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arXiv:1007.2569v1 [math.DG] 15 Jul 2010
HIGHER-DIMENSIONAL OSSERMAN METRICS WITH
NON-NILPOTENT JACOBI OPERATORS
E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO
Abstract. We exhibit Osserman metrics with non-nilpotent Jacobi operators
and with non-trivial Jordan normal form in neutral signature (n,n) for any
n ≥ 3. These examples admit a natural almost para-Hermitian structure and
are semi para-complex Osserman with non-trivial Jordan normal form as well;
they neither satisfy the third Gray identity nor are they integrable.
1. Introduction
A pseudo-Riemannian manifold (M,g) is said to be Osserman if the eigenvalues
of the Jacobi operatorsgJ(X) : Y →gR(Y,X)X are constant on the unit pseudo-
sphere bundles S±(TM,g). Any isotropic space is Osserman and the converse is
true in the Riemannian (dim M ?= 16) [6, 25, 26] and Lorentzian [1, 15] settings.
However, there exist many non-symmetric Osserman pseudo-Riemannian metrics
in other signatures (cf. [17, 20] and the references therein). Since the eigenvalue
structure need not determine the conjugacy class of a self-adjoint operator in the
indefinite setting, a pseudo-Riemannian manifold is called Jordan-Osserman if the
Jordan normal form of the Jacobi operators is constant on S±(TM,g). Osserman
metrics are Einstein and thus of constant sectional curvature in dimensions 2 and
3. The special significance of the four-dimensional case relies on the fact that a
four-dimensional algebraic curvature tensor is Osserman if and only if it is Einstein
and self-dual; the classification of all four-dimensional Osserman metrics of neutral
signature (2,2) is almost complete [2, 5, 8, 10, 11, 14, 18, 19].
The situation is much more difficult in higher dimensions where only some par-
tial results are known [17]. The structure of a Jordan-Osserman algebraic curvature
tensor strongly depends on the signature (p,q) of the metric tensor. For example,
the Jacobi operators of a spacelike Jordan-Osserman algebraic curvature tensor are
necessarily diagonalizable whenever p < q [21]. In the neutral case (p = q), the
Jordan normal form can be arbitrarily complicated [22]. However in the geomet-
ric setting, less is known as with the exception of some six-dimensional examples
of Osserman metrics with non-nilpotent Jacobi operators [5], all previously known
examples of Osserman metrics have either diagonalizable or nilpotent Jacobi opera-
tors (see [17, 20, 23] and the references therein). There are, of course, other natural
operators beside the Jacobi operator that one could examine – see, for example, the
discussion of the spectral geometry of the skew-symmetric curvature operator [24].
The purpose of this paper is to investigate further the construction in [5], showing
that for any affine Osserman manifold (M,D), the cotangent bundle T∗M equipped
with the modified Riemannian extension is an Osserman manifold whose Jacobi
operators are, in general, neither diagonalizable nor nilpotent.
Key words and phrases. Affine connection, almost para-Hermitian, Einstein, Jacobi operator,
non-integrable para-complex structure, modified Riemannian extension, Osserman manifold, third
Gray identity, Walker metric.
2010 Mathematics Subject Classification. 53C50, 53B30.
Supported by projects MTM2009-07756 and INCITE09 207 151 PR (Spain).
1
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2 E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO
1.1. Affine Osserman manifolds. Let (M,D) be an n-dimensional affine mani-
fold, i.e., D is a torsion-free connection on the tangent bundle of a smooth manifold
M of dimension n. LetDR(X,Y ) := DXDY − DYDX− D[X,Y ]be the associated
curvature operator. We say that (M,D) is affine Osserman if the Jacobi operators
are nilpotent [16], i.e. 0 is the only eigenvalue ofDJ(·) on TM.
There are corresponding local notions which are important; an affine manifold
(M,D) is said to be affine Osserman at P ∈ M ifDJ(·) is nilpotent on TPM.
Similarly a pseudo-Riemannian manifold (M,g) is said to be Osserman at P ∈ M
if the eigenvalues ofgJ(·) are constant on S±(TPM,g). Clearly (M,D) is affine
Osserman if and only if (M,D) is affine Osserman at every point P ∈ M. Similarly
(M,g) is Osserman if and only if (M,g) is Osserman at every point P ∈ M and if
the eigenvalue structure and eigenvalue multiplicities are independent of P.
1.2. Riemannian extensions. Let N := T∗M be the cotangent bundle of an
n-dimensional manifold M, let σ : N → M be the natural projection, and let
Z(N) be the zero section. If x = (x1,...,xn) are local coordinates on M, let
x′= (x1′,...,xn′) be the associated dual coordinates on the fiber where we expand
a 1-form ω as ω = xi′dxi; we shall adopt the Einstein convention and sum over
repeated indices henceforth. The following natural distribution will play a crucial
role in our analysis:
Y := Span{∂x1′,...,∂xn′} = ker(σ∗).
For each vector field X = Xi∂xion M, the evaluation map ιX(P,ω) = ω(XP)
defines a function on N which, in local coordinates, is given by
ιX(xi,xi′) = xi′Xi.
Vector fields on N are characterized by their action on functions ιX; the complete
lift XCof a vector field X on M to N is characterized by the identity
XC(ιZ) = ι[X,Z],for all Z ∈ C∞(TM).
Moreover, since a (0,s)-tensor field on N is characterized by its evaluation on com-
plete lifts of vector fields on M, for each tensor field S of type (1,1) on M, we define
a 1-form ιS on N which is characterized by the identity
(ιS)(XC) = ι(SX).
Let (M,D) be an affine manifold. The Riemannian extension gDis the pseudo-
Riemannian metric on N of neutral signature (n,n) characterized by the identity:
gD(XC,YC) = −ι(DXY + DYX).
If u and v are cotangent vectors, let u ◦ v :=1
2(u ⊗ v + v ⊗ u). Expand
D∂xi∂xj=DΓijℓ∂xℓ
to define the Christoffel symbolsDΓ of D. One then has:
gD= 2dxi◦ dxi′− 2xk′DΓijkdxi◦ dxj.
Riemannian extensions were originally defined by Patterson and Walker [27] and
further investigated in relating pseudo-Riemannian properties of N with the affine
structure of the base manifold (M,D). Moreover, Riemannian extensions were also
considered in [16] in relation to Osserman manifolds (see also [9]).
The modified Riemannian extension is the neutral signature metric on N defined
by (see [5] for a more general construction)
gN:= ιId◦ιId+gD.
In a system of local coordinates one has
(1) gN= 2dxi◦ dxi′+ {xi′xj′ − 2xk′DΓijk}dxi◦ dxj.
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HIGHER-DIMENSIONAL OSSERMAN METRICS3
The manifold (N,gN) is a Walker manifold where the parallel degenerate distribu-
tion is in this instance given by Y [29]. There is a canonical almost para-Hermitian
structure J, i.e. a linear map of TN so that J2= Id and J∗gN= −gN, which will
play a crucial role in our analysis. In local coordinates, it is given by
(2)
J : ∂xi→ ∂xi− {xi′xj′ − 2xk′DΓijk}∂xj′
and
J : ∂xi′→ −∂xi′.
The case that D is flat is of particular interest. Let˜CP be para-complex projec-
tive space of constant para-holomorphic sectional curvature +1. Then [5]:
Theorem 1.1. If D is flat, then (N,gN) is isomorphic to˜CP.
Para-complex projective space˜CP is Jordan-Osserman with diagonalizable Ja-
cobi operators. Let g˜CP(ξ,ξ) = ±1. Then the eigenvalues ofg˜CPJ(ξ) are ±(0,1,1
with multiplicities (1,1,2n − 2), respectively. If (M,D) is affine Osserman, then
gN can be viewed as a deformation of g˜CP. This introduces Jordan normal form
into the Jacobi operator, but does not change the eigenvalue structure in the affine
Osserman context:
4)
Theorem 1.2. Let (M,D) be an affine manifold.
(1) If (M,D) is affine Osserman at P ∈ M, then (N,gN) is Osserman at any
Q ∈ σ−1(P). The eigenvalues ofgNJ(·) on S±(TQN,gN) are ±(0,1,1
multiplicities (1,1,2n − 2), respectively.
(2) If (M,D) is affine Osserman, then (N,gN) is Osserman.
4) with
Let P(N) be the bundle over N of non-degenerate J-invariant tangent 2-planes.
If π ∈ P(N), choose ξ ∈ S+(π,gN) and, following [28], define the para-complex
Jacobi operator to be:
gNJ(π) :=gNJ(ξ) −gNJ(Jξ);
this operator is independent of the particular ξ chosen. Higher order Jacobi op-
erators of this nature were first considered by Stanilov and Videv [28] in the real
setting. One says that (N,gN,J) is semi para-complex Osserman ifgNJ(·) has con-
stant eigenvalues on P(N); if additionallygNJ(π) commutes with J for all π ∈ P(N),
then (N,gN,J) is said to be para-complex Osserman – this implies D is flat by
Theorem 1.6 so this condition is not particularly interesting in the setting we are
considering.
Theorem 1.3. Let (M,D) be an affine manifold. Let π ∈ P(N). The eigenvalues
ofgNJ(π) are (1,1
2) with multiplicities (2,2n−2), respectively, and any Jordan block
forgNJ(π) has size at most 2 × 2; (N,gN,J) is semi para-complex Osserman.
These examples provide genuinely new phenomena. The following result shows
that Jordan normal form ofgNJ can be quite complicated; it also shows that (N,gN)
need not be Jordan-Osserman:
Theorem 1.4. Let r ≥ 2 and let U be an r×r lower triangular matrix. There exists
an affine Osserman manifold (M,D) of dimension r + 1, there exists Q ∈ Z(N),
and there exist ξi∈ S+(TQN,gN) for i = 1,2 so that:
(1)gNJ(ξ1) is diagonalizable.
(2) Relative to a suitable basis for TQN,
gNJ(ξ2) = 0 · Id1⊕1 · Id1⊕(1
4· Idr+U) ⊕ (1
4· Idr+Ut).
There also are non-trivial examples in the para-complex setting:
Theorem 1.5. Let n ≥ 3. There exists an affine Osserman manifold (M,D) of
dimension n so that (N,gN,J) is not Jordan semi para-complex Osserman, and so
that the para-complex Jacobi operators are not always diagonalizable.
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4 E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO
One says that an almost para-Hermitian manifold (A,gA,J) satisfies the third
Gray identity if
(3)
gAR(X,Y,Z,W) =gAR(JX,JY,JZ,JW) for all X,Y,Z,W .
An almost para-Hermitian manifold (A,gA,J) is integrable if there exist local coor-
dinates (u1,...,un,v1,...,vn) centered at any given point of A so that
J∂ui= ∂vi
and
J∂vi= ∂ui
or, equivalently [7], if the Nijenhuis tensor NJvanishes where
(4)NJ(X,Y ) := [X,Y ] − J[JX,Y ] − J[X,JY ] + [JX,JY ].
Theorem 1.6. Let (M,D) be an affine manifold. The following conditions are
equivalent:
(1) (M,D) is flat.
(2) (N,gN,J) is integrable.
(3) (N,gN,J) satisfies the third Gray identity.
(4) JgNJ(π) =gNJ(π)J for all π ∈ P(N).
Theorem 1.2 was first discovered in low dimensions using a computer assisted
calculation; subsequently the general case was derived. Here is a brief outline to
the paper. In Section 2, we prove Theorem 1.2 and Theorem 1.3. In Section 3,
we construct various examples to demonstrate Theorem 1.4 and Theorem 1.5. We
conclude the paper in Section 4 by establishing Theorem 1.6.
2. The eigenvalue structure
We begin our discussion with a technical result. Although well known, we include
the proof to keep our discussion as self-contained as possible. If D is an arbitrary
connection on TM, the torsion tensor T ∈ Λ2(T∗M) is defined by:
T (X,Y ) := DXY − DYX − [X,Y ].
Lemma 2.1. Let D be an arbitrary connection on TM. Let P ∈ M. The following
conditions are equivalent:
(1) There exist local coordinates x = (x1,...,xn) centered at P soDΓ(P) = 0.
(2) The torsion tensor T = 0 vanishes at P.
Proof. Let x = (x1,...,xn) be a system of local coordinates on M. The torsion
tensor T vanishes at P if and only ifDΓijk(P) =DΓjik(P). In particular, if there
exists a coordinate system whereDΓ(P) = 0, then necessarily T vanishes at P.
Thus Assertion (1) implies Assertion (2). Conversely, assume that Assertion (2)
holds. Define a new system of coordinates by setting:
zi= xi+1
2aijkxjxk
where aijk= aikj remains to be chosen. As ∂xj= ∂zj+ aljixi∂zl,
D∂xi∂xj(0) = D∂zi∂zj(0) + alji∂zl(0).
Assertion (1) now follows by setting alij:=DΓijl; the fact that alij= aljiis exactly
the assumption that D is torsion-free at P.
?
Let (M,D) be an affine manifold, let Q ∈ N = T∗M, and let P := σQ ∈ M. As
D is torsion-free, we may apply Lemma 2.1 to make a change of coordinates on M
so thatDΓ(P) = 0. Letg˜CPR be the curvature tensor of the metric
g˜CP:= 2dxi◦ dxi′+ xi′xj′dxi◦ dxj.
This metric is not invariantly defined but depends on the coordinates chosen.
We note gN(Q) = g˜CP(Q). We set2R :=gNR−g˜CPR. Let {gNJ,g˜CPJ,2J,DJ,gDJ}
be the associated Jacobi operators.
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Lemma 2.2. Let (M,D) be an affine manifold, let Q ∈ N, and let P = σQ.
Let ξ ∈ TQN, and let a = σ∗(ξ) ∈ TPM. Then we have, relative to the natural
coordinate frame {∂x1,...,∂xn,∂x1′,...,∂xn′} for TN, that:
2J(ξ) =
?
DJ(a)
⋆
0
DJ(a)t
?
.
Proof. DecomposegNR =g˜CPR+gDR+ER whereER is an additional term measuring
the interactions between the metrics g˜CPand gD in the combined metric gN of
Equation (1). By [4] (page 54 Equation (3.6)),gDJ(ξ) has the form given in the
Lemma. We will complete the proof by showing that the additional interaction
terms define a Jacobi operator withEJ(ξ) : Span{∂xi} → Y, i.e.
EJ(ξ) =
?
0
⋆
0
0
?
.
We use dimensional analysis. Define:
Deg(xi) = −1,
Deg(xi′) = +1,
Deg(DΓ⋆⋆⋆) = +1.
Deg(dxi) = −1,
Deg(dxi′) = +1,
Deg(∂xi) = +1,
Deg(∂xi′) = −1,
We consider the rescaling xi→ c−1xi; this induces a dual rescaling xi′ → cxi′. If
Θ is a tensor of degree k, then Θ → ckΘ under this rescaling. For example, the
metrics g˜CP, gD, and gN are all homogeneous of degree 0; thus they are invariant
under this rescaling.
It is clear that the Christoffel symbols of the first kind decouple:
gNΓ⋆⋆⋆=g˜CPΓ⋆⋆⋆+gDΓ⋆⋆⋆.
SinceDΓ vanishes at P, we must have at least one ∂xiderivative ofDΓ in computing
gNR; thus any variable involvingDΓ has degree at least +2. In raising indices in
the metric gN rather than in the metric gD, we must take into consideration the
xi′xj′dxi◦dxjterm; thus interactions of this form involvingDΓ contribute terms of
degree at least 2 + 2 = 4 toER. We also have interaction terms which are bilinear
ing˜CPΓ⋆⋆⋆ andgDΓ⋆⋆⋆ after an index is raised in each factor. Such terms are at
least quadratic in {xi′} and linear in ∂xiDΓ and consequently have total degree at
least +4. We therefore conclude that any monomial of the interaction tensorER has
total degree at least +4. The degree ofERw1w2w3w4is ±1 ± 1 ± 1 ± 1; such a term
has degree at most +4 and the degree is exactly +4 if and only if w1∈ {1,...,n},
w2∈ {1,...,n}, w3∈ {1,...,n}, and if w4∈ {1′,...,n′}. ConsequentlyER defines
a Jacobi operator mapping Span{∂xi} to Y.
?
Let ξ ∈ S+(TQN,gN) and let ξ1:= Jξ ∈ S−(TQN,gN). Let Eλ(ξ) (resp. Eλ(ξ1))
be the eigenspaces ofg˜CPJ(ξ) (resp.g˜CPJ(ξ1)) for the eigenvalue λ ∈ {0,1,1
for λ ∈ {0,−1,−1
4}). Set
4} (resp.
(5)
E0(ξ) = ξ · R = E−1(ξ1),
E 1
E1(ξ) = ξ1· R = E0(ξ1),
4(ξ) = {E0(ξ) ⊕ E1(ξ)}⊥= {E−1(ξ1) ⊕ E0(ξ1)}⊥= E−1
S(ξ) := Y ∩ E 1
4(ξ1),
4(ξ),U(ξ) := E0(ξ) ⊕ Y .
We then have TQN = E0(ξ) ⊕ E1(ξ) ⊕ E 1
4(ξ).
Lemma 2.3. Let (M,D) be an affine manifold. Let Q ∈ N. Let ξ ∈ S+(TQN,gN).
(1) Y = (ξ1− ξ) · R + S(ξ).
(2)2J(ξ)Y ⊂ S(ξ).
(3)g˜CPJ(ξ)U(ξ) ⊂ U(ξ) and2J(ξ)U(ξ) ⊂ U(ξ).
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6 E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO
Proof. Equation (2) implies ξ1− ξ ∈ Y. We can choose an orthonormal basis for
E 1
4(ξ) of the form
{e+
1,...,e+
n−1,Je+
iare timelike. We prove Assertion (1) by
1,...,Je+
n−1}
where the e+
noting that we have the following basis for Y:
iare spacelike and the Je+
{ξ − ξ1,e+
1− Je+
1,...,e+
n−1− Je+
n−1}.
Suppose that Assertion (2) fails. We argue for a contradiction. Choose η ∈ Y
so that2J(ξ)η / ∈ S(ξ). By Lemma 2.2,2J(ξ)η ∈ Y. By Assertion (1), there exists
c ?= 0 so that
2J(ξ)η = c(ξ − ξ1) + η1
forη1∈ S(ξ).
Thus cξ ∈ E1(ξ)+Range(2J(ξ))+E 1
establishes Assertion (2). To prove Assertion (3), express:
4(ξ) ⊂ E0(ξ)⊥which is false; this contradiction
(6)U(ξ) = E0(ξ) ⊕ Y = ξ · R ⊕ (ξ1− ξ) · R ⊕ S(ξ) = ξ · R ⊕ ξ1· R ⊕ S(ξ).
Asg˜CPJ(ξ)ξ = 0, asg˜CPJ(ξ)ξ1= ξ1, and as S(ξ) ⊂ E 1
As2J(ξ)ξ = 0 and as2J(ξ)Y ⊂ Y,2J(ξ) preserves U(ξ) as well.
4(ξ),g˜CPJ(ξ) preserves U(ξ).
?
We examine the eigenvalue structure:
Lemma 2.4. Let (M,D) be an affine manifold. Let Q ∈ N. Let ξ ∈ S+(TQN,gN).
Assume (M,D) is affine Osserman at P = σ(Q). If there is 0 ?= η ∈ TQN ⊗RC
withgNJ(ξ)η = µη, then:
(1) If η ?∈ U(ξ) ⊗RC, then µ =1
4.
(2) If η ∈ U(ξ) ⊗RC and if η ?∈ S(ξ) ⊗RC, then µ = 0 or µ = 1.
(3) If η ∈ S(ξ) ⊗RC, then µ =1
4.
(4) Spec{gNJ(ξ)} ⊂ {0,1,1
4}.
Proof. By Lemma 2.2,2J(ξ) is nilpotent since (M,D) is affine Osserman at P. By
Lemma 2.3, U(ξ) is preserved byg˜CPJ(ξ) and by2J(ξ). Thus, there are induced
operatorsg˜CP˜ J(ξ),2˜ J(ξ), andgN˜ J(ξ) =g˜CP˜ J(ξ) +2˜ J(ξ) on the quotient space:
V(ξ) := {TQN/U(ξ)} ⊗RC.
If η / ∈ U(ξ) ⊗RC, then ˜ η ∈ V(ξ), ˜ η ?= 0 andgN ˜ J(ξ)˜ η = µ˜ η. By Equation (6),
V(ξ) = {E 1
4(ξ)/S(ξ)} ⊗RC.
Consequently,g˜CP˜ J(ξ) =1
gN˜ J(ξ) has only the eigenvalue1
To prove Assertion (2), suppose there exists 0 ?= η ∈ U(ξ) ⊗RC such that
η / ∈ S(ξ) ⊗RC andgNJ(ξ)η = µη. By Lemma 2.3, S(ξ) is preserved byg˜CPJ(ξ) and
2J(ξ). Thus there are induced operators that we again denote byg˜CP˜ J(ξ),2˜ J(ξ),
andgN˜ J(ξ) =g˜CP˜ J(ξ) +2˜ J(ξ) on the quotient space:
4Id. Since2˜ J(ξ) is nilpotent andgN˜ J(ξ) =1
4. Thus µ =1
4Id+2˜ J(ξ),
4. This establishes Assertion (1).
W(ξ) := {U(ξ)/S(ξ)} ⊗RC.
Since ˜ η ?= 0, µ is an eigenvalue ofgN˜ J(ξ). By Equation (6), W(ξ) =˜ξ · R ⊕˜ξ1· R.
By Lemma 2.3,2J(ξ)ξ = 0 and2J(ξ)ξ1=2J(ξ)(ξ1−ξ) ∈ S(ξ) and thus2˜ J(ξ) = 0.
Sinceg˜CP˜ J(ξ)˜ξ = 0 andg˜CP˜ J(ξ)˜ξ1 =˜ξ1 we havegN˜ J(ξ)˜ξ = 0 andgN˜ J(ξ)˜ξ1 =˜ξ1.
Thus µ ∈ {0,1}. Assertion (2) follows.
To prove Assertion (3), we note thatg˜CPJ(ξ) =1
nilpotent and preserves S(ξ). Assertion (4) follows from Assertions (1)-(3).
4Id on S(ξ) and that2J(ξ) is
?
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HIGHER-DIMENSIONAL OSSERMAN METRICS7
Proof of Theorem 1.2. Let (M,D) be an affine manifold which is affine Osserman
at P ∈ M. Choose local coordinates on M soDΓ(P) = 0. Let D0 be the flat
torsion-free connection defined on a neighborhood of P whose Christoffel symbols
vanish in these coordinates. Set
Dε:= εD + (1 − ε)D0
to define a 1-parameter family of metrics gε
g0
P. Thus Lemma 2.4 implies Spec{gε
tiplicities are unchanged during this perturbation as well. Taking ε = 0 yields the
desired multiplicities and establishes Assertion (1) of Theorem 1.2 for ξ spacelike.
We now use results of [17] to see that spacelike Osserman implies timelike Osserman
and to relate the eigenvalues and eigenvalue multiplicities on S+(TQN,gN) to the
eigenvalues and eigenvalue multiplicities on S−(TQN,gN); alternatively, of course,
one could simply proceed directly as well. This proves Assertion (1) of Theorem
1.2; Assertion (2) follows from Assertion (1).
Ninterpolating between g1
N= gN and
N= g˜CP. SinceDεR(P) = ε·DR(P), all the connections Dεare affine Osserman at
NJ(ξ)} ⊂ {0,1,1
4} for all ε so the eigenvalue mul-
?
Proof of Theorem 1.3. Let (M,D) be an affine manifold, let Q ∈ N, let π ∈ PQ(N),
and let ξ ∈ S+(π,gN). Let a = σ∗ξ = σ∗J(ξ). By Lemma 2.2,
?
⋆
?
⋆π
2J(ξ) =
DJ(a)0
DJ(a)t
?
,
2J(Jξ) =
?
DJ(a)
⋆1
0
DJ(a)t
?
,
2J(π) =2J(ξ) −2J(Jξ) =
00
0
?
.
Thus Range(2J(π)) ⊂ Y,2J(π)Y = 0, and2J(π) is nilpotent. Choose a basis
{e1,e2,f1,...,f2n−2} for TQN so Span{e1,e2} is the +1 eigenspace ofg˜CPJ(π) and
so Span{f1,...,f2n−2} is the +1
(gNJ(π) − Id)ei=2J(π)ei∈ Yand
2eigenspace ofg˜CPJ(π). We compute:
(gNJ(π) −1
2Id)fi=2J(π)fi∈ Y .
By Equation (5),
g˜CPJ(π) = Id on (ξ − Jξ) · R
and
g˜CPJ(π) =1
2Id on S(ξ).
Consequently by Lemma 2.3,g˜CPJ(π)Y ⊂ Y. Since2J(π) = 0 on Y, this implies
gNJ(π)Y ⊂ Y. We may now conclude:
Range{(gNJ(π) − Id) · (gNJ(π) −1
2Id)} ⊂ Y .
Since2J(π) = 0 on Y,gNJ(π) =g˜CPJ(π) on Y and consequently by Equation (5)
and Lemma 2.3,
(gNJ(π) − Id) · (gNJ(π) −1
(gNJ(π) − Id)2·(gNJ(π) −1
2Id)Y = {0}
2Id)2= {0}.
so
Consequently Spec{gNJ(π)} ⊂ {1
blocks. As in the proof of Theorem 1.2, we set Dε:= εD+(1−ε)D0to construct a 1-
parameter family of semi para-complex Osserman metrics gε
g1
N= g˜CP. Since the eigenvalues are unchanged, the eigenvalue
multiplicities are unchanged. Consequently1
and 1 is an eigenvalue of multiplicity 2.
2,1} andgNJ(π) has only 1 × 1 or 2 × 2 Jordan
Ninterpolating between
N= gN and g0
2is an eigenvalue of multiplicity 2n−2
?
3. Examples
Throughout Section 3, we will take M = Rnfor some n and we will consider
a point Q ∈ Z(N) so thatDΓ(P) = 0 where P = σ(Q).
computations greatly. We will list only the possibly non-zero components of various
tensors up to the obvious Z2symmetries.
This simplifies the
Page 8
8 E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO
Lemma 3.1. Let (M,D) be an affine manifold, let Q ∈ Z(N), and let P = σ(Q).
Assume thatDΓ(P) = 0.
(1) The possibly non-zero curvatures ofgDR(Q) aregDRijkl′(Q) =DRijkl(P).
(2) The non-zero curvatures ofg˜CPR(Q) are:
g˜CPRi,i′,i′,i(Q) = −1,
g˜CPRi,j′,i′,j(Q) =g˜CPRi,i′,j′,j(Q) = −1
2for i ?= j .
(3)gNR(Q) =g˜CPR(Q) +gDR(Q).
Proof. Let (u1,...,un) be coordinates on a pseudo-Riemannian manifold (U,gU).
Expand gU= gabdua◦dub. Suppose that the 1-jets of the functions gabvanish at a
point S of U. We then have
gURabcd(S) =1
2{∂ua∂ucgbd+ ∂ub∂udgac− ∂ua∂udgbc− ∂ub∂ucgad}(S).
We apply this observation to the setting at hand. Since x′andDΓ vanish at Q
and P, respectively, the 1-jets of gD, of g˜CP, and of gN vanish at Q. We establish
Assertion (1) by computing:
gDRijkl′(Q) =1
2{∂xj∂xℓ′(−2xh′DΓikh) − ∂xi∂xℓ′(−2xh′DΓjkh)}(Q)
DΓjkl− ∂xj
= {∂xi
DΓikl}(P) =DRijkl(P).
The proof of Assertion (2) and of Assertion (3) is similar.
?
Proof of Theorem 1.4. Let r ≥ 2 and let M := Rr+1. Let {x0,...,xr} be the usual
coordinates on M and {x0′,...,xr′} the dual fiber coordinates on T∗M. We let
indices a,b,c,d range from 1 through r and indices i,j,k,l range from 0 through
r. Let Uabbe a lower triangular matrix, i.e. Uab= 0 for b ≤ a. Let θ = θ(x0) be
a smooth function of 1 variable. Define a torsion-free connection D on TM with
non-zero Christoffel symbols:
DΓ0ab=DΓa0b= θUab.
The curvature is given by
DRijkl= ∂xi{DΓjkl} − ∂xj{DΓikl} + {DΓicl}{DΓjkc} − {DΓjcl}{DΓikc}.
Without loss of generality, we suppose i < j. The first term can play a role only
if i = k = 0. The second term plays no role. The third can play a role only if
i = k = 0. The final term plays no role. Thus possibly non-zero curvatures are:
DR0a0b= ∂x0{DΓa0b} + {DΓ0cb}{DΓa0c} = ∂x0θ · Uab+ θ2· UcbUac.
Let X ∈ TPM. As we must have 0 < a < b in the above relation,
DJ(X)∂xi∈ Span{∂xi+1,...,∂xr}.
Consequently (M,D) is affine Osserman. Assume θ(0) = 0 and ∂x0θ(0) = −1. We
set P = 0 and take Q = (0,0). We may then apply Lemma 3.1 to see:
gNR(Q) =g˜CPR(Q) +gDR(Q),
gNR(∂xi,∂xi′,∂xi′,∂xi)(Q) = −1,
gNR(∂xi,∂xj′,∂xi′,∂xj)(Q) =gNR(∂xi,∂xi′,∂xj′,∂xj)(Q) = −1
2
(i ?= j),
gNR(∂xi,∂xj,∂xk,∂xd′)(Q) =DRijkd(P).
First, take ξ1 :=
exhibits trivial Jordan normal form; the curvature of D plays no role. Next, we
consider ξ2:=
1
√2(∂x1+ ∂x1′). ThengNJ(ξ1) =g˜CPJ(ξ1) is diagonalizable and
1
√2(∂x0+ ∂x0′) and Jξ2=
1
√2(∂x0− ∂x0′). Then:
Page 9
HIGHER-DIMENSIONAL OSSERMAN METRICS9
g˜CPJ(ξ2)ξ2= 0,
g˜CPJ(ξ2)∂xa=1
g˜CPJ(ξ2)Jξ2= Jξ2,
g˜CPJ(ξ2)∂xa′=1
Proof of Theorem 1.5. Let n ≥ 3, let M = Rn, let P = 0, and let Q = (0,0). Let
θ = θ(x1) be a smooth function of 1 variable. Let θ1:= ∂x1θ; we suppose θ(0) = 0
and θ1(0) ?= 0. Let D be the affine connection whose only non-zero Christoffel
symbol isDΓ223= θ. Since θ = θ(x1) and since the only non-zero covariant
derivative is D∂x2∂x2= θ∂x3,DJ is nilpotent and (M,D) is affine Osserman as the
only non-zero curvature is:
gDJ(ξ2)ξ2= 0,
gDJ(ξ2)∂xa= Uab∂xb,
gDJ(ξ2)Jξ2= 0,
gDJ(ξ2)∂xa′= Uba∂xb′.
4∂xa,
4∂xa′,
?
DR(∂x1,∂x2)∂x2= θ1∂x3.
By Lemma 3.1,
gNR(∂xi,∂xi′,∂xi′,∂xi)(Q) = −1,
gNR(∂xi,∂xj′,∂xi′,∂xj)(Q) =gNR(∂xi,∂xi′,∂xj′,∂xj)(Q) = −1
gNR(∂x3′,∂x2,∂x2,∂x1)(Q) = θ1.
2
(i ?= j),
Let ξ :=
0,gNJ(ξ) =g˜CPJ(ξ) andgNJ(πξ) =g˜CPJ(πξ) are diagonalizable. Next consider
η :=1
Jη =1
1
√2(∂x1+∂x1′) ∈ S+(TQN,gN); Jξ =
1
√2(∂x1−∂x1′). As2J(ξ) =2J(Jξ) =
2(∂x1+ ∂x3+ ∂x1′+ ∂x3′) ∈ S+(TQN,gN),
2(∂x1+ ∂x3− ∂x1′− ∂x3′) ∈ S−(TQN,gN).
The only non-trivial components are provided by:
gDJ(η)∂x2=1
2θ1∂x2′,
gDJ(Jη)∂x2= −1
2θ1∂x2′,
gDJ(πη)∂x2= θ1∂x2′.
Since π2:= Span{∂x2,∂x2′} is contained both in the
in the1
exhibit non-trivial Jordan normal form.
1
4eigenspace ofg˜CPJ(η) and
2eigenspace ofg˜CPJ(πη), this analysis shows that bothgNJ(η) andgNJ(πη)
?
4. The third Gray identity, integrability, flatness, and para-complex
Osserman
In Section 4.1, we show (M,D) is flat implies (N,gN,J) is integrable and satisfies
the third Gray identity. In Section 4.2, we show (N,gN,J) is integrable implies
(M,D) is flat. In Section 4.3, we show (N,gN,J) satisfies the third Gray identity
implies (M,D) is flat. In Section 4.4, we show JgNJ(π) =gNJ(π)J for all π ∈ P(N)
if and only if (N,gN,J) satisfies the third Gray identity. This will complete the
proof of Theorem 1.6.
4.1. Flat geometry. If (M,D) is flat, then (N,gN,J) is isomorphic to˜CP by
Theorem 1.1;˜CP is integrable and satisfies the third Gray identity.
4.2. Integrability. Let (M,D) be an affine manifold. Suppose that the Nijenhuis
tensor NJof Equation (4) vanishes for the manifold (N,gN,J). Let P ∈ M. Choose
local coordinates on M so thatDΓ(P) = 0. Let Q ∈ σ−1(P). Then:
J∂xi= ∂xi− {xi′xa′ − 2xb′DΓiab}∂xa′,
J∂xj= ∂xj− {xj′xc′ − 2xd′DΓjcd}∂xc′,
[∂xi,∂xj] = 0,
J[J∂xi,∂xj] = 2xb′∂xjDΓiab∂xa′,
J[∂xi,J∂xj] = −2xb′∂xiDΓjab∂xa′,
[J∂xi,J∂xj]Q= {2xb′∂xiDΓjab− 2xb′∂xjDΓiab}Q∂xa′
+{xi′xa′∂xa′(xj′xc′) − xj′xa′∂xa′(xi′xc′)}Q∂xc′.
Page 10
10 E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO
NJ(∂xi,∂xj)(Q) = 4xb′DRijab(P)∂xa′.
Since NJ = 0, we conclude (M,D) is flat. We note that NJ always vanishes on
Z(N). Thus for this computation it is necessary to take Q arbitrary.
4.3. The third Gray identity. Suppose that (N,gN,J) satisfies the third Gray
identity which is given in Equation (3). Let Q ∈ Z(N) and let P = σ(Q). Choose
coordinates on M soDΓ(P) = 0. We apply Lemma 3.1. Since˜CP satisfies the
third Gray identity, we concludegDR satisfies the third Gray identity at Q. Thus
gDR(∂xi,∂xj,∂xk,∂xl′)(Q) =gDR(J∂xi,J∂xj,J∂xk,J∂xl′)(Q)
gDR(∂xi,∂xj,∂xk,−∂xl′)(Q) = −gDR(∂xi,∂xj,∂xk,∂xl′)(Q).
ConsequentlygDR(Q) = 0. By Lemma 3.1, this impliesDR(P) = 0. Since Q, and
hence P, was arbitrary, (M,D) is flat.
=
4.4. The commutation relation JgAJ(·) =gAJ(·)J. The third Gray identity in
the complex setting is crucial – see, for example, the discussion in [3, 12, 13]; a purely
algebraic computation shows that this condition is equivalent to the condition that
gAJ(π) commutes with the almost complex structure for every complex 2-plane
π. This computation extends to show that an almost para-Hermitian manifold
(A,gA,J) satisfies the third Gray identity if and only ifgAJ(π)J = JgAJ(π) for all
π ∈ P(A). This completes the proof of Theorem 1.6.
?
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C-L, G-R, V-L: Department of Geometry and Topology, Faculty of Mathematics,
University of Santiago de Compostela, 15782 Santiago de Compostela, Spain
E-mail address: estebcl@edu.xunta.es, eduardo.garcia.rio@usc.es,
ravazlor@edu.xunta.es
G: Mathematics Department, University of Oregon, Eugene, Oregon 97403, USA
E-mail address: gilkey@uoregon.edu
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