riemannian manifolds in which certain curvature operator has constant eigenvalues along each circle

Annals of Global Analysis and Geometry 15: 157–171, 1997. 157c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Riemannian Manifolds in which Certain CurvatureOperator has Constant Eigenvalues Along EachCircle

STEFAN IVANOV? and IRINA PETROVA??

Department of Geometry, Faculty of Mathematics and Informatics, University of Sofia, bul. JamesBouchier 5, 1126 Sofia, Bulgaria(E-mail: [email protected])

Abstract. Riemannian manifolds for which a natural curvature operator has constant eigenvalues oncircles are studied. A local classification in dimensions two and three is given. In the 3-dimensionalcase one gets all locally symmetric spaces and all Riemannian manifolds with the constant principalRicci curvatures r1 = r2 = 0, r3 6= 0, which are not locally homogeneous, in general.

Key words: circles, constant eigenvalues of the curvature operator, curvature homogeneous spaces,locally symmetric spaces

1991 Mathematics Subject Classification: 53C15, 53C55, 53B35

1. Introduction

Curvature is a fundamental notion of differential geometry. A useful techniqueto describe the curvature along a geodesic in a Riemannian manifold is the useof the Jacobi operator. In [1] Berndt and Vanhecke have introduced the so calledC-spaces andP-spaces as a generalizations of locally symmetric spaces. A C-space(respectivelyP-space) is by definition a Riemannian manifold for which the Jacobioperator has constant eigenvalues (respectively parallel eigenspaces) along everygeodesic.

In the present note we describe the curvature along a unit circle using thecurvature operator. A unit circle in a Riemannian manifold is a smooth curve withfirst curvature equal to one and all other curvatures equal to zero (cf. Nomizu andYano [8]). We consider Riemannian manifold for which natural curvature operatorsdefined along every unit circle have constant eigenvalues along the circle. Such aRiemannian manifold is called an O-space. For example, every locally symmetricspace is an O-space.

? The author was supported by Contract MM 423/1994 from the Ministry of Science and Educationof Bulgaria and by Contract 219/1994 from the University of Sofia “St. Kl. Ohridski”.?? The author was supported by Contract MM 413/1994 from the Ministry of Science and Education

of Bulgaria.

158 S. IVANOV AND I. PETROVA

An equivalent point of view is the following. Let c(t) be a unit circle on aRiemannian manifold (M; g) and let Tc(t)M denote the tangent space at everypoint c(t). We may consider the “extrinsic circle flow” on the orthonormal 2-framebundle of M whose orbits are defined by t �! ( _c(t); n(t)) 2 Tc(t)M � Tc(t)M,where _c(t) and n(t) denote the tangent vector field and the first normal field of aunit circle c(t), respectively. An O-space is a space such that the skew-symmetriccurvature operatorsR( _c(t); n(t)) on Tc(t)M have constant eigenvalues along everyorbit.

The main result of this note is the following

THEOREM 1.1. Let (M; g) be a 3-dimensional O-space of class C1. Then M islocally isometric almost everywhere (that is on an open and dense subset of M) toone of the following spaces:

(i) a space of constant Riemannian sectional curvature;(ii) a Riemannian product of the form M2 � R, where M2 is a 2-dimensional

Riemannian manifold of constant sectional curvature.(iii) a Riemannian manifold with constant principal Ricci curvaturesr1 = r2 = 0,

r3 6= 0.Conversely, any 3-dimensional Riemannian manifold of type (i), (ii) or (iii) is anO-space.

Milnor has studied in [7] 3-dimensional Lie groups with left invariant Riemannianmetrics with constant principal Ricci curvatures r1 = r2 = 0, r3 6= 0. He hasfound a lot of examples of such spaces which are locally homogeneous but notlocally symmetric (SU(2) with its special left invariant metric for example). ByTheorem 1.1 these spaces are examples ofO-spaces which are locally homogeneousbut not locally symmetric.

According to Singer [10] a Riemannian manifold (M; g) is called curva-ture homogeneous if for any pair of points p; q 2 M there is a linear isom-etry F : TpM �! TqM between the corresponding tangent spaces such thatF �Rq = Rp (where R denotes the curvature tensor of type (0; 4)). It is wellknown that a 3-dimensional Riemannian space is curvature homogeneous iff it hasconstant principal Ricci eigenvalues. Recently, Kowalski has constructed in [5] anexplicit example of Riemannian 3-manifold with constant principal Ricci curva-tures r1 = r2 = 0, r3 6= 0 which is not locally homogeneous. By Theorem 1.1this is an example of O-space which is curvature homogeneous but not locallyhomogeneous. We describe a special kind of it.

EXAMPLE 1 [5]. We consider R3 with the following nonhomogeneous Riemannianmetric

ds2 = e�2ydz2 +h(eydx+ z(e�y � x2ey)dz

i2+ (dy + 2xzdz)2;

where x; y; z are the standard coordinates in R3. Its principal Ricci curvatures aregiven by: r1 = r2 = 0, r3 = �2.

CONSTANT EIGENVALUES ON CIRCLES 159

By Theorem 4 in [1] this O-space is not a C-space. We show (see Remark 2 atthe end of this note) that this example is not also aP-space. Moreover, Theorem 1.1and Theorem 7 in [1] imply (see also Remark 2) that a 3-dimensional Riemannianmanifold M is locally symmetric iff M is an O-space and M is a P-space simulta-neously. On the other hand, a 3-dimensional Riemannian manifold M which is bothO- and P-space may not be a locally symmetric space. (For example, SU(2) withits special left invariant metric is anO-space and it is also a C-space by Theorem 4in [1].)O-spaces can be regarded as a nontrivial generalization of locally symmetric

spaces since every locally symmetric space is an O-space but, in dimension 3,there is a family ofO-spaces which are not even locally homogeneous and dependessentially on two arbitrary functions of one variable [5]. Theorem 1.1 impliesthat every 3-dimensional O-space is curvature homogeneous but there is a lot ofcurvature homogeneous (and even locally homogeneous) 3-manifolds which arenot O-spaces [5, 6, 11].

2. Characterizations of O-spaces

Let (M; g) be an n-dimensional Riemannian manifold and r the Levi–Civitaconnection of the metric g. We denote by TpM the tangential space at a point pof M. The curvature operator is defined by R(X;Y ) = [rX ;rY ] � r[X;Y ] forevery smooth vector fields X;Y;Z; V on M. The curvature tensor of type (0; 4) isdefined by R(X;Y;Z; V ) := g(R(X;Y )Z; V ).

A smooth curve c(t) : I� ! M; I� = (��; �), � > 0, with tangent vector field_c, parameterized by the arc length, is said to be a circle of curvature k1 if its firstcurvature k1 is a constant different from zero and all other curvatures are zeros [8].A unit circle is a circle with curvature k1 = 1. For a unit circle we have

r _c _c = n; r _cn = � _c: (2.1)

The unit vector field n is the first normal of c(t). All other normals orthogonalto _c and n are parallel along c(t). The differential equations for a unit circle isequivalent to

r _cr _c _c+ _c = 0: (2.2)

If dim M � 2 then for every point p 2 M and every two orthonormal vectorsu; v 2 TpM there exists locally a unique unit circle c(t), parameterized by the arclength and satisfying the following initial conditions

c(0) = p; _c(0) = u; (r _c _c)(0) = v: (2.3)

Let c(t) be a unit circle on M. We consider the families of smooth skew-symmetriclinear operators along c(t) defined by

�c(X) := R( _c;r _c _c)X; �0c(X) := (r _cR)( _c;r _c _c)X (2.4)


for every vector field X along c(t).It follows from (2.1) and (2.4) that

�0c(X) = r _c�c(X)� �c(r _cX): (2.5)

We have the following characterization of locally symmetric spaces:

PROPOSITION 2.1. For an n-dimensional (n � 2) Riemannian manifold (M; g)the following conditions are equivalent:

(i) (M; g) is a Riemannian locally symmetric space;(ii) For every unit circle c on M we have �0c = 0;

(iii) For every unit circle c on M we have �c � r _c = r _c � �c;(iv) For every unit circle c on M the operator �c transforms a vector field parallel

along c to a vector field parallel along c;(v) For every unit circle c on M the matrix of �c with respect to a basis of vector

fields parallel along c is invariant, i.e. the matrix has constant entries.Proof. The implication (i) ! (ii) is clear.To prove the converse, let u; v be two orthonormal vectors at a point p 2 M. Let

c(t) be the unit circle on M determined by the initial conditions (2.3). For everyw 2 TpM, we get from �0c = 0 the equality

(ruR)(u; v)w = 0: (2.6)

By the well known properties of the Riemannian curvature operator we can con-clude that (2.6) is valid for arbitrary u; v; w 2 TpM. If z 2 TpM, replacing v byv + z in (2.6) and using the second Bianchi identity we get rR = 0 which proves(i).

The equivalence of (ii), (iii), (iv) and (v) follows from (2.5). 2

We recall some of the basic properties of the skew-symmetric linear operators.Let A be a skew-symmetric linear operator on a Euclidean vector space E. If

A has a real eigenvalue then it has to be zero. If A possesses a nonzero eigenvalueof multiplicity m then it has to be of the form

p�1a, a 2 R. Then �p�1ais also an eigenvalue of A of the same multiplicity. Let A has eigenvalues 0,p�1a1,�p�1a1, : : : ,

p�1al,�p�1al of multiplicitiesm;m1;m1; : : : ;ml;ml,

respectively. Then there exists an orthonormal basis Z1, : : : , Zm, X(1)1 , Y (1)

1 , : : : ,

X(1)m1 , Y (1)

m1 , : : : , X(l)1 , Y (l)

1 , X(l)ml , Y

(l)ml of E such that

A(Zp) = 0; p = 1; : : : ;m; (2.7)

A(X(j)s ) = ajY

(j)s ; A(Y (j)

s ) = �ajX(j)s ; j = 1; : : : ; l; s = 1; : : : ;mj:

Such a basis of E is known as a Jordanian basis for A.


DEFINITION 2.1. A Riemannian manifold (M; g) is said to be an O-space if forany unit circle c(t) the operator �c has constant eigenvalues along c(t).

DEFINITION 2.2. A Riemannian manifold (M; g) is said to be a T -space if forany unit circle c(t) the operator �c has Jordanian basis parallel along c(t) (at leastlocally around almost every point of c(t)).

Remark 1. It follows from Proposition 2.1 that a smooth Riemannian manifoldM is a locally symmetric Riemannian manifold iff M is an O-space and M is aT -space simultaneously.

Due to this observation it is natural to study theO-spaces and the T -spaces sep-arately. This leads to two natural generalizations of locally symmetric Riemannianmanifolds.

In this paper we consider the O-spaces. The T -spaces we study in [4].Our further considerations are based on the following simple observation: If a

skew-symmetric linear operatorA on an Euclidean vector spaceE has eigenvalues0,p�1a1, �p�1a1, : : : ,

p�1al, �p�1al then the operator A2 is self-adjoint

linear operator and it has eigenvalues 0, �a21, : : : , �a2

l .

PROPOSITION 2.2. Let (M; g) be an n-dimensional (n � 2) Riemannian mani-fold. Then M is anO-space iff the self-adjoint operator�2

c has constant eigenvaluesalong every unit circle c on M.

Further, for each point p 2 M and for any two orthonormal tangent vectors u; v atp we define two skew-symmetric endomorphisms �u;v , �0u;v of TpM by

�u;v = R(u; v); �0u;v = (ruR)(u; v):

Then the endomorphisms �2u;v = �u;v ��u;v and (�2

u;v)0 = �0u;v ��u;v+�u;v ��0u;v

are self-adjoint endomorphisms of TpM.We have

PROPOSITION 2.3. A Riemannian manifold (M; g)(dim M � 2) is anO-space ifffor each point p 2 M and for any two vectorsu; v at p there exists a skew-symmetricendomorphism Tu;v of TpM such that:

(�2u;v)

0 = �2u;v � Tu;v � Tu;v � �2

u;v: (2.8)

Proof. We notice that if (2.8) holds for orthonormalu; v then it holds for arbitraryu; v. Let (M; g) be an O-space and u; v orthonormal vectors at a point p 2 M. Letc(t) be the unit circle on M determined by the initial conditions (2.3) andA(t) theself-adjoint endomorphism of TpM which is obtained by a parallel translation of�2c along c from c(t) to p. Applying Lemma 4 of [1] to the operatorA(t), we obtain

the endomorphism Tu;v of TpM.To prove the converse, let c(t) be a unit circle determined by (2.3). Let A(t)

be defined as above and let T (t) be the skew-symmetric endomorphism of TpM


which is obtained by parallel translation of T _c;r_c _c along c from the point c(t) to the

point p. ThenA0 = A �T �T �A and Lemma 4 of [1] implies that the eigenvaluesof A, therefore of �2

c, are constant. By Proposition 2.2, M is an O-space. 2

Further we have

THEOREM 2.4. Let (M; g) be a smooth Riemannian manifold of dimension n

(n � 2). Then the following conditions are equivalent:(i) M is an O-space;(ii) For each point p 2 M and for any two tangent vectors u; v at p there existsa skew-symmetric endomorphism Tu;v of TpM such that

�0u;v = �u;v � Tu;v � Tu;v � �u;v: (2.9)

Proof. To prove (i) ! (ii) we need the following

LEMMA 2.5. Let c(t) be a unit circle and �t be a smooth family of skew-symmetriclinear operators along c(t). If the eigenvalues of �t are constant then there existslocally a smooth Jordanian basis for �t along c(t).

Proof of the Lemma 2.5. By the conditions of the lemma the eigenvalues ofthe family of self-adjoint operators �2

t are also constant. Then �2t is locally Ck-

diagonalizable, say �t(Ei(t)) = �a2iEi(t); ai 2 R; i = 1; : : : ; l. Then by the

standard procedure we obtain locally a smooth Jordanian basis B for �t, whichproves the lemma.

Let M be an O-space and u; v tangent vectors at a point p 2 M. Without lossof generality we may assume that u; v are orthonormal. Let c(t); t 2 I , be the unitcircle determined by the initial conditions (2.3). By Lemma 2.5 we can find locallya smooth Jordanian basis B of �c along the circle c(t). We consider the family �tof skew-symmetric linear operators along c(t), defined by �t(w) := (r _cw)t; forw 2 B, t 2 J � I . We shall prove that for each w 2 B and for each t 2 J thefollowing equality holds:

(r _cR)( _c;r _c _c)w = �(R( _c;r _c _c)w)�R( _c;r _c _c)�(w): (2.10)

Using (2.2), we obtain from (2.4) the equality

�0c(w) = r _c(�cw)� �c�(w): (2.11)

It is easy to verify that the following equality holds:

r _c(�cw) = �(�c(w)): (2.12)

Now, (2.10) follows from (2.11) and (2.12). Set Tu;v(t) := �t. Then (2.10) implies(2.9).

For the converse, if (ii) holds then it is easy to check that (2.8) also holds. ThenProposition 2.3 implies (i) which completes the proof of the theorem. 2

As an immediate consequence of Theorem 2.4 we get


COROLLARY 2.6. If (M; g) admits an (1; 2) tensor field T written as TXZ , skew-symmetric with respect to the metric g and such that

(rXR)(X;Y )Z = TXR(X;Y )Z �R(X;Y )TXZ

for all tangent vectors X;Y;Z at every point of M, then (M; g) is an O-space.

COROLLARY 2.7. Let M = M1 � : : : � Mr be a locally reducible smooth Rie-mannian manifold and each of Mi is smooth. Then M is an O-space iff each of Mi

is an O-space.

3. Two and Three Dimensional O-Spaces

In this section we treat the classification problem for O-spaces of dimension twoand three.

For the 2-dimensional case the eigenvalues of �c arep�1s and�p�1s, where

s is the scalar curvature. Thus we have

PROPOSITION 3.1. A connected 2-dimensional smooth Riemannian manifold isan O-space iff it is of constant sectional curvature.

The key point for the 3-dimensional case is the following

PROPOSITION 3.2. Let (M; g) be a 3-dimensional Riemannian manifold. ThenM is an O-space iff the following identity holds

R(X;Y; Y;X)(rXR)(X;Y; Y;X) +R(X;Y; Y; Z)(rXR)(X;Y; Y; Z)

+R(Y;X;X;Z)(rXR)(Y;X;X;Z) = 0 (3.13)

for any orthonormal vectors X;Y;Z at any point p 2 M.Proof. Since M is 3-dimensional then M is an O-space iff for any unit circle

c(t)

trace(�2c) = const: (3.14)

Let c(t) be the unit circle determined by the initial conditions (2.3), n its firstnormal given by (2.2) and b its second normal. The second normal b is parallelvector field along c(t) and _c; n; b form an orthonormal basis along the circle c(t).The equality (3.14) is equivalent to the following equation

_c�g(R( _c; n) _c;R( _c; n) _c) + g(R( _c; n)n;R( _c; n)n) + g(R( _c; n)b;R( _c; n)b)

�= 0:

By (2.1) the latter equality can be written as

g((r _cR)( _c; n) _c;R( _c; n) _c) + g((r _cR)( _c; n)n;R( _c; n)n)

+ g((r _cR)( _c; n)b;R( _c; n)b) = 0:


Evaluating this equality at the point p we obtain (3.13). 2

In every point p 2 M we consider the Ricci operator Ric as a linear self-adjointoperator on the tangential space TpM. Let be the subset of M on which thenumber of distinct eigenvalues of Ric is locally constant. This set is open and denseon M. We can chooseC1 eigenvalue functions of Ric on , say r1; r2; r3, in sucha way such that they form the spectrum of Ric at each point of (see for example[1, 5, 9]). We fix a point p 2 . Then there exists a local orthonormal frame fieldE1, E2, E3 on an open connected neighborhood U of p such that

Ric(Ei) = riEi; i = 1; 2; 3: (3.15)

We set !ij;k = g(rEiEj; Ek), i; j; k 2 f1; 2; 3g.

We obtain from Proposition 3.2 the following technical result

PROPOSITION 3.3. Let (M; g) be a 3-dimensional Riemannian manifold. Let be an open and dense set of M and r1, r2, r3 smooth functions forming at eachpoint of the spectrum of Ric. Let U � and let E1, E2, E3 be orthonormalvector fields on U satisfying the condition (3.15). If M is an O-space then, for alldistinct i; j; k 2 f1; 2; 3g, the following formulae are valid on U :

Ei(ri) = 0; Ei(rj � rk) = 0; (3.16)

Ei(rj(rk � ri)) = �2rj(rk � ri)!kk;i ; (3.17)

rj(rk � ri)!kk;i = rk(rj � ri)!jj;i ; (3.18)

ri(rj � rk)!ij;k = 0: (3.19)

Proof. For a 3-dimensional Riemannian, manifold the curvature tensor R isgiven by

R(X;Y )Z = ric(Y;Z)X � ric(X;Z)Y + g(Y;Z)Ric(X)� g(X;Z)Ric(Y )

� 12s[g(Y;Z)X � g(X;Z)Y ]; X; Y; Z 2 TpM; p 2 M; (3.20)

where ric and s denote the Ricci tensor and the scalar curvature of R, respectively.Using (3.20), we see that (3.13) is equivalent to

[s� 2ric(Z;Z)][2(rX ric)(X;X) + 2(rXric)(Y; Y )�X(s)]

+ 4ric(X;Z)(rX ric)(X;Z) + 4ric(Y;Z)(rXric)(Y;Z) = 0; (3.21)

for any orthonormal tangent vectors X;Y;Z .Let A = (a1; a2; a3), B = (b1; b2; b3), C = (c1; c2; c3) be three orthonormal

vectors in R3. We set X = aiEi, Y = biEi, Z = ciEi (the Einstein summation


convention is assumed). Substituting these vectors into (3.21), we obtain after somecalculations

[r1 + (2(c3)2 � 1)r2 + (2(c2)2 � 1)r3]X(r1)

+ [r2 + (2(c3)2 � 1)r1 + (2(c1)2 � 1)r3]X(r2)

+ [r3 + (2(c2)2 � 1)r1 + (2(c1)2 � 1)r2]X(r3)

� 4a1c1c2r3(rE1ric)(E1; E2)� 4a2c1c2r3(rE2ric)(E2; E1)




� 4a3c1c2r3(rE3ric)(E1; E2) = 0: (3.22)

If we take A = (1; 0; 0), C = (0; 1; 0) then (3.22) implies

E1(r1 � r2 + r3) = 0: (3.23)

Choosing A = (1; 0; 0), C = (0; 0; 1) we get from (3.22) that

E1(r1 + r2 � r3) = 0: (3.24)

The equalities (3.23) and (3.24) imply E1(r1) = E1(r2 � r3) = 0. Similarly weget all other equalities in (3.16).

Further, we take A = (1; 0; 0), C = (0;p

2=2;p

2=2). Using (3.16) and (3.22)we get r1(rE1ric)(E2; E3) = 0. We obtain (3.19) in an analogous way.

Since the vectors A(a1; a2; a3), B(b1; b2; b3), C(c1; c2; c3) are orthonormal,the vectors A0 = (�a1; a2; a3), B0 = (�b1; b2; b3), C 0 = (�c1; c2; c3) are alsoorthonormal vectors. We take consequently a1 =

p2=2, a2 = 0, a3 =

p2=2,

c1 =p

2=2, c2 = 0, c3 = �p2=2 and a1 = 0, a2 =p

2=2, a3 =p

2=2,c1 =

p3=3, c2 =

p3=3, c3 = �p3=3. ReplacingA, B, C byA0, B0, C 0 in (3.22),

subtracting the last equality from (3.22) and taking into account (3.16) and (3.19)we derive

�(�r1 + r2 + r3)E1(r2) = 2r2(rE3ric)(E3; E1) = 2r3(rE2ric)(E2; E1):

Similarly, for distinct i; j; k = 1; 2; 3 we have

�(�ri + rj + rk)Ei(rj) = 2rj(rEkric)(Ek; Ei) = 2rk(rEj

ric)(Ej ; Ei):

The latter equations are equivalent to the equations:

�(�ri + rj + rk)Ei(rj) = 2rj(rk � ri)!kk;i = 2rk(rj � ri)!jj;i

for distinct i, j, k = 1; 2; 3. Taking into account (3.16), we get (3.17) and(3.18). 2


4. Proof of Theorem 1.1

Let be an open and dense set in M and r1; r2; r3 smooth functions on repre-senting the spectrum of the Ricci operator. Let � be the set of all points p of where each of the functions ri, ri � rj , i; j = 1; 2; 3, i 6= j (separately) is eitheridentically zero in a neighborhood of p or nonzero in a whole neighborhood. Then� is open and dense in M.

Let p 2 M and let U � � be a neighborhood of p. We can find smoothorthonormal vector fieldsE1,E2,E3 onU satisfying (3.15). We have the followingpossibilities for the functions r1(r2 � r3), r2(r3 � r1), r3(r1 � r2) on U :

Case A. All of them are identically equal to zero;Case B. One of them is identically equal to zero and the others are nonzero

everywhere on U ;Case C. All of them are nonzero everywhere on U .CASE A: In this case we have (up to ordering) two possibilities for the functions

r1, r2, r3:

r1 = r2 = r3 and r1 = r2 = 0; r3 6= 0:

In the first case (U; g) is an Einstein manifold and hence it is of constant sectionalcurvature. In the second case we get r3 = const: 6= 0 from (3.16).

CASE B: Let us assume

r1(r2 � r3) = 0; r2(r3 � r1) 6= 0; r3(r1 � r2) 6= 0: (4.25)

Using (4.25), we get from (3.18) and (3.19) that

!33;1 = !22;1; !23;1 = !32;1 = !11;2 = !11;3 = 0: (4.26)

Taking into account (4.26) we calculate

0 = ric(E2; E1) = g(R(E2; E3)E3; E1) = E2(!33;1);

0 = ric(E3; E1) = g(R(E3; E2)E2; E1) = E3(!22;1): (4.27)

Because of (4.26) and (4.27), we can apply a result of Hiepko [3] to obtain that Uis locally a warped product of type N1 �� N2 with a 1-dimensional base N1 and a2-dimensional leaf N2 where � > 0 is a smooth function on N1. Let us denote by� and � the natural projections � : N1 �N2 ! N1 and � : N1 �N2 ! N2. Thenwe have

g = ��gN1 + (��)2��gN2 ;

where gN1 and gN2 are the metric on N1 and N2, respectively. By a simple calcu-lation we obtain

r1 = �2E2

1(��)

��; r2 = r3 =

��k

(��)2 �(E1(�

��))2

(��)2 � E21(�

��)

��; (4.28)


where k is the sectional curvature ofN2 with respect to gN2 . Using (4.28) we derivefrom (3.16) and (3.17) that

r1 = const:; E1(r3(r3 � r1)) = �2r3(r3 � r1)!33;1: (4.29)

Further, using (4.29) and (4.28) we calculate

E1(r3(r3 � r1)) = (2r3 � r1)E1(r3) = 2(r3 � r1)(2r3 � r1)!33;1: (4.30)

By (4.25), r3 � r1 6= 0 on U . Let p 2 U and !33;1(p) 6= 0. Then there existsa neighborhood of p, say W � U , such that !33;1 6= 0 everywhere on W . Theequalities (4.30) and (4.29) imply r3 = (1=3)r1 = const: Then (4.29) impliesr3 = r1 = 0 on W , but this is impossible because of (4.25). Hence, !33;1 = 0 onU . Analogously, !22;1 = 0 on U . Therefore M = N1 � N2 [3, p. 209]. Since Mis an O-space then N2 is of constant sectional curvature by Proposition 3.1 andCorollary 2.7.

CASE C: We may assume

ri(rj � rk) 6= 0; (4.31)

for distinct i; j; k;2 f1; 2; 3g. We get !ij;k = 0 for distinct i; j; k = 1; 2; 3 from(3.19). Then we can apply Lemma 8 of [1] to conclude that there is a local coordinatesystem (U; x) of M and positive functions �i(x1; x2; x3); i = 1; 2; 3 such that themetric g is given in the form

g = �21(x1; x2; x3)dx

21 + �2

2(x1; x2; x3)dx22 + �2

3(x1; x2; x3)dx23: (4.32)

and the coordinate vectors Xi := @=@xi are eigenvectors of the Ricci tensoreverywhere on U . We may assume Xi = �iEi, i = 1; 2; 3.

Following [1] we set

�s = ln(�s); �s;t =@�s

@xt; �s;tq =

@2�s

@xt@xq;

Ric(Xt) =3X

q=1

Sqt xq; s; t; q = 1; 2; 3:

Then for distinct i; j; k we have [1].

Sii =1�2j

[�i;j�j;j � �2i;j � �i;jj � �i;j�k;j]

+1�2k

[�i;k�k;k � �2i;k � �i;kk � �i;k�j;k]

+1�2i

[�j;i�i;i � �2j;i � �j;ii + �k;i�i;i � �2

k;i � �k;ii]; (4.33)


�i;jk = �i;j�j;k + �i;k�k;j � �i;j�i;k; (4.34)

�k;j + �j!kk;j = 0: (4.35)

We shall prove that the following system of nonlinear partial differential equationsholds locally for distinct i; j; k;= 1; 2; 3

�i;jk = 0;

�i;j�i;k � �i;j�j;k � �i;k�k;j = 0;

�i;ij + 2�i;j�j;k = 0: (4.36)

We get from (4.35) and (3.18) that

ri(rk � rj)�k;j = rk(ri � rj)�i;j (4.37)

for distinct i; j; k.LetV be the set of all points p ofU where each of the functions �i;j, i; j = 1; 2; 3,

i 6= j is either identically zero in a neighborhood of p or nonzero in a wholeneighborhood. Then V is open and dense in U . If �i;j = 0 around a point p 2 V ,then �i;ij = �i;ji = 0 and it follows from (4.31) and (4.37) that the last twoequations of (4.36) are fulfilled. The first equation of (4.36) follows from (4.34)and the second equation from (4.36).

Now, let �i;j 6= 0 around a point p 2 V . Then, from (4.31) and (4.37), we get

�i;j

�k;j=ri(rk � rj)

rk(ri � rj)= 1� rj(rk � ri)

rk(rj � ri): (4.38)

Replacing j by k in (4.37) we get ri(rj�rk)�j;k = rj(ri�rk)�i;k. Then we obtain

ri(rj � rk)(�j;k � �i;k) + rk(rj � ri)�i;k = 0: (4.39)

Using (4.38), (4.31) and (4.39) we see that the second equation of (4.36) is valideverywhere on V . Then (4.34) and the second equation of (4.36) imply the firstequation of (4.36). By (3.16), (3.17) and (4.38) we obtain:

�i;j

�k;jXi(rj(rk � ri)) = �Xi

"�i;j

�k;j

#rk(rj � ri): (4.40)

Using (3.17), (4.35) and (4.37), we get from (4.40) that

2�i;j

�k;j�j;i = �Xi

"�i;j

�k;j

#:

The latter equation and the first equation of (4.36) imply the third equation of(4.36).

The system (4.36) is well known: it arises from a quantum mechanical problems;on the other hand it is shown in [1] that this system describes 3-dimensional P-spaces with distinct Ricci eigenvalues. The complete solution of this system hasbeen given by Eisenhart in [2]. He has obtained four kinds of solutions, namely:


(E1) �21 = 1, �2

2 = G(x2)�(x1)(�(x2) + (x3)); �23 = H(x3)�(x1)(�(x2) +

(x3)), where � is a function of x1, G;� are functions of x2, H , arefunctions of x3 and G, H , � > 0;

(E2) �21 = 1, �2

2 = �2(x1), �23 = 2(x1), where � and are functions of x1;

(E3) �21 = �2

2 = (�(x1) + (x2)), �23 = j�(x1) + (x2)j, where � is a function

of x1 and is a function of x2.(E4) �2

i = Fi(xi)jxi � xjjjxi � xkj, for distinct i; j; k = 1; 2; 3, where Fi arefunctions of xi.

To complete the proof of Theorem 1.1 we show below that any metric of theform (4.32) with anyone of the solutions E1, E2, E3, E4 does not appear as ametric of an O-space.

CASE E1. In this case it is proved in [1] that U is precisely the warped productof a 1-dimensional base and a 2-dimensional leaf. Hence, the number of distincteigenvalues of the Ricci operator Ric is at most two, which contradicts to (4.31).

CASE E2. In this case we have �1 = �1;1 = �1;2 = �1;3 = �2;2 = �2;3 = �3;2 =�3;3 = 0. Using (4.33) we calculate

r1 = S11 = ��2

2;1 � �2;11 � �23;1 � �3;11;

r2 = S22 = ��2

2;1 � �2;11 � �2;1�3;1;

r3 = S33 = ��2

3;1 � �3;11 � �3;1�2;1: (4.41)

Using (3.16) we obtain from (4.41)

�22;1 + �2;11 = B = const:; �2

3;1 + �3;11 = C = const: (4.42)

Now, (4.41) takes the form:

r1 = �B � C; r2 = �B � �2;1�3;1; r3 = �C � �2;1�3;1: (4.43)

The conditions (4.31) imply

B 6= C; B 6= �C: (4.44)

We obtain from (3.17) and (4.37) that

(�2;1�3;1)2(�3;1 � �2;1) = B2�3;1 � C2�2;1: (4.45)

X1(�2;1�3;1)2 = 2((�2;1�3;1)

2 �B2)�3;1: (4.46)

We claim that

�2;1 � �3;1 6= 0; �2;1 6= 0; �3;1 6= 0: (4.47)

Indeed, if �2;1 = 0 at a point of V , then the equality (4.42) implies B = 0 and,from (4.43) we get r2 = 0 which contradicts to (4.31). Similarly we prove that


�3;1 6= 0. Further, if �2;1 = �3;1 at a point of V then (4.45) impliesB2 = C2, whichcontradicts to (4.44).

Keeping in mind (4.47) we get from (4.45) and (4.46):

(�2;1�3;1)2 =

B2�3;1 � C2�2;1

�3;1 � �2;1: (4.48)

(�2;1�3;1)0 =

B2 �C2

�3;1 � �2;1: (4.49)

Differentiating (4.48) and using (4.49) and (4.42) we get:

�2;1�3;1 =B�3;1 � C�2;1

�3;1 � �2;1: (4.50)

From (4.48), (4.49) and (4.50), keeping in mind (4.47), we get B2 = C2 which isimpossible because of (4.44).

CASE E3. In this case we have (see the calculations in [1]):

�2;1 =�0

2(�+ ); �3;1 =

�0

2��2;2 =

0

2(�+ ); �3;2 =

0

2 ): (4.51)

r2 � r3 = T ; r3 � r1 = �T; r1 � r2 = �(�+ )T;

where

T =1

4(�+ )3

(1�

"2�00(�+ )� (�0)2

�(3�+ )

#

� 1

"2 00(�+ )� ( 0)2

(3 + �)

#):

In any point of V at least one of the functions �0 or 0 is different from zero (if�0 = 0 = 0, then T = 0 and r1 = r2 = r3, which is impossible because of(4.31)).

Let p 2 V and �0(p) 6= 0. Then, by (4.51), we have at p:

�2;1

�3;1=

�

�+ =r3 � r1

r2 � r16= 0:

Comparing this with (4.38) we get r2 = r3 at p, which contradicts to (4.31).CASE E4. In this case we have [1, p. 78] for all distinct i; j 2 f1; 2; 3g

�i;j =1

xj � xi; ri � rj = (xi � xj)T;

T =�1;2;3

�2F1

�1

x1�x3+ 1

x1�x2

��

1F1

�0�(x2 � x3)

2

4(x1 � x2)2(x2 � x3)2(x3 � x1)2 : (4.52)


If at some point p 2 V we have T (p) = 0, then ri = rj which is impossible on V .Let T (p) 6= 0. We derive from (4.52) that

rk � ri

rj � ri=�j;i

�k;i:

Similar arguments as in the case E3 show that the case E4 is also impossible.It is easy to verify that any manifold of type (i), (ii) and (iii) is an O-space.

Especially, in the case (iii) we check that r1 = r2 = 0 and r3 = const: alwayssatisfies Equation (3.22), which is equivalent to (3.13). This completes the proofof Theorem 1.1. 2

Remark 2. It follows from the proof of Theorem 7 in [1] that if a 3-dimensionalP-space M has two distinct Ricci eigenvalues then it is locally isometric to awarped product N1 �� N2 with a 1-dimensional base N1 and a 2-dimensional leafN2 where � > 0 is a smooth function onN1. If moreover, we assume that M is alsoanO-space and r1 = const:, r2 = r3 it follows from the proof of Theorem 1.1 thatM is locally isometric to the Riemannian productN1 �N2 and by Theorem 1.1 Mhas to be a locally symmetric Riemannian manifold. Hence, theO-space describedin Example 1 is not a P-space.

Acknowledgement

We would like to thank the referee for his remarks and helpful suggestions.

References

1. Berndt, J. and Vanhecke, L.: Two natural generalizations of locally symmetric spaces, J. Diff.Geom. and Appl. 2 (1992), 57–80.

2. Eisenhart, L. P.: Separable system of Stackel, Ann. of Math. 35 (1934), 284–305.3. Hiepko, S.: Eine innere Kennzeichnung der verzerrten Produkte, Math. Ann. 241 (1979), 209–

215.4. Ivanov, S. and Petrova, I.: Curvature operator with parallel Jordanian basis on circles, Riv. Mat.

Univ. Parma, to appear.5. Kowalski, O.: A classification of Riemannian manifolds with constant principal Ricci curvatures

r1 = r2 6= r3, Nagoya Math. J. 132 (1993), 1–36.6. Kowalski, O. and Prufer, F.: On Riemannian 3-manifolds with distinct constant Ricci eigenvalues,

Math. Ann. 300 (1994), 17–28.7. Milnor, J.: Curvature of left-invariant metrics on Lie groups, Adv. in Math. 21 (1976), 163–170.8. Nomizu, K. and Yano, K.: On circles and spheres in Riemannian geometry, Math. Ann. 210

(1974), 163–170.9. Shabo, Z.: Structure theorems on Riemannian spaces satisfying R(X;Y ) � R = 0. I. The local

version, J. Diff. Geom. 17 (1982) 531–582.10. Singer, I. M.: Infinitesimally homogeneous spaces, Commun. Pure Appl. Math 13 (1960), 685–

697.11. Spiro, A. and Tricerri, F.: 3-dimensional Riemannian metrics with prescribed Ricci principal

curvatures, J. Math. Pure Appl. 74 (1995), 253–271.

Communicated by O. Kowalski(Received April 22, 1996; revised version November 7, 1996)

riemannian manifolds in which certain curvature operator has constant eigenvalues along each circle

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