On curvature tensors of Norden and metallic pseudo-Riemannian manifolds

Abstract We study some properties of curvature tensors of Norden and, more generally, metallic pseudo-Riemannian manifolds. We introduce the notion of J-sectional and J-bisectional curvature of a metallic pseudo-Riemannian manifold (M, J, g) and study their properties.We prove that under certain assumptions, if the manifold is locally metallic, then the Riemann curvature tensor vanishes. Using a Norden structure (J, g) on M, we consider a family of metallic pseudo-Riemannian structures {Ja,b}a,b∈ℝ and show that for a ≠ 0, the J-sectional and J-bisectional curvatures of M coincide with the Ja,b-sectional and Ja,b-bisectional curvatures, respectively. We also give examples of Norden and metallic structures on ℝ2n.


Introduction
Let (M, g) be a pseudo-Riemannian manifold. A metallic pseudo-Riemannian structure J on M is a gsymmetric ( , )-tensor eld on M such that J = pJ + qI, for some p and q real numbers, [1], [7]. In particular, for p = and q = − , J is a Norden structure on M, [10], [2].
In this paper we study some di erential geometrical properties of Norden and, more generally, metallic pseudo-Riemannian manifolds. We de ne nearly locally metallic and nearly Kähler-Norden manifolds, extending a result of Gray, [5], about -dimensional nearly Kähler manifolds, [6], (Proposition 3.2). We focus then on curvature tensors, we introduce the notion of J-sectional and J-bisectional curvature of a metallic pseudo-Riemannian manifold (M, J, g) and study their properties. We prove that under certain assumptions, if the manifold is locally metallic, then the Riemann curvature tensor vanishes (Theorem 4.8). In the case of Norden manifolds we give a formula that express the sectional curvature with respect to the J-sectional curvature and some other terms (Proposition 4.10). This formula is the analogue of Vanhecke's formula for almost hermitian manifolds, [14], however in the Kähler-Norden case it is not possible to simplify the other terms like in [14]. In this sense, we recall that Norden metrics are necessarily neutral metrics, [3], and this class arises naturally in N = string theory, [11], [4]. There are profound di erences between Riemannian and neutral geometry, Hermitian and Norden geometry and this is one of the di erences. Then, using a Norden structure (J, g) on M, we consider a family of metallic pseudo-Riemannian structures {J a,b } a,b∈R and show that for a ≠ , the J-sectional and J-bisectional curvatures of M coincide with the J a,b -sectional and J a,b -bisectional curvatures, respectively. We also give examples of Norden and metallic structures on R n and describe the geometrical meaning of the sign of p + q.
Let us compute: From the hypothesis (∇ X J)Y + (∇ Y J)X = , we get: Moreover: Then (∇ X J)Y = . Thus ∇J = and the proof is complete.

Curvature tensors of a metallic pseudo-Riemannian manifold
In this section, we give some properties of the Riemann curvature tensor of a metallic pseudo-Riemannian manifold (M, J, g) and introduce the notion of J-sectional and J-bisectional curvature in this setting.

. Riemannian curvature
Let (M, J, g) be a locally metallic pseudo-Riemannian manifold with J = pJ + qI, p, q real numbers. Denoting by R the Riemann curvature tensor of g: we obtain: From a direct computation, we successively get: = g(R(Z, W)X, JY) = R(X, JY , Z, W), and also: From (2) and (3) we obtain: and we can state: is a locally metallic pseudo-Riemannian manifold, then the Riemann curvature tensor R of g de ned by R(X, Y , Z, W) := g(R(X, Y)Z, W) satis es the identities: for any X, Y, Z, W ∈ C ∞ (TM). .

J-sectional curvature
An analogous of the holomorphic sectional curvature will be de ned on a metallic pseudo-Riemannian manifold (M, J, g) for non degenerate plane sections as follows.

De nition 4.2.
Let X be a non zero tangent vector eld and let π X,JX be the plane generated by X and JX. If g(X, X)g(JX, JX) − [g(X, JX)] ≠ , then π X,JX is called non degenerate and we de ne the J-sectional curvature as: De nition 4.4. Let X and Y be two linearly independent tangent vector elds and let π X,JX and π Y ,JY be the planes generated by X and JX, and Y and JY, respectively. If π X,JX and π Y ,JY are non degenerate, we de ne the J-bisectional curvature as: Proof. For a locally metallic Riemannian manifold the following identities hold: Thus for any X, Y ∈ C ∞ (TM).
Since q ≠ , J is invertible and J − = q J − p q I, therefore, by using the RM-hypothesis we get Substituting R(X, JY , X, JY) we get In particular, we get and replacing X by JX: If ( p + q q + ) ≠ , that is if p ≠ (q − ) and p ≠ , q ≠ − , R must be zero. are no restrictions on the curvature. Moreover, we remark that the case p = , q = − , is the case of Norden structures. Kähler-Norden manifolds are RM-manifolds, but not necessarily at, [8], [9], [12].

. Curvature tensor of Norden manifolds
Let (M, J, g) be a Norden manifold and let R be the Riemann curvature tensor of g. Denote by: for any X, Y ∈ C ∞ (TM).
Proposition 4.10. The following formula holds: for any X, Y ∈ C ∞ (TM).

Proof. A direct computation gives:
Thus: Then, by using rst Bianchi's identity and substituting it, we get the statement.
Corollary 4.11. If (M, J, g) is an RM-manifold, then we obtain the following formula: for any X, Y ∈ C ∞ (TM).

Remark 4.12.
The above formula is the formula in Theorem 1 of [14], however here it is not possible to express the sectional curvature with the respect to the J-sectional curvature. Namely, in Kähler-Norden case, λ doesn't vanish. Actually, in Kähler-Norden case, k J is zero and the formula is just an identity.

Metallic pseudo-Riemannian structures and Norden structures
In [1], we showed that any metallic pseudo-Riemannian structure (J, g) on M satisfying J = pJ + qI, where p and q are real numbers with p + q < , de nes two Norden structures: and let g be the neutral metric on R de ned by M. J ∈ R( ) de nes a Norden structure on (R , g) if and only if J = −I and t JM = MJ. A direct computation gives the expression of J: for some real number a. If we denote by Ja ± ,n the matrices of order n with n diagonal blocks equal to Ja ± : . Examples of locally metallic pseudo-Riemannian structures on (R n , g n ) Let Ja ± ,n be the at Kähler-Norden structures on R n de ned above, with respect to the neutral metric gn, and let α, β be real numbers. Then J α,β,n := αJa ± ,n + βI is a family of at locally metallic pseudo-Riemannian structures on (R n , gn). In block matrix form, we have the following expression: . Examples of locally metallic Riemannian structures on (R n , < , >) De nition 6.1. A ∈ R(n) is called a metallic matrix if there exist two real numbers p and q such that A = pA + qI, where I is the identity matrix.
We remark that the condition α − pα − q ≤ implies p + q ≥ and Remark 6.2. The sign of p + q characterizes the geometry of a metallic structure (M, J, g) with J = pJ + qI. Namely, p + q < is a necessary and su cient condition in order that J de nes a Norden structure on (M, g). In particular, in this case, M is even dimensional and g must be a neutral metric.