Riesgo Financiero y Riesgo de Insolvencia

The Ricci curvature Ric of the Levi-Civita connection on a Riemannian manifoldM is a symmetric two-tensor. The Einstein condition requires the Ricci tensor to be a multiple of the metric. In this case, the second Bianchi identity constrains the scalar curvature to be constant in dimensionn > 2 [KN96], and so the Ricci tensor is parallel. Gray generalized the Einstein condition by studying the subrepresentation W of T∗⊗S2_T∗_{of tensors}ϕ that satisfy the relation ∑

iϕ(x,ei,ei) = 2 ∑iϕ(ei,ei,x)

with respect to any orthonormal basis{ei∣ 1 ≤ i ≤ n} of T [Gra78, Chapter 3]. For

ϕ = ∇ Ric this corresponds to the identity d tr Ric = 2δ Ric, so ∇ Ric is a section of the associated vector bundleW M. If ∇ Ric takes values in the subbundle CM withC = T∗⊗S2

0T

∗∩W , then d tr Ric = 0 that is the scalar curvature scal is con-

stant [DN69, Proposition. 2.3] The tensor product decomposition of T∗⊗S2_T∗_is

(S3

reducible representationsA = S[2,1]_T∗, B = S3

0T∗withA ⊕ B = C and Q = C⊥⊂ W

[Gra78, Theorem 3.1].

The condition∇ Ric ∈ Γ(A) is equivalent to ∇ Ric having no component in Q ⊕ B. This condition means that d0Ric is zero and d tr Ric= 2δ Ric = 0. Hence, Ric is a

Killing tensor in this case. A manifold whose Ricci tensor is a Killing tensor is said to be of classA [Gra78, (2.1)]. If∇ Ric ∈ Γ(A⊕Q)M then the Ricci tensor is conformal Killing. Such manifolds are often calledAC⊥ _{manifolds. A detailed exposition of}

this topic can be found in [Bes87, Chapter 16, Section G]. What follows is a short summary of the results presented there and an overview of Jelonek’s results on this topic.

9.55 Theorem ([DN69, Corollary 2.4]) If(M,д) is a two-dimensional Riemannian

manifold of classA then д has constant curvature. 9.55 ◂

Proof. If∇ Ric has values in A ⊂ C it has constant scalar curvature. The statement is true because on a surface, scalar curvature coincides with sectional curvature of

the only available two-plane. 

The next statement is an application ofTheorem 6.4, a corollary to the Weitzenböck formula.

9.56 Theorem ([Bes87, p. 16.54]) Let(M,д) be a compact Riemannian manifold of classA. If д has non-positive sectional curvature, then Ric is parallel. Moreover, if there is a point onM at which the sectional curvature is not vanishing on any

two-plane thenд is Einstein. 9.56 ◂

Proof. Since the trace free part of the Ricci tensor Ric0is conformal Killing it must

be parallel byTheorem 6.4. Because∇ Ric has values in A ⊂ C, the scalar curvature is constant. Thus, Ric= Ric0+scal_n д is parallel. Furthermore, if there is a point in

M at which the sectional curvature is strictly negative on any two-plane then the

above theorem implies that Ric0 = 0. Hence, Ric = scal_n д. 

9.57 Definition ([KPV96, §3]) A Riemannian manifold(M,д) is a D’Atri space if each local geodesic symmetrysp∶ exp(tξ ) ↦ exp(−tξ ) for ξ ∈ TpM with ∣ξ ∣ = 1

9.58 Theorem ([Bes87, Theorem 16.55], [KPV96, §3.4, §4.1]) Every D’Atri space (M,д) is of class A and every naturally reductive Riemannian homogeneous mani-

fold is a D’Atri space. 9.58 ◂

The statement that every naturally reductive Riemannian manifold is of classA can also be found in [HMS16, Proposition 4.5]. Gray gave the first examples of homogeneous manifolds of classA with non-parallel Ricci curvature.

9.59 Example ([Gra78, Theorem 4.3]) Embed O(2) into O(4) by Q(u,v) = (Qu,v)

forQ ∈ O(2) and (u,v) ∈ R2_{× R}2 _{= R}4_{. ThenM = O(4)/O(2) is of class A and}

does not have parallel Ricci tensor. 9.59 ◂

Note, that this Stiefel manifold is a principal O(2)-bundle over the Grassmannian. It is equipped with a fibre bundle metric making it the total space of a Riemannian submersion. Such metrics were described in the last section.

9.60 Example ([Gra78, Theorem 4.4]) The action ofI, J, K ∈ H on S3_{⊂ H induces}

three nowhere vanishing vector fields onS3. LetϕI,ϕJ,ϕK be the corresponding dual one-forms. Fora,b,c ∈ R ∖ {0} the symmetric two tensor дa,b,c = a2ϕ_I2+b2ϕ2_J+c2ϕ_K2

defines a Riemannian metric onS3_{. The Riemannian space(S}3_,д

a,b,c) is of class A

and has non-parallel Ricci curvature if and only if two of the numbersa2,b2_andc2

are equal and the third number is different. 9.60 ◂

Once again, this a Riemannian submersion of the circle bundleS3_{over CP}1_{, equipped}

with a bundle metric. Another source of homogeneous examples can be found in [Gra78].

9.61 Definition ([Kow80, Definition 0.5]) A Riemannian manifold(M,д) is a 3- symmetric space if for every pointp ∈ M there is an isometry θp ofд with θp3= id

that has an isolated fixed pointp.

A Riemannian manifold(M,д) is a nearly Kähler manifold, if there is a complex structureJ ∈ Γ(End(T)) with J2 _{= − id such that (∇}

xJ)x = 0 for all x ∈ T M. 9.61 ◂

Every 3-symmetric space is a homogeneous almost complex manifold [Gra72, Proposition 3.2, Corollary 5.3].

9.62 Theorem ([Gra78, Theorem 7.1]) Let M be a nearly Kähler, 3-symmetric

space thenM is of class A. 9.62 ◂

Gray shows that under the assumption of the theorem holds∇xρ(x,x) = ρ((∇xJ)x,x)

9.63 Example ([Gra78, p. 272])

U(4)/(U(2) × U(1) × U(1))

is a nearly Kähler, 3-symmetric space with non-parallel Ricci curvature. 9.63 ◂

In consequence of the following fact, it is necessary to look at the non-Kähler case to get non-trivial examples.

9.64 Theorem ([Gra78, Theorem 6.1]) If a Kähler manifold is of classA then it

has parallel Ricci tensor. 9.64 ◂

The proof given by Gray is very short and is repeated here for the convenience of the curious reader.

Proof. The Ricci tensor of a Kähler manifold commutes with the complex structure. [Bes87, 2.45 Proposition] Hence, the Ricci form defined byρ(x,y) = Ric(Jx,y) is a J-invariant two-form. In addition, the Ricci form of a Kähler manifold is closed [Bes87, 2.47 Proposition]. Therefore one has

0= d ρ(x,y,z) = ∇xρ(y,z) + ∇yρ(z,x) + ∇zρ(x,y).

Specialising toz = Jx and using the definition of ρ and the J-invariance of Ric one gets

0= ∇xRic(y,x) − ∇yRic(x,x) + ∇JxRic(Jx,y).

On the other hand, the Killing equation 0= ∇uRic(v,w)+∇vRic(w,u)+∇wRic(u,v)

specialises forw = u to

2∇uRic(u,v) = −∇vRic(u,u).

Using the latter once for(u,v) = (x,y) and a second time for (u,v) = (Jx,y) to simplify the former yields

0= −1

2∇yRic(x,x) − ∇yRic(x,x) − 1

2∇yRic(Jx, Jx) = −2∇yRic(x,x). Since∇yRic is symmetric for everyy the result follows from polarization. 

9.65 Proposition ([Jel07, Corollary to Proposition 1]) If the Ricci tensor of a four- dimensional, almost Kähler manifold of class A is hermitian then it is parallel.

9.65 ◂

The proof is given in the cited article. It is based on the fact that in dimension four a self-adjoint endomorphism commuting with the complex structure has at most two eigenvalues.

The first non-homogeneous examples of classA with non-parallel Ricci tensor were provided by Jelonek in [Jel95]. It is a Riemannian submersion on aS1_-principal

bundle with bundle metric as introduced in Lemma 9.33. It can be seen as a generalization of Gray’sExample 9.60.

9.66 Example ([Jel95, Theorem 3.3, Corollary 3.7]) Let(M,д,ω) be a n-dimen- sional Kähler-Einstein manifold with scalar curvatures < 0 as well as n > 2 and π∶ P → M be the S1_{-principal bundle with connection one-formθ and defining}

curvature form dθ = −_2ns π∗ω. There is a bi-invariant metric ˇд on S1_{such that the}

bundle metric

ˆ

д = π∗д + θ∗д_ˇ

makes (P, ˆд) a manifold of class A with non-parallel Ricci tensor. Moreover, if (M,д,ω) is a compact locally non-homogeneous Kähler-Einstein manifold then so

is(P, ˆд). 9.66 ◂

Jelonek gave also a source which provides possible base manifolds: The existence of locally non-homogeneous Kähler-Einstein manifolds(M,д,ω) are guaranteed by the famous theorem of Calabi-Yau [Joy00, Chapter 5].

The construction of the example above uses the characterization of a Killing tensor with two constant eigenvalues as introduced in Chapter 5 and the ansatz of a Riemannian submersion metric on a principal bundle, presented inSection 9.2. In a series of several articles Jelonek considers variations of that idea. The following examples are all Riemannian submersions with a classA-metric and non-parallel Ricci curvature. In [Jel98b] the Kähler-Einstein condition is relaxed to almost Hodge manifolds. These are almost Kähler manifolds with Kähler form proportional to an integer cohomology class. The articles [Jel98a; Jel99a] consider SO(3)-principal bundles and twistor bundles over a self-dual Einstein base with further generaliza- tion in [Jel01]. Yet another variation is [Jel00;Jel01] where the base is changed to a quaternionic Kähler manifold. Furthermore, there are the examples of Pedersen and Tod and Zborowski. These are tori-bundle over products of Kähler-Einstein

[PT99] or almost Hodge manifolds [Zbo13;Zbo15]. It is worth mentioning that the examples of Zborowski have more then two distinct eigenvalues.

At last it should not go unnoticed that there are also manifolds with conformal Killing Ricci tensor.

9.67 Example ([Jel99b, Theorem 3.3]) Let(M, [д]) be a compact Einstein-Weyl manifold. Then there is a metric tensorд ∈ [д] such that (M,д) is a proper AC⊥_-

manifold. That is, the corresponding Ricci tensor is a conformal Killing tensor,

which is neither parallel nor Killing. 9.67 ◂