Four-dimensional semi-Riemannian Szab\'o manifolds

In this paper, we prove that the deformed Riemannian extension of any affine Szab\'o manifold is a Szab\'o pseudo-Riemannian metric and vice-versa. We proved that the Ricci tensor of an affine surface is skew-symmetric and nonzero everywhere if and only if the affine surface is Szab\'o. We also find the necessary and sufficient condition for the affine Szab\'o surface to be recurrent. We prove that for an affine Szab\'o recurrent surface the recurrence covector of a recurrence tensor is not locally a gradient.


INTRODUCTION
Let T * M be the cotangent bundle of n-dimensional manifold M with a torsion free affine connection ∇.Patterson and Walker [17] introduced the notion of Riemannian extensions and showed how to construct a pseudo-Riemannian metric on the 2n-dimensional cotangent bundle of any n-dimensional manifold with a torsion free connection.Afifi [1] studied the local properties of Riemannian extension of connected affine spaces.Riemannian extensions were also studied by Garcia-Rio et al. [11] for Osserman manifolds.One of the author Diallo [6] find the fruitful results for the Riemannian extension of an affine Osserman connection on 3-dimensional manifolds.In [12], the authors generalized the Riemannian extension to the deformed Riemannian extensions.In the recent paper [7], we construct example of pseudo-Riemannian Szabó metrics of signature (2, 2) by using the deformed Riemannian extension, whose Szabó operators are nilpotent.The Riemannian extension can be constructed with the help of the coefficients of the torsion free affine connection.For Riemannian extensions, also see [15,16,19].For deformed Riemannian extensions, also see [2,3,8].
In this paper, we study the deformed Riemannian extensions of affine Szabó manifold.Our paper is organized as follows.In the section 2, we recall some basic definitions and results on the deformed Riemannian extension.In section 3, we provide some known results on affine Szabó manifolds.We proved that the Ricci tensor of an affine surface is skew-symmetric and nonzero everywhere if and only if affine surface is Szabó.We also find the necessary and sufficient condition for the affine Szabó surface to be recurrent.We prove that for an affine Szabó recurrent surface the recurrence covector of a recurrence tensor is not locally a gradient.Finally in section 4, we prove that the deformed Riemannian extension of any affine Szabó manifold is a Szabó pseudo-Riemannian metric and vice-versa.
Throughout this paper, all manifolds, tensors fields and connections are always assumed to be C ∞ -differentiable.

DEFORMED RIEMANNIAN EXTENSIONS
Let T * M be the cotangent bundle of n-dimensional affine manifold M with torsion free affine connection ∇ and let π : T * M → M be the natural projection defined by is a basis for the cotangent space (T * M ) (p,ω) .For more details on the geometry of cotangent bundle, see [21].
The Riemannian extension g ∇ is the pseudo-Riemannian metric on T * M of neutral signature (n, n) characterized by the identity [12] where X C is a complete lift of the vector field X on M and the function ιX : T * M −→ R defined by ιX(p, ω) = ω(X p ).For more details, see [12].In the locally induced coordinates (u i , u i ′ ) on π −1 (U ) ⊂ T * M , the Riemannian extension [17] is expressed by where Γ k ij are the coefficients of the torsion free affine connection ∇ with respect to (U, u i ) on M .
Riemannian extensions provide a link between affine and pseudo-Riemannian geometries, therefore by using the properties of the Riemannian extension g ∇ , we investigate the properties of the affine connection ∇.Like, (M, ∇) is locally symmetric if and only if (T * M, g ∇ ) is locally symmetric.In the same way, (M, ∇) is projectively flat if and only if (T * M, g ∇ ) is locally conformally flat [4].
Let φ be a symmetric (0, 2)-tensor field on an affine manifold (M, ∇).In [4], the authors introduced a deformation of the Riemannian extension by means of a symmetric (0, 2)-tensor field φ on M .They considered the cotangent bundle T * M equipped with the metric g ∇ + π * φ, which is called the deformed Riemannian extension.
The deformed Riemannian extension denoted g (∇,φ) is the metric of neutral signature (n, n) on the cotangent bundle given by In local coordinates the deformed Riemannian extension is given by where Γ k ij are the coefficients of the torsion free affine connection ∇ and φ ij are the local components of the symmetric (0, 2)-tensor field φ.Equivalently, Note that the crucial terms g (∇,φ) (∂ i , ∂ j ) now no longer vanish on the 0-section, which was the case for the Riemannian extension, the Walker distribution is the kernel of the projection from T * M : In the deformed Riemannian extension, the tensor φ plays an important role.If the underlying connection is flat, the deformed Riemannian extension need not be flat [12].Deformed Riemannian extensions have nilpotent Ricci operator therefore they are Einstein if and only if they are Ricci flat.So deformed Riemannian extension can be used to construct non-flat Ricci flat pseudo-Riemannian manifolds [4].

THE AFFINE SZABÓ MANIFOLDS
Let (M, ∇) be an affine manifold and X ∈ Γ(T p M ).The affine Szabó operator S ∇ (X) [18] with respect to X is a function from T p M to T p M , p ∈ M defined by for any vector field Y and where R ∇ is the curvature operator of the affine connection ∇.The affine Szabó operator satisfies S ∇ (X)X = 0 and S ∇ (βX) = Let (M, ∇) be an affine manifold and p ∈ M .(M, ∇) is said to affine Szabó at p ∈ M if the affine Szabó operator S ∇ has the same characteristic polynomial for every vector field X on M .If (M, ∇) is affine Szabó at each p ∈ M , then (M, ∇) is known as affine Szabó.For more details, see [10].Now, we give a known result for later use.
Proof.It easy to show that the Ricci tensor of ∇ is skew-symmetric.
A Lagrangian L : U → R in a manifold Σ is a function on a nonempty open set U ⊂ T Σ.A Lagrangian L : U → R gives rise to equations of motion, which are the Euler-Lagrange equations, imposed on curves t → y(t) ∈ Σ, the velocity t → v(t) ∈ T Σ, lies entirely in U .A fractional-linear function in a twodimensional real vector space Π is a rational function of the form α/β, defined on a nonempty open subset of Π\ ker β, where α, β ∈ Π * are linearly independent functionals.By using [5,Th 11.1] and Theorem 3.3, we have Theorem 3.5.Let ∇ be an torsion-free affine connection on a surface Σ.If every point in T Σ\Σ has a neighborhood U with a fractional-linear Lagrangian L : U → R such that the solutions of the Euler-Lagrange equations for L coincide with those geodesics of ∇ which, lifted to T Σ, lie in U , then (Σ, ∇) is affine Szabó.Definition 3.6.[20] A tensor field T is said to be recurrent if there exists a 1-form α such that ∇T = α ⊗ T , where ∇ is an affine connection.In particular, an affine surface (Σ, ∇) is said to be recurrent if its Ricci tensor is recurrent.Theorem 3.7.Let (Σ, ∇) be an affine Szabó surface.Then (Σ, ∇) is recurrent if and only if around each point there exists a coordinate system (U, u h ) with the non-zero components of ∇ are Proof.Consider the Ricci tensor ρ = ρ a + ρ s , where ρ a is the antisymmetric part of ρ and ρ s is the symmetric part of ρ.Then by using Theorem 3.3, we can say that (Σ, ∇) is an affine Szabó if and only if the Ricci tensor of ∇ is skew-symmetric and nonzero everywhere.Then it follows from [20, Th 4.2] that one of the three possibility for a non-flat recurrent affine surface is the one in which around each point there exists a coordinate system (U, u h ) with the non-zero components of ∇ are By using the result of [20, Th 2.2] and Theorem 3.3, we can say that Theorem 3.8.Let (Σ, ∇) be an affine Szabó recurrent surface.Then the recurrence covector of a recurrence tensor is not locally a gradient.

THE DEFORMED RIEMANNIAN EXTENSIONS OF AN AFFINE SZABÓ MANIFOLD
A pseudo-Riemannian manifold (M, g) is said to be Szabó if the Szabó operators S(X) = (∇ X R)(•, X)X has constant eigenvalues on the unit pseudo-sphere bundles S ± (T M ).Any Szabó manifold is locally symmetric in the Riemannian [18] and the Lorentzian [14] setting but the higher signature case supports examples with nilpotent Szabó operators (cf.[13] and the references therein).Now, we will prove the following result: Theorem 4.1.Let (M, ∇) be a 2-dimensional smooth torsion-free affine manifold.Then the following assertions are equivalent: (1) (M, ∇) is an affine Szabó manifold.
Proof.Let Γ k ij be the coefficients of the torsion free affine connection ∇ and φ ij denote the local components of φ.Then, the deformed Riemannian extension of the torsion free affine connection ∇ is the pseudo-Riemannian metric tensor on T * M of signature (2, 2) given by A straightforward calculation shows that the non-zero Christoffel symbols Γγ αβ of the Levi-Civita connection are given as follows where (i, j, k, l, r = 1, 2) and (i ′ = i + 2, j ′ = j + 2, k ′ = k + 2, r ′ = r + 2).The non-zero components of the curvature tensor of (T * M, g (∇,φ) ) up to the usual symmetries are given as follows (we omit R h ′ kji , as it plays no role in our considerations) kji are the components of the curvature tensor of (M, ∇).(For more details, see [4].)Let X = α i ∂ i + α i ′ ∂ i ′ be a vector field on T * M .Then the matrix of the Szabó operator S( X) with respect to the basis {∂ i , ∂ i ′ } is of the form S( X) = S ∇ (X) 0 t S ∇ (X) .
2n around (p, ω) ∈ T * M , where ω i are components of covectors ω in each cotangent space T * p M , p ∈ U with respect to the natural coframe {du i }.Let ∂ i =