then obtain, with ϱ̌ = \frac{τ^{3}}{n^{2}} and ∇ϱ = 0,

− \frac{64}{21}g(ϱ ⊗ ϱ, R̄) + \frac{32}{7}g(ϱ, Ṙ) − \frac{110}{63}Ř − \frac{200}{63} ˇR̄ −  (3.5)
− \frac{45}{14}||∇R||^{2} − \frac{30}{7}g(ΔR, R) = const.

Further, on an Einstein manifold we also have (see [9, p.191])

g(ϱ, Ṙ) = (τ/n)||R||^{2} ,
g(ϱ ⊗ ϱ, R̄) = τ^{3}/n^{2} ,
g(ΔR, R) = (2τ/n)||R||^{2} − Ř − 4 ˇR̄

and so, (3.5) yields

\frac{160}{63}Ř + \frac{880}{63} ˇR̄ − \frac{45}{14}||∇R||^{2} = const.  (3.6)

Next, we suppose that (M, g) is four-dimensional. Then, since ||R||^{2} is constant, from (2.2) and (2.3) we have

\{ \begin{matrix}Ř = −τ^{3}/12 + (2τ/3)||R||^{2} + (1/3)||∇R||^{2} , \\ ˇR̄ = τ^{3}/48 − (τ/24)||R||^{2} + (1/6)||∇R||^{2} , \end{matrix}  (3.7)

and hence, ||∇R||^{2} is constant. □

Next, we prove our main results.

Theorem 3.3. Let (M,g) be a four-dimensional 2-stein space. If (M,g) is ball-homogeneous, then (M,g) is flat or locally isometric to a rank-one symmetric space.

Proof. First we observe that (M, g) is a pointwise Osserman space (Proposition 2.4). Further, from (2.2), (2.3), (2.4) and (2.5) we obtain, with respect to a Singer-Thorpe basis,

\{ \begin{matrix}a + b + c = τ/4 , \\ ab + bc + ca = − (1/32)||R||^{2} + (5/192)τ^{2} , \\ 96abc = τ^{3}/12 − (1/6)τ||R||^{2} + (1/6)||∇R||^{2} . \end{matrix}  (3.8)

Since τ, ||R||^{2} and ||∇R||^{2} are constant (Lemma 3.2), it follows that a, b and c are constant on M. Hence, (M, g) is a globally Osserman space and this completes the proof because of Proposition 3.5. □

We note that the hypothesis of ball-homogeneity can be replaced by the condition ||∇R||^{2} is constant.

Lemma 3.4. Let (M,g) be a four-dimensional Kähler-Einstein space. If (M,g) is ball-homo-geneous, then (M,g) is locally symmetric.

Proof. Since (M, g, J) is a Kähler-Einstein space, we may consider for T_{m}M an adapted Singer-Thorpe basis as in Lemma 2.6. From (2.4) and (2.6) we then obtain

\{ \begin{matrix}τ = 4(a + b + c) , \\ ||R||^{2} = 8(a^{2} + 3b^{2} + 3c^{2} + 2bc) , \\ ˇR̄ = 48bc(a − b − c) . \end{matrix}  (3.9)

Further, formulas (3.7) are still valid and hence, from (3.9) we have

\{ \begin{matrix}τ = 4(a + b + c) , \\ ||R||^{2} = 8(a^{2} + 3b^{2} + 3c^{2} + 2bc) , \\ τ^{3}/48 − (τ/24)||R||^{2} + (1/6)||∇R||^{2} = 48bc(a − b − c) \end{matrix}  (3.10)

where τ, ||R||^{2} and ||∇R||^{2} are constant on M. Now, we substitute a from the first equation in (3.10) into the other equations. We then have

\{ \begin{matrix}(b + c)^{2} − (τ/8)(b + c) − bc = const. , \\ bc(τ/8 − b − c) = const. \end{matrix}  (3.11)

It follows that bc and b + c are constant, that is, b and c, and hence also a, are constant on M. Then, (M, g) is curvature homogeneous and so Proposition 2.2 yields that (M, g) is locally symmetric. □

Now, we follow the same method as in [7] to prove

Theorem 3.5. Let (M,g,J) be a connected four-dimensional Hermitian Einstein space. If (M,g) is ball-homogeneous, then (M,g) is locally symmetric.

Proof. For any four-dimensional Hermitian manifold we have (see [7])

2g((∇_{X}J)Y, Z) = ω(JX)g(X, Z) + ω(Y)g(X, JZ) −  (3.12)
− ω(JZ)g(X, Y) − ω(Z)g(X, JY)

for all smooth vector fields X, Y, Z on M and where ω = δΩ ∘ J, Ω being the Kählerform of (M, g, J). The form ω is related to τ and τ^{∗} by the formula

τ − τ^{∗} = 2δω + ||ω||^{2} .  (3.13)

Now, put

\{ \begin{matrix}K(x) & = & x(τ^{∗})H(x) , \\ N(x) & = & x(τ^{∗^{2}})g(x, x)g(x, x) , \\ P (x) & = & (∇_{x}R)(x, Jx, x, Jx) − ω(x)H(x) − \\ & & − ∑_{i=1}^{4} ω(Je_{i}) R(x, e_{i}, x, Jx) g(x, x) \end{matrix}  (3.14)

for x ∈ T_{m}M, m ∈ M. Using (2.7), (3.12) and (3.14) we obtain by differentiation (see also [7])

64xF (x) − 24K(x) + 8(τ − 3τ^{∗})P (x) + 3N(x) = 0 .  (3.15)

Next, let D denote the Euclidean Laplacian of T_{m}M. We first observe that

xF (x) = 2 ∑_{a,b} R_{xaxb}∇R_{xaxb} = 2G(x) .  (3.16)

Then, since (M, g) is Einsteinian and hence Ricci-parallel, we have [17]

D^{2}G(x) = 12x||R||^{2} .  (3.17)

In our case, since ||R||^{2} is constant, from (3.17) we get D^{2}(xF (x)) = 24x||R||^{2} = 0 . Calculating D^{2}K(x), D^{2}P (x) and D^{2}N(x) and making use of (3.16) we then obtain

(τ − 3τ^{∗})(τ + 3τ^{∗})ω(x) = 0 .  (3.18)

Now, put

M_{0} = {m ∈ M / (τ − 3τ^{∗})(τ + 3τ^{∗}) ≠ 0 at m} ,
M_{1} = {m ∈ M / τ + 3τ^{∗} = 0 at m} ,
M_{2} = {m ∈ M / τ − 3τ^{∗} = 0 at m} .

If M_{0} ≠ ∅, then, since ω(x) = 0 on M_{0}, (M_{0}, g, J) is Kählerian and hence locally symmetric (Lemma 3.4). Then (3.13) implies τ = τ^{∗} = const. . So, M_{0} = M̄_{0} and hence M_{0} = M. Next, if M_{0} = ∅, we have M = M_{1} ∪ M_{2} .

We consider first the case τ ≠ 0. Then, M_{1} ∩ M_{2} = ∅ and hence M = M_{1} or M = M_{2} since M is connected. First, assume M = M_{2}. Then from (2.7) we obtain that F (x) does not depend on x ∈ T_{m}M, that is, (M, g) is a 2-stein space and hence it is locally symmetric (Theorem 3.3). Next, let M = M_{1}. Following [7] we can construct an adapted Singer-Thorpe basis { e_{1}, e_{2}, e_{3}, e_{4} } of T_{m}M and with respect to such a basis we have

α = −τ/12 − a , β = τ/6 − b , γ = τ/6 − c .  (3.19)

From (2.2), (2.3), (2.4) and (3.19) we then obtain

\{  \begin{matrix}(a + τ/8) + b + c = 3τ/8 , \\ 16\{(a + τ/8)^{2} + b^{2} + c^{2}\} = 5τ^{2}/12 + ||R||^{2} , \\ 16\{4[(a + τ/8)^{3} + b^{3} + c^{3}] − τ[(a + τ/8)^{2} + b^{2} + c^{2}]\} = \\ = (2τ/3)||R||^{2} − (13/14)τ^{3} + (1/3)||∇R||^{2} . \end{matrix}  (3.20)

Since τ, ||R||^{2} and ||∇R||^{2} are constant, it follows from (3.20) that a, b and c are constant. So, (M, g) is curvature homogeneous and hence locally symmetric.

Finally, if τ = 0, from (3.18) we set τ^{∗} = 0 since M_{0} = ∅ and hence ω ≠ 0. Then (2.7) implies that (M, g) is 2-stein and thus locally symmetric. □

Remark 3.6. Since a D’Atri space is always ball-homogeneous [10], Theorems 3.3 and 3.5 extend the analogous results for D’Atri spaces ([18] and [7], respectively). In that case Lemma 3.4 holds without the Einstein condition [17].

Remark 3.7. It is clear that Lemma 3.4 may be replaced by the following: Let (M,g,J) be a four-dimensional Kähler manifold. Then it is locally symmetric if and only if it is Ricci-parallel and ball-homogeneous.

4. Four-dimensional Einstein C-spaces

Let (M, g) be a connected Riemannian manifold, m a point of M and x ∈ T_{m}M a unit vector. Denote by γ : r ↦ exp(rx) the geodesic with arc length r tangent to x at m. Further, let R_{γ} = R(γ̇, ·)γ̇ be the Jacobi operator along γ. (M, g) is called a C-space if the eigenvalues of R_{γ} are constant along γ, for each geodesic γ of (M, g). From this definition we obtain easily the following

Lemma 4.1. Let (M,g) be a C-space. Then

∇_{x}ϱ_{xx} = 0 ,  (4.1)
G(x) = ∑_{a,b} R_{xaxb}∇_{x}R_{xaxb} = 0 ,  (4.2)
L(x) = ∑_{a,b,c} R_{xbxc}R_{xcxa}∇_{x}R_{xaxb} = 0 ,  (4.3)

for each m ∈ M and x ∈ T_{m}M unit vector.

Proof. Since trR_{x} = ϱ_{xx} is constant along the geodesic tangent to x at m, we have at once ∇_{x}ϱ_{xx} = 0. Further, we observe that trR_{x}^{2} = ∑ R_{xaxb}^{2} and trR_{x}^{3} = ∑ R_{xaxb}R_{xbxc}R_{xcxa} . Then (4.2) and (4.3) follow in the same way as (4.1). □

Lemma 4.2. Let (M,g) be a Ricci-parallel C-space. Then ||R||^{2} is constant on M.

Proof. From Lemma 4.1 it follows that G = 0. Since (M, g) is Ricci-parallel, (3.18) holds and hence ||R||^{2} is constant on M. □

Remark 4.3. If (M, g, J) is a Kählerian C-space, then ||R||^{2} is constant on M. In fact, (M, g, J) is a Kähler manifold with ∇_{x}ϱ_{xx} = 0. Then, (M, g) is Ricci-parallel (see [16]) and so the result follows from Lemma 4.2.

Now, we can prove our main results about four-dimensional C-spaces.

Theorem 4.4. Let (M,g) be a four-dimensional 2-stein space. If (M,g) is a C-space, then (M,g) is locally flat or locally isometric to a rank-one symmetric space.

Proof. We proceed as in the proof of Theorem 3.3. Since (M, g) is Einsteinian, it is Ricciparallel and hence ||R||^{2} is constant (Lemma 4.2). Moreover, since (M, g) is 2-stein, we have (see [18], p.221)

D^{3}L(x) = 40x||∇R||^{2} .  (4.4)

But L(x) = 0 (Lemma 4.1) and hence ||∇R||^{2} is constant. So, a, b and c are constant on M and we can conclude, as in the proof of Theorem 3.3, that (M, g) is flat or locally isometric to a rank-one symmetric space. □

Theorem 4.5. A four-dimensional Kählerian C-space is locally symmetric.

Proof. First, we observe that (M, g, J) is Ricci-parallel (Remark 4.3). So, if (M, g) is locally reducible, it is locally symmetric.

Now, let (M, g) be locally irreducible. Then it is Einsteinian. We consider first the case τ ≠ 0. Let m be a point of M, x ∈ T_{m}M a unit vector and γ the geodesic tangent to x at m. Since (M, g) is a C-space, we have G(γ̇) = 0 (formula (4.2)). But G(γ̇) = (1/2)∇_{γ̇(r)}F (γ̇(r)). Then, F (γ̇(r)) is constant along γ. Since ||R||^{2} is constant, from (2.8) it follows that H(γ̇(r)) is constant along γ. Thus, by deriving along γ, we obtain

∇_{γ̇(r)}R_{γ̇(r)Jγ̇(r)γ̇(r)Jγ̇(r)} = ∇_{γ̇(r)}H(γ̇(r)) = 0 ,  (4.5)

that is, ∇_{x}R_{xJxxJx} = 0 for each x ∈ T_{m}M and hence, (M, g) is locally symmetric [16, Lemma 10].

If τ = 0, (2.8) implies that (M, g, J) is 2-stein and hence locally symmetric (Theorem 4.4). □

Theorem 4.6. Let (M,g,J) be a four-dimensional Hermitian Einstein space. If (M,g) is a C-space and ||∇R||^{2} is constant on M, then (M,g) is locally symmetric.

Proof. We may follow the same argument as for Theorem 3.5. In this case, formula (3.15) is simpler, since xF (x) = 2G(x) = 0. □

If we suppose that M is compact, we can delete the hypothesis “||∇R||^{2} is constant on M ” in Theorem 4.6. In fact, we have

Theorem 4.7. A compact, four-dimensional, Hermitian Einstein C-space is locally symmetric.

Proof. We can follow again the proof of Theorem 3.5 up to the case M = M_{1}. For this case we consider the following two possibilities:

a) τ = 0. Then, τ^{∗} = −τ/3 = 0 and (2.7) implies that (M, g) is 2-stein and hence locally symmetric.

b) τ ≠ 0. We shall prove that this case does not occur. Indeed, because of the compactness of M we can integrate (3.13) on M. We have

∫_{M}(τ − τ^{∗}) dM = ∫_{M} ||ω||^{2}dM ≥ 0 .  (4.6)

Since τ + 3τ^{∗} = 0, (4.6) yields

∫_{M} τ ≥ 0 .  (4.7)

But since τ is a nonvanishing constant, (4.7) implies τ > 0. Hence, τ^{∗} = − τ/3 < 0. Then (M, g) is Kählerian [15]. So, ω = 0 and because of (3.13), τ = τ^{∗}. Since τ + 3τ^{∗} = 0, we have τ = 0 which contradicts the hypothesis. □

Remark 4.8. It follows from Proposition 2.7 and Theorems 3.3 and 4.4 that any fourdimensional Ricci-flat Kähler manifold which is ball-homogeneous or a C-space is locally flat. This implies that a K_{3}-space (see [2]) cannot be ball-homogeneous nor a C-space.

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Received September 15, 1996