Relating the curvature tensor and the complex Jacobi operator of an almost Hermitian manifold

Abstract

Let J be a unitary almost complex structure on a Riemannian manifold (M,g). If x is a unit tangent vector, let P be the associated complex line spanned by x and by Jx. We show that if (M,g) is Hermitian or if (M,g) is nearly Kaehler, then either the complex Jacobi operator (JC(P)y=R(y,x)x+R(y,Jx)Jx) or the complex curvature operator (RC(P)y=R(x,Jx)y) completely determine the full curvature operator; this generalizes a well known result in the real setting to the complex setting. We also show this result fails for general almost Hermitian manifold.

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arXiv:math/0611605v1 [math.DG] 20 Nov 2006
RELATING THE CURVATURE TENSOR AND THE COMPLEX
JACOBI OPERATOR OF AN ALMOST HERMITIAN MANIFOLD
M. BROZOS-V´AZQUEZ, E. GARC´IA-R´IO AND P. GILKEY
Abstract. Let J be a unitary almost complex structure on a Riemannian
manifold (M,g). If x is a unit tangent vector, let π := Span{x,Jx} be the
associated complex line in the tangent bundle of M.
operator and the complex curvature operators are defined, respectively, by
J(π) := J(x) + J(Jx) and R(π) := R(x,Jx). We show that if (M,g) is
Hermitian or if (M,g) is nearly K¨ ahler, then either the complex Jacobi operator
or the complex curvature operator completely determine the full curvature
operator; this generalizes a well known result in the real setting to the complex
setting. We also show this result fails for general almost Hermitian manifolds.
The complex Jacobi
1. Introduction
We shall let M := (M,g) be a Riemannian manifold of dimension m. Let
R(x,y) := ∇x∇y− ∇y∇x− ∇[x,y],
R(x,y,z,w) := g(R(x,y)z,w)
be the curvature operator and the curvature tensor, respectively; R has the sym-
metries:
R(x,y,z,w) = R(z,w,x,y) = −R(y,x,z,w),
R(x,y,z,w) + R(y,z,x,w) + R(z,x,y,w) = 0.
(1.a)
It is convenient to work in the algebraic setting. Let V be a vector space of
dimension m. We say that A ∈ ⊗4(V∗) is an algebraic curvature tensor if A has
the symmetries given in Equation (1.a). We consider a model M := (V,?·,·?,A)
where ?·,·? is an auxiliary positive definite inner product on V . Every model is
geometrically realizable; given a model M, one can construct a Riemannian manifold
M so that M is isomorphic to (TPM,gP,RP) for some point P ∈ M.
Given a model M, one uses the inner product ?·,·? to raise indices and define
an associated curvature operator A. The Jacobi operator J : y → A(y,x)x is
characterized by the identity
?J(x)y,z? = A(y,x,x,z).
The Jacobi operator determines the full curvature tensor. The following theorem
is well known; Assertion (2) in the geometric setting is an immediate consequence
of the corresponding Assertion (1) in the algebraic setting:
Theorem 1.1.
(1) Let Mi = (Vi,?·,·?i,Ai) be models for i = 1,2. Suppose there exists an
isometry θ : (V1,?·,·?1) → (V2,?·,·?2) so that JM2(θx)θ = θJM1(x) for all
x ∈ V1. Then θ∗A2= A1.
(2) Let Mi= (Mi,gi) be Riemannian manifolds for i = 1,2. Suppose there is
an isometry θ : (TPM1,g1) → (TQM2,g2) so that JM2,Q(θx)θ = θJM1,P(x)
for all x ∈ TPM1. Then θ∗R2(Q) = R1(P).
Key words and phrases. complex curvature operator, complex Jacobi operator, almost Hermit-
ian manifold, Hermitian manifold, nearly K¨ ahler manifold.
2000 Mathematics Subject Classification. 53C15, 53C55.
1

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2M. BROZOS-V´AZQUEZ, E. GARC´IA-R´IO AND P. GILKEY
If J is a unitary almost complex structure on a Riemannian manifold (M,g), then
C := (M,g,J) is said to be an almost Hermitian manifold. In the algebraic setting,
C := (V,?·,·?,J,A) is said to be a complex model if J is a unitary complex structure
and if A is an algebraic curvature tensor. Any point P of an almost Hermitian man-
ifold C determines a corresponding complex model C(C,P) := (TPM,gP,JP,RP) in
a natural fashion.
Let C be a complex model. The Ricci tensor ρ and the ⋆-Ricci tensor ρ⋆are
defined by contracting indices. If {e1,...,em} is an orthonormal basis for V , then
m
?
i=1
ρ(x,y) =
R(ei,x,y,ei)andρ⋆(x,y) :=
m
?
i=1
R(x,Jei,Jy,ei).
We note that ρ⋆is not in general a symmetric 2-tensor; however one does have
that ρ⋆(x,y) = ρ⋆(y,x) if the compatibility condition given below in Lemma 1.4
is satisfied. The scalar curvature τ and the ⋆-scalar curvature τ⋆are defined by a
final contraction:
τ =
m
?
i=1
ρ(ei,ei)andτ⋆=
m
?
i=1
ρ⋆(ei,ei).
We say a 2-dimensional subspace π of V is a complex line if Jπ = π.
CP(V,J) be the complex projective space of complex lines in V . If π ∈ CP(V,J),
let S(π) be the set of unit vectors in π. Let x ∈ S(π) for π ∈ CP(V,J). Then one
has π = πx := Span{x,Jx}. The holomorphic sectional curvature Q(π), complex
Jacobi operator J(π), and complex skew-symmetric curvature operator R(π) are
then defined for π ∈ CP(V,J), respectively, by setting
Q(π) := A(x,Jx,Jx,x),
R(π) := R(x,Jx)
We shall be considering several important families of almost Hermitian manifolds.
C is said to be Hermitian if the Nijenhuis tensor vanishes, i.e. if
[X,Y ] + J[JX,Y ] + J[X,JY ] − [JX,JY ] = 0
Equivalently, see [10], this means that we can find local holomorphic coordinates
zν = xν+√−1yν for 1 ≤ ν ≤
transition functions relating two such coordinate systems are then complex analytic.
There are other natural assumptions that may be imposed and which define other
important families. For example, one says that C is nearly K¨ ahler if (∇xJ)x = 0 for
all tangent vectors x; we refer to [9] for further information concerning this class
of manifolds. We say that C is almost K¨ ahler if the two form Ω(x,y) := ?Jx,y? is
closed; we refer to [3] for a survey and to [2], [8] and [11] for some recent results
concerning this class of manifolds.
The following result, which generalizes Theorem 1.1 to the complex setting, is the
central result of this paper. It shows that the full curvature tensor is determined
either by the complex Jacobi operator or by the complex curvature operator in
certain natural geometric contexts:
Let
(1.b)
J(π) := J(x) + J(Jx),
for any x ∈ S(π).
for allX,Y .
1
2m so that J∂xν= ∂yνand J∂yν= −∂xν; the
Theorem 1.2. For i = 1,2, let Ci = (Mi,gi,Ji) be either Hermitian or nearly
K¨ ahler manifolds. Let θ : (TPM1,g1,J1) −→ (TQM2,g2,J2) be a complex isometry.
The following assertions are equivalent:
(1) θJC1,P(π) = JC2,Q(θπ)θ for all π ∈ CP(TPM1,J1,P).
(2) θRC1,P(π) = RC2,Q(θπ)θ for all π ∈ CP(TPM1,J1,P).
(3) θ∗RC2,Q= RC1,P.
Our result for almost K¨ ahler setting manifolds is a bit weaker:

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THE COMPLEX JACOBI OPERATOR3
Theorem 1.3. Let C = (M,g,J) be an almost K¨ ahler manifold. Assume either
that J(π) = 0 for all π ∈ CP(TM,J) or that R(π) = 0 for all π ∈ CP(TM,J).
Then M is flat.
Let C be a complex model. There is a basic compatibility condition we work
with that relates the structures J and A:
Lemma 1.4. Let C = (V,?·,·?,J,A) be a complex model. The following conditions
are equivalent and if any is satisfied, we shall say that C is a compatible complex
model.
(1) J∗A = A, i.e. A(x,y,z,t) = A(Jx,Jy,Jz,Jt) for all x,y,z,t ∈ V .
(2) J(π)J = JJ(π) for all π ∈ CP(V,J).
(3) A(π)J = JA(π) for all π ∈ CP(V,J).
We note, see Lemma 3.1, that if C = (M,g,J) is a nearly K¨ ahler manifold,
then C(C,P) = (TPM,gP,JP,RP) is a compatible complex model for any point
P ∈ M. In general, a manifold satisfying this condition at every point is known in
the literature as a RK-manifold.
What is perhaps rather surprising is that Theorem 1.2 does not have a purely
algebraic analogue even if we impose the compatibility condition of Lemma 1.4:
Theorem 1.5. If m ≡ 0 mod 4, there exists a complex model C = (V,?·,·?,J,A)
with A ?= 0 so that J(π) = 0 and so that A(π) = 0 for every π ∈ CP(V,J).
Let C = (M,g,J) be an almost Hermitian manifold. We let UC be the bundle
of complex isometries of TM; the fibers of this bundle are the associated unitary
group of the fibers. If Θ ∈ C∞{UC} and if P ∈ M, then θP := Θ(P) is a complex
isometry of (TPM,gP,JP) for any P ∈ M. We show that Theorem 1.2 fails in the
almost Hermitian context by establishing the following result:
Theorem 1.6. Let m ≡ 0 mod 4. There exists an almost Hermitian manifold C
and there exists Θ ∈ C∞{UC} so that for any point P in M we have:
(1) θPJC(π) = JC(θPπ)θP for all π ∈ CP(TPM,J).
(2) θPRC(π) = RC(θPπ)θP for all π ∈ CP(TPM,J).
(3) θ∗
PRP ?= RP.
Here is a brief outline to this paper. Section 2 is algebraic in nature. We begin
by establishing Lemma 1.4. Next, in Lemma 2.1, we prove a result of Vanhecke
[17] which expresses R(x,y,y,x) for a compatible complex model in terms of the
holomorphic sectional curvature Q defined in Equation (1.b) and in terms of an
additional tensor
(1.c)λ(x,y) = R(x,y,y,x) − R(x,y,Jy,Jx).
The identity of Lemma 2.1 is polarized to establish a result of Sato [12] in Lemma
2.2. We then turn to a study of the complex Jacobi operator by studying the
condition J(π) = 0 for all π ∈ CP(V,J) in Lemma 2.3 and show this implies that ρ
and ρ∗both vanish. We then prove Theorem 1.5. We conclude Section 2 by relating
Lemma 2.3 to the curvature decompositions of Gray [7] in Lemma 2.4.
In Section 3 we apply the results of Section 2 to the geometric context. We begin
by recalling certain results of Gray and Yano concerning Hermitian, nearly K¨ ahler
and almost K¨ ahler manifolds. These results are then applied to prove Theorems 1.2
and 1.3. The construction used to establish Theorem 1.5 in the algebraic setting is
then used to prove Theorem 1.6 in the geometric setting.
There are many example in the literature of almost K¨ ahler manifolds which are
not K¨ ahler [1, 5, 18]. Also, a great interest has been shown in finding conditions
for an almost K¨ ahler manifold to be K¨ ahler. For example, the Goldberg conjecture
states: A compact Einstein almost K¨ ahler manifold is K¨ ahler. This conjecture has

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4M. BROZOS-V´AZQUEZ, E. GARC´IA-R´IO AND P. GILKEY
generated extensive literature, see for example [14, 15]. Many other conditions have
been studied which might imply that an almost K¨ ahler manifold is K¨ ahler, see [2, 4]
for example. We conclude the paper in Section 4 with two related results.
We shall adopt the following notational conventions. The curvature tensor and
curvature operator of a Riemannian manifold will be denoted by R and R, re-
spectively; an algebraic curvature tensor and the corresponding algebraic curvature
operator will be denoted by A and A, respectively. A real Riemannian manifold and
a real model will be denoted by M = (M,g) and M = (V,?·,·?,A), respectively. An
almost Hermitian manifold and a complex model will be denoted by C = (M,g,J)
and C = (V,?·,·?,J,A), respectively. The Jacobi operator will be denoted by J;
we will subscript as appropriate when more than one curvature operator is under
consideration.
In this preprint, we have chosen to give full details of many algebraic computa-
tions in the interests of keeping matters as self-contained as possible for the conve-
nience of the reader; the version that will be submitted for publication will be a bit
shorter as we shall omit many of these computations if they are available elsewhere.
2. Algebraic Results
Proof. We first establish Lemma 1.4. Suppose first that Assertion (1) holds, i.e.
that A(x,y,z,t) = A(Jx,Jy,Jz,Jt) for all x, y, z, t. Then
(2.a)A(y,x,x,Jz) + A(y,Jx,Jx,Jz) = −A(Jy,x,x,z) − A(Jy,Jx,Jx,z),
which implies ?J(πx)y,Jz? = −?J(πx)Jy,z? and, hence, J(πx)J = JJ(πx). As-
sume conversely that J(πx)J = JJ(πx) or equivalently that Equation (2.a) holds
for all x. Polarizing this identity and replacing z by −Jz yields
A(y,x,w,z) + A(y,w,x,z) + A(y,Jx,Jw,z) + A(y,Jw,Jx,z)
=A(Jy,x,w,Jz) + A(Jy,w,x,Jz) + A(Jy,Jx,Jw,Jz) (2.b)
+A(Jy,Jw,Jx,Jz).
Interchanging arguments 1 ↔ 2 and 3 ↔ 4 in the curvature tensors then yields:
A(x,y,z,w) + A(w,y,z,x) + A(Jx,y,z,Jw) + A(Jw,y,z,Jx)
=A(x,Jy,Jz,w) + A(w,Jy,Jz,x) + A(Jx,Jy,Jz,Jw)
+A(Jw,Jy,Jz,Jx).
If we interchange x and y and we interchange z and w in this identity, we get
A(y,x,w,z) + A(z,x,w,y) + A(Jy,x,w,Jz) + A(Jz,x,w,Jy)
=
+
A(y,Jx,Jw,z) + A(z,Jx,Jw,y) + A(Jy,Jx,Jw,Jz)
A(Jz,Jx,Jw,Jy).
(2.c)
Adding (2.b) and (2.c) and simplifying yields:
(2.d)A(y,x,w,z) + A(y,w,x,z) = A(Jy,Jx,Jw,Jz) + A(Jy,Jw,Jx,Jz)
We permute the indices in Equation (2.d) to change y → x → w → y. This yields:
(2.e)A(x,w,y,z) + A(x,y,w,z) = A(Jx,Jw,Jy,Jz) + A(Jx,Jy,Jw,Jz).
We add 2(2.d) and (2.e) and use the Bianchi identity to see
3A(y,w,x,z) = A(y,x,w,z) + 2A(y,w,x,z) + A(x,w,y,z)
=
=
A(Jy,Jx,Jw,Jz) + 2A(Jy,Jw,Jx,Jz) + A(Jx,Jw,Jy,Jz)
3A(Jy,Jw,Jx,Jz).
The desired identity now follows.

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THE COMPLEX JACOBI OPERATOR5
We now prove that Assertion (1) implies Assertion (3). We have:
?JA(πx)y,z? = −?A(πx)y,Jz? = −A(x,Jx,y,Jz)
−A(Jx,JJx,Jy,JJz) = A(x,Jx,Jy,z) = ?A(πx)Jy,z?.
Thus JA(πx) = A(πx)J as desired.
We finally show that Assertion (3) implies Assertion (1). We have
=
JA(x,Jx) = A(x,Jx)J,
?JA(x,Jx)z,w? − ?A(x,Jx)Jz,w? = 0,
A(x,Jx,z,Jw) + A(x,Jx,Jz,w) = 0.
⇒
⇒
Polarizing yields an identity for all x,y,z,w:
0=A(y,Jx,z,Jw) + A(x,Jy,z,Jw) + A(y,Jx,Jz,w)
+A(x,Jy,Jz,w).
Interchange the first two arguments in the first and third term to see:
0=−A(Jx,y,z,Jw) + A(x,Jy,z,Jw) − A(Jx,y,Jz,w)
A(x,Jy,Jz,w).+
Replace (x,w) by (Jx,Jw) to show:
(2.f)
0=
+
−A(x,y,z,w) − A(Jx,Jy,z,w) + A(x,y,Jz,Jw)
A(Jx,Jy,Jz,Jw).
Interchange the first two arguments with the final two arguments:
0=
+
−A(z,w,x,y) − A(z,w,Jx,Jy) + A(Jz,Jw,x,y)
A(Jz,Jw,Jx,Jy).
Change notation to interchange x and z, and y and w, to see:
(2.g)
0=
+
−A(x,y,z,w) − A(x,y,Jz,Jw) + A(Jx,Jy,z,w)
A(Jx,Jy,Jz,Jw).
We add Equations (2.f) and (2.g) to conclude
−A(x,y,z,w) + A(Jx,Jy,Jz,Jw) = 0
and complete the proof that Assertion (3) implies Assertion (1).
?
The following result is due to Vanhecke [17]; it was originally stated in a purely
geometrical setting. Let Q and λ be defined by Equations (1.b) and (1.c), respec-
tively.
Lemma 2.1. Let C = (V,?·,·?,A,J) be a compatible complex model. Then
32A(x,y,y,x)=3Q(x + Jy) + 3Q(x − Jy) − Q(x + y) − Q(x − y)
−
Proof. First note that
4Q(x) − 4Q(y) + 4{5λ(x,y) + λ(x,Jy)}.
Q(x + y)=
=
A(x + y,Jx + Jy,Jx + Jy,x + y)
A(x,Jx,Jx,x) + A(x,Jy,Jy,x)
+A(y,Jx,Jx,y) + A(y,Jy,Jy,y)
+2A(x,Jx,Jx,y) + 2A(x,Jy,Jy,y) + 2A(x,Jx,Jy,x)
+2A(y,Jx,Jy,y) + 2A(x,Jx,Jy,y) + 2A(x,Jy,Jx,y).
Hence