Complex Osserman Algebraic Curvature Tensors and Clifford Families

Abstract

We classify the algebraic curvature tensors which are both Osserman and complex Osserman in all but a finite number of exceptional dimensions.Information concerning the possible eigenvalue structures, which is provided by methods of algebraic topology, plays a central role in the analysis.

Full text

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arXiv:math/0608749v1 [math.DG] 30 Aug 2006
COMPLEX OSSERMAN ALGEBRAIC CURVATURE TENSORS
AND CLIFFORD FAMILIES
M. BROZOS-V´AZQUEZ AND P. GILKEY
Abstract. We use methods of algebraic topology to study the eigenvalue
structure of a complex Osserman algebraic curvature tensor. We classify the
algebraic curvature tensors which are both Osserman and complex Osserman
in all but a finite number of exceptional dimensions.
1. Introduction
In recent years, the Osserman problem has played an important role in the under-
standing of curvature. The real setting has been studied previously; in this paper,
we study the complex setting. We introduce the following notational conventions.
Let ∇ be the Levi-Civita connection of a Riemannian manifold (M,g) and let R be
the associated Riemann curvature tensor:
R(x,y,z,w) := g((∇x∇y− ∇y∇x− ∇[x,y])z,w).
The Jacobi operator JRand the skew-symmetric curvature operator R are charac-
terized by the identities:
(1.a)g(JR(x)y,z) = R(y,x,x,z)
Motivated by the seminal paper of Osserman [13], one says that (M,g) is Osserman
if the eigenvalues of JRare constant on the sphere bundle S(M,g) of unit tangent
vectors. Since the local isometries of a local two-point homogeneous manifold act
transitively on S(M,g), such manifolds are Osserman. Osserman wondered if the
converse was also true, that is, are Osserman manifolds necessarily local two-point-
homogeneous spaces. This question has been called the Osserman conjecture by
subsequent authors and has been also considered in the pseudo-Riemannian context;
in this paper, we will only work in the Riemannian context and refer to [5, 9] for a
discussion of the pseudo-Riemannian setting.
andg(R(x,y)z,w) = R(x,y,z,w).
1.1. Algebraic curvature tensors. It turned out to be convenient to work in a
purely algebraic context in studying the Osserman conjecture. Let M := (V,?·,·?,R)
be a model. This means that V is a vector space of dimension n which is equipped
with a positive definite inner product ?·,·? and that R ∈ ⊗4V∗is an algebraic
curvature tensor, i.e. R satisfies the Riemannian curvature tensor identities:
R(x,y,z,t) = −R(y,x,z,t) = R(z,t,x,y),
R(x,y,z,t) + R(y,z,x,t) + R(z,x,y,t) = 0.
One uses Equation (1.a) to define the Jacobi operator and skew-symmetric curvature
operator in this setting as well; M is said to be Osserman if the eigenvalues of JR(·)
are constant on the sphere S(V,?·,·?) of unit vectors in V . Clearly, if (M,g) is a
Riemannian manifold, and if P ∈ M, then MP := (TPM,gP,RP) defines a model.
Conversely, every model is geometrically realizable.
Key words and phrases. algebraic curvature tensor, complex Osserman model, Jacobi operator,
Osserman conjecture.
2000 Mathematics Subject Classification. 53C20.
1

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2M. BROZOS-V´AZQUEZ AND P. GILKEY
1.2. The Osserman conjecture. This conjecture for Riemannian manifolds was
established by Chi [3] in dimensions n ≡ 1 (mod 2), n ≡ 2 (mod 4) and n = 4.
Subsequent work by Nikolayevsky [11, 12] has established the Osserman conjecture
in dimensions n ?= 16; the case n = 16 is still open. Nikolayevsky used a two
step approach following the discussion in [6]. He first showed that any Osserman
model is given by a Clifford family as specified in Equation (1.e) of Section 1.7
below except in dimension 16. He then used the Second Bianchi Identity to prove
the necessary integrability results to show any Osserman manifold of dimension
n ?= 16 was locally isometric to a rank 1 symmetric space or was flat. Note that
the algebraic classification fails if n = 16; indeed the curvature tensor of the Cayley
plane is Osserman but it is not given by a Clifford family, i.e. it is not expressible
in the form given in Equation (1.e).
1.3. The higher order Jacobi operator. There are other related questions. One
may follow the discussion of Stanilov and Videv [14] to define a higher order Jacobi
operator as follows. Let {e1,...,ep} be an orthonormal basis for a p-plane P. Set
JR(P) =
p
?
i=1
JR(ei);
this is independent of the particular orthonormal basis chosen. If p = 1, one recovers
the ordinary Jacobi operator. Furthermore, if p = n, then ρ := JR(V ) is the
Ricci operator; thus the higher order Jacobi operator can also be thought of as
a generalization of the Ricci operator to lower dimensional subspaces. One says
that a model M is p-Osserman if the eigenvalues of JR(P) are constant on the
Grassmannian Grp(V ) of p-planes. The geometry is very rigid in this setting. If
p = 1 or if p = n − 1, then M is p-Osserman if and only if M is Osserman. Thus
these values of p may be excluded from consideration. If 2 ≤ p ≤ n − 2, then it is
known [7] that M is p-Osserman if and only if M has constant sectional curvature
c, i.e. that R = cR0where R0is given by:
(1.b)R0(x,y,z,t) := ?x,t??y,z? − ?x,z??y,t?.
1.4. Complex geometry. In this paper, we will consider a complex analogue of
these questions. Let J denote an Hermitian almost complex structure on (V,?·,·?);
this means that J is an isometry of (V,?·,·?) with J2= −id. A 2-plane is said to
be holomorphic if it is J-invariant and a real linear transformation T of V is said to
be complex linear if TJ = JT. We let CP(V,?·,·?,J) be the set of all holomorphic 2
planes. If x ∈ S(V,?·,·?) is a unit vector, let πx:= Span{x,Jx}. The natural map
x → πxdefines the Hopf fibration from S(V,?·,·?) to CP(V,?·,·?,J). Let
JR(πx) := JR(x) + JR(Jx)
be the complex Jacobi operator; this is the restriction of the higher order Jacobi
operator to the set of complex 2-planes. The following result is well known, for
example see [9]. Conditions (2), (3) of the Lemma simply mean that the operator
under consideration is complex linear.
Lemma 1.1. We say that R and J are compatible if any of the following equivalent
conditions are satisfied:
(1) R(x,y,z,t) = R(Jx,Jy,Jz,Jt) for all x,y,z,t ∈ V .
(2) JR(πx)J = JJR(πx) for all x ∈ S(V,?·,·?).
(3) R(x,Jx)J = JR(x,Jx) for all x ∈ S(V,?·,·?).
Note that the curvature R and the almost complex structure J of a K¨ ahler
manifold are compatible. Thus this is a very natural geometric condition.

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COMPLEX OSSERMAN ALGEBRAIC CURVATURE TENSORS AND CLIFFORD FAMILIES 3
1.5. The complex Osserman condition. Instead of the 2-Osserman condition
where the eigenvalues are constant on the Grassmannian of 2-planes, we consider
a natural weaker condition with constant eigenvalues on the space of holomorphic
planes, CP(V,?·,·?,J).
Definition 1.2. Let V := (V,?·,·?,J,R). We say that V is a complex model if
?·,·? is a positive definite inner product on V , if J is an Hermitian almost complex
structure on (V,?·,·?), and if R is an algebraic curvature tensor on (V,?·,·?). We
say that V is complex Osserman if
(1) V is a complex model.
(2) J and R are compatible, i.e. JR(πx) is complex linear for all x ∈ S(V,?·,·?).
(3) The eigenvalues of JR(πx) are constant on CP(V,?·,·?,J).
We shall also sometimes simply say that R is complex Osserman in this situation.
1.6. The canonical curvature tensor. In addition to the tensor of constant sec-
tional curvature +1 defined in Equation (1.b), it is useful to consider the tensor
(1.c)RΨ(x,y,z,t) := ?x,Ψt??y,Ψz? − ?x,Ψz??y,Ψt? − 2?x,Ψy??z,Ψt?
where Ψ is a skew-symmetric endomorphism of (V,?·,·?). Such tensors play an
important role in studying the space of all algebraic curvature tensors. For exam-
ple, Fiedler [4] has shown that tensors of this form span the space of all algebraic
curvature tensors. In this paper, we shall study tensors of this form where the endo-
morphism in question defines an Hermitian almost complex structure on (V,?·,·?).
We note for future reference that
(1.d)JR0(x)y = y − ?y,x?andJRΨ(x)y = 3?y,Ψx?Ψx.
1.7. Algebraic curvature tensors given by Clifford families. We say that
a set F = {J1,...,Jκ} of Hermitian almost complex structures on (V,?·,·?) is a
Clifford family of rank κ if they are subject to the commutation rules
JiJj+ JjJi= −δijid.
We say that a model (V,?·,·?,R) is given by a Clifford family F of rank κ if there
exist constants ciwith ci?= 0 for 1 ≤ i ≤ κ so that
(1.e)R = c0R0+ c1RJ1+ ... + cκRJκ.
We shall also sometimes say that R is given by a Clifford family in this setting. The
relations of Equation (1.d) yield that:
(1.f)JR(x)y = c0{y − ?y,x?x} + 3c1?y,J1x?J1x + ... + 3cκ?y,Jκx?Jκx.
From this it follows immediately that
(1.g)
JR(πx)y = c0{2y − ?y,x?x − ?y,Jx?Jx}
+
?
i=1
κ
3ci{?y,Jix?Jix + ?y,JiJx?JiJx}.
1.8. Reparametrizing Clifford families. Let A = (Aij) ∈ O(κ) be an orthogo-
nal matrix. Set
˜F := {˜Ji= Ai1J1+ ··· + AiκJκ}.
This new Clifford family is said to be a reparametrization of F; this defines an
equivalence relation on the collection of Clifford families.

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4M. BROZOS-V´AZQUEZ AND P. GILKEY
1.9. Summary of results. In this paper we begin the study of complex Osserman
manifolds by concentrating on the analysis of complex Osserman models. In Section
2, we give necessary and sufficient conditions so that a model V = (V,?·,·?,J,R) is
complex Osserman and we show that R is necessarily Einstein if V is complex Osser-
man. We also give a topological result in Theorem 2.4 which controls the eigenvalue
structure of JR(πx) if V is complex Osserman. In Section 3, we recall results of
Adams on the existence of Clifford families and discuss some reparametrization
results. We also present some examples of complex Osserman models and show
Theorem 2.4 is sharp.
Work of Nikolayevsky shows that any Osserman model (V,?·,·?,R) is given by a
Clifford family except in dimension 16. We divide our study into two cases depend-
ing on the rank κ of the structure in question.
We study the case κ > 3 in Section 4 and show:
Theorem 1.3. Let V := (V,?·,·?,J,R) where R = c0R0+ c1RJ1+ ... + cκRJκis
given by a Clifford family of rank κ ≥ 4 on a vector space V of dimension n. The
following assertions hold:
(1) Let c0= 0. If κ = 4,5, assume n ≥ 2κand, if κ ≥ 6, assume n ≥ κ(κ−1).
Then V is not complex Osserman.
(2) Let c0?= 0. If κ = 4 assume n ≥ 32, if κ = 5,6,7 assume n ≥ 2κ, if κ ≥ 8
assume n ≥ κ(κ − 1). Then V is not complex Osserman.
Note that, as a consequence of Lemma 3.1 below, the hypothesis n ≥ κ(κ−1) in
Theorem 1.3 is not a restriction when κ ≥ 16. Consequently, there are only a finite
number of possibly exceptional dimensions and ranks when κ ≥ 4.
Section 5 is devoted to the study of Clifford families of lower rank. Results in
this section are summarized in the following theorem:
Theorem 1.4. Let V := (V,?·,·?,J,R). Let F = {Ji} be a Clifford family on a
vector space V of dimension n. Let ci?= 0 be given for 1 ≤ i ≤ κ where κ ≤ 3.
(1) Rank κ = 0. Let R = c0R0. Then V is complex Osserman.
(2) Rank κ = 1. Let R = c0R0+ c1RJ1.
(a) If c0= 0, then V is complex Osserman if and only if JJ1= ±J1J.
(b) If c0 ?= 0, then V is complex Osserman if and only if J = ±J1 or
JJ1= −J1J.
(3) Rank κ = 2. Let R = c0R0+c1RJ1+c2RJ2. Then V is complex Osserman
if and only if there exists a reparametrization {˜J1,˜J2} of F so that one has
R = c0R0+ ˜ c1R˜ J1+ ˜ c2R˜ J2and so that one of the following holds:
(a) c0= 0, J˜J1=˜J1J and J˜J2= −˜J2J.
(b) Either J =˜J1or J =˜J1˜J2.
(4) Rank κ = 3. Let R = c0R0+ c1RJ1+ c2RJ2+ c3RJ3.
(a) Assume n ≥ 12. If c0= 0, then V is complex Osserman if and only
if there exists a reparametrization {˜J1,˜J2,˜J3} of F so that one has
R = ˜ c1R˜ J1+ ˜ c2R˜ J2+ ˜ c3R˜ J3and that J =˜J1or J =˜J2˜J3.
(b) Assume n ≥ 16. If c0?= 0, then V is complex Osserman if and only
if there exists a reparametrization {˜J1,˜J2,˜J3} of F so that one has
R = c0R0+ ˜ c1R˜ J1+ ˜ c2R˜ J2+ ˜ c3R˜ J3, J =˜J1, and˜J1˜J2˜J3= id.
Remark 1.5. From Theorem 1.4 we obtain the following geometric conclusions:
(1) Let (M,g) be a manifold of constant sectional curvature. Then (M,g) is
complex Osserman with respect to any Hermitian almost complex structure
J.
(2) Let (M,g,J) be a K¨ ahler manifold which has constant holomorphic sec-
tional curvature. Then (M,g,J) is complex Osserman with respect to J.

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COMPLEX OSSERMAN ALGEBRAIC CURVATURE TENSORS AND CLIFFORD FAMILIES 5
(3) Let (M,g,{J1,J2,J3}) be a quaternionic K¨ ahler manifold which has con-
stant quaternionic sectional curvature, where {J1,J2,J3} forms a locally de-
fined quaternionic structure. Then, for any J ∈ Span{J1,J2,J3}, (M,g,J)
is complex Osserman.
Note that if (M,g) is Osserman of dimension different from 16, then it is isometric
to one of these three examples or is flat [3, 11, 12].
2. Algebraic preliminaries
In this section we present some foundational results. Our first result is the well
known observation:
Lemma 2.1. Let Vi:= (V,?·,·?,Ri), with i = 1,2, be models. If JR1(x) = JR2(x)
for all x in V , then R1= R2.
What is perhaps somewhat surprising is that this observation fails for the complex
Jacobi operator as we shall see in Theorem 3.6. Let Spec{JR(πx)} be the spectrum
of JR(πx) and let Eλ(πx) be the eigenspace associated to the eigenvalue λ of JR(πx).
Since JR(πx) is self-adjoint, JR(πx) is diagonalizable with real eigenvalues. Thus
we have an orthogonal direct sum decomposition
V = ⊕λEλ(πx)
The following lemma is an immediate consequence of
Lemma 1.1 and provides a criterion for complex Osserman curvature tensors:
for any x ∈ S(V,?·,·?).
Lemma 2.2. V = (V,?·,·?,J,R) is complex Osserman if and only if
(1) JEλ(πx) = Eλ(πx) for all πx∈ CP(V,?·,·?,J) and λ ∈ Spec{JR(πx)}.
(2) Spec{JR(πx)} = Spec{JR(πy)} for all πx,πy∈ CP(V,?·,·?,J).
An model (V,?·,·?,R) is said to be Einstein if ρ(·,·) = c?·,·? for a constant c,
where by ρ we denote the Ricci tensor. In general, p-Osserman models are Einstein.
This result generalizes to become:
Lemma 2.3. Let V = (V,?·,·?,J,R) be complex Osserman. Then V is Einstein.
Proof. Assume that V is complex Osserman. Let x ∈ S(V,?·,·?). As R is compati-
ble, R(y,Jx,Jx,z) = R(Jy,x,x,Jz) and thus JR(Jx) = −JJR(x)J. Consequently
ρ(x,x) = Tr{JR(x)} = Tr{JR(Jx)} =1
is independent of x ∈ S(V,?·,·?). This implies ρ(·,·) = c?·,·?. Consequently V is
Einstein.
2Tr{JR(πx)} =1
2
?
λλdim{Eλ(πx)}
?
Methods of algebraic topology can be used to control the eigenvalue structure of
a complex Osserman model. In particular, no more than 3 distinct eigenvalues may
occur.
Theorem 2.4. Let V = (V,?·,·?,J,R) be complex Osserman. If JR(πx) is not a
multiple of the identity (i.e. if JR(πx) has at least 2 distinct eigenvalues), then:
(1) If n ≡ 2 (mod 4), there are 2 eigenvalues with multiplicities (n − 2,2).
(2) If n ≡ 0 (mod 4), then one of the following holds:
(a) There are 2 eigenvalues with multiplicities (n − 2,2).
(b) There are 2 eigenvalues with multiplicities (n − 4,4).
(c) There are 3 eigenvalues with multiplicities (n − 4,2,2).
Proof. Let V := CP(V,?·,·?,J) × V be the trivial bundle over projective space.
Lemma 2.2 shows that the eigenspaces
Eλi(π) := {v ∈ V : JR(π)v = λiv}
have constant rank and patch together to define smooth vector bundles Eλi(π)
over CP(V,?·,·?,J) where {λ0,...,λk} denote the distinct eigenvalues of JR(π) for