# Four-Dimensional Semi-Riemannian Szabó Manifolds

Authors: Abdoul Salam Diallo; Punam Gupta
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we prove that the deformed Riemannian extension of any affine Szabó manifold is a Szabó pseudo-Riemannian metric and vice versa. We prove that the Ricci tensor of an affine surface is skew-symmetric and nonzero everywhere if and only if the 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.


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## Body

## 1. Introduction

LetT∗M be the cotangent bundle of n-dimensional manifold M with a torsion-free affine connection ∇. Patterson and Walker [1] 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 [2] studied the local properties of Riemannian extension of connected affine spaces. Riemannian extensions were also studied by García-Río et al. [3] for Osserman manifolds. Diallo [4] found the fruitful results for the Riemannian extension of an affine Osserman connection on 3-dimensional manifolds. In [5], the authors generalized the Riemannian extension to the deformed Riemannian extensions. In the recent paper [6], 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 [7–9]. For deformed Riemannian extensions, also see [10–12].In this paper, we study the deformed Riemannian extensions of affine Szabó manifold. Our paper is organized as follows. In Section2, 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 prove that the Ricci tensor of an affine surface is skew-symmetric and nonzero everywhere if and only if the 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 beE∞-differentiable.


## 2. Deformed Riemannian Extensions

LetT∗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(1)πp,ω=p∈Mandp,ω∈T∗M.A system of local coordinatesU,ui,i=1,…,n around p∈M induces a system of local coordinates π−1U,ui,ui′=ωi,i′=n+i=n+1,…,2n, around p,ω∈T∗M, where ωi are components of covectors ω in each cotangent space Tp∗M, p∈U, with respect to the natural coframe dui. Let ∂i=∂/∂ui and ∂i′=∂/∂ωi,i=i,…,n, then, at each point p,ω∈T∗M,(2)∂1p,ω,…,∂np,ω,∂1′p,ω,…,∂n′p,ω,is a basis for the cotangent space T∗Mp,ω. For more details on the geometry of cotangent bundle, see [13].The Riemannian extensiong∇ is the pseudo-Riemannian metric on T∗M of neutral signature n,n characterized by the identity [5].(3)g∇XC,YC=−ι∇XY+∇YX,where XC is a complete lift of the vector field X on M, and the function ιX:T∗M⟶ℝ is defined by(4)ιXp,ω=ωXp.For more details, see [5]. In the locally induced coordinates ui,ui′ on π−1U⊂T∗M, the Riemannian extension [1] is expressed by(5)g∇=−2uk′Γijkδijδij0,with respect to the basis ∂1,…,∂n,∂1′,…,∂n′i,j,k=1,…,n;k′=k+n, where Γijk are the coefficients of the torsion-free affine connection ∇ with respect to U,ui on M.Riemannian extensions provide a link between affine and pseudo-Riemannian geometries. Therefore, by using the properties of the Riemannian extensiong∇, we investigate the properties of the affine connection ∇. Similarly, 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 [14].Letϕ be a symmetric 0,2-tensor field on an affine manifold M,∇. In [14], 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 denotedg∇,ϕ which is the metric of neutral signature n,n on the cotangent bundle given by(6)g∇,ϕ=g∇+π∗ϕ.In local coordinates, the deformed Riemannian extension is given by(7)g∇,ϕ=ϕiju−2uk′Γijkδijδij0,with respect to the basis ∂1,…,∂n,∂1′,…,∂n′,i,j,k=1,…,n;k′=k+n, where Γijk are the coefficients of the torsion-free affine connection and ∇ and ϕij are the local components of the symmetric 0,2-tensor field ϕ. Equivalently,(8)g∇,ϕ∂i,∂j=ϕiju−2uk′Γijk,g∇,ϕ∂i,∂j′=δij,g∇,ϕ∂i′,∂j′=0.Note that the crucial termsg∇,ϕ∂i,∂j now no longer vanish on the 0-section, which was the case for the Riemannian extension, and the Walker distribution is the kernel of the projection from T∗M:(9)D=kerπ∗=Span∂i′.In the deformed Riemannian extension, the tensorϕ plays an important role. If the underlying connection is flat, the deformed Riemannian extension need not be flat [5]. 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 nonflat Ricci pseudo-Riemannian manifolds [14].


## 3. The Affine Szabó Manifolds

LetM,∇ be an affine manifold and X∈ΓTpM. The affine Szabó operator S∇X [15] with respect to X is a function from TpM to TpM, p∈M, defined by(10)S∇XY=∇Xℛ∇Y,XX,for any vector field Y and where ℛ∇ is the curvature operator of the affine connection ∇. The affine Szabó operator satisfies S∇XX=0 and S∇βX=β3SX, for β∈ℝ∗. If Y=∂m, for m=1,2,…,n, and X=∑iαi∂i, we have(11)S∇X∂m=∑i,j,k=1nαiαjαk∇iℛ∇∂m,∂j∂k,where ∇i=∇∂i.LetM,∇ 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 [16].Now, we give a known result for later use.Theorem 1.
(see [17]). Let M,∇ be an n-dimensional affine manifold and p∈M. Then, M,∇ is affine Szabó at p∈M if and only if the characteristic polynomial of the affine Szabó operator S∇ is PλS∇X=λn, for every X∈TpM.We have a complete description of affine Szabó surfaces.Theorem 2.
(see [17]). Let Σ=M,∇ be an affine surface. Then, Σ is affine Szabó at p∈M if and only if the Ricci tensor of M,∇ is cyclic parallel at p∈M.Next, we investigate some particular case. The curvature of an affine surface is encoded by its Ricci tensor. We fix coordinatesu1,u2 on Σ and let ∇∂i∂j=Γijk∂k, for i,j,k=1,2, where Γijk=Γijku1,u2. Then, a straightforward calculation shows that the components of the curvature tensor ℛ are given by(12)ℛ∂1,∂2∂1=ρ21∂1−ρ11∂2andℛ∂1,∂2∂2=ρ22∂1−ρ12∂2,where ρij,i,j=1,2 are the components of the Ricci tensor given(13)ρ21=∂1Γ121−∂2Γ111+Γ121Γ122−Γ112Γ221,ρ11=−∂1Γ122−∂2Γ112+Γ112Γ121+Γ122Γ122−Γ111Γ122−Γ112Γ222,ρ22=∂1Γ221−∂2Γ121+Γ111Γ221+Γ121Γ222−Γ121Γ121−Γ122Γ221,ρ12=−∂1Γ222−∂2Γ122+Γ112Γ221−Γ121Γ122.LetX=α1∂1+α2∂2 be a vector field on Σ. It is easy to check that the affine Szabó operator SX expresses, with respect to the basis ∂1,∂2, as(14)S∇X=ABCD.where the coefficients A, B, C, and D are given by(15)A=α12α2∂1ρ21−Γ111+Γ122ρ21−Γ121ρ11−Γ112ρ22+α1α22∂2ρ21+∂1ρ22−Γ121+Γ222ρ21−ρ12+ρ21Γ121−Γ221ρ11−3Γ122ρ22+α23∂2ρ22−2Γ222ρ22−ρ12+ρ21Γ221,B=α12α2−∂1ρ11+2Γ111ρ11+ρ12+ρ21Γ112+α1α22−∂2ρ11−∂1ρ12+3Γ121ρ11+Γ112ρ22+ρ12+ρ21Γ122+Γ111+Γ122ρ12+α23−∂2ρ12+Γ221ρ11+Γ122ρ22+Γ121+Γ222ρ12,C=α13−∂1ρ21+Γ111+Γ122ρ21+Γ121ρ11+α12α2−∂2ρ21−∂1ρ22+Γ121+Γ222ρ21+Γ221ρ11+3Γ122ρ22+ρ12+ρ21Γ121+α1α22−∂2ρ22+2Γ222ρ22+ρ12+ρ21Γ221,D=α13∂1ρ11−2Γ111ρ11−ρ12+ρ21Γ112+α12α2∂2ρ11+∂1ρ12−3Γ121ρ11−Γ112ρ22−Γ111+Γ122ρ12−ρ12+ρ21Γ122+α1α22∂2ρ12−Γ221ρ11−Γ122ρ22−Γ121+Γ222ρ12.Its characteristic polynomial is given by(16)PλS∇X=λ2−λA+D+AD−BC.Here, we investigate affine surfaces whose Ricci tensor is skew-symmetric.Theorem 3.
Let∇ be an torsion-free affine connection on a surface Σ. Then, the Ricci tensor of ∇ is skew-symmetric and nonzero everywhere if and only if Σ,∇ is affine Szabó.Proof.
If the Ricci tensor of∇ is skew-symmetric, that is, ρ11=ρ22=0 and ρ12=−ρ21. Then, the Szabó operator is nilpotent.
Conversely, ifΣ,∇ is affine Szabó, then the trace and determinant of (14) will be zero, which is possible only if ρ11=ρ22=0 and ρ12=−ρ21.The investigation of affine connections with skew-symmetric Ricci tensor on surfaces has been extremely attractive and fruitful over the recent years. We refer to the paper [18] by Derdzinski for further details. Taking into account the simplified Wong’s theorem ([19], Theorem 4.2) given in [18], we have the following.Theorem 4.
If every point of an affine surfaceΣ has a neighborhood U with coordinates u1,u2 in which the component functions of a torsion-free affine connection ∇ are Γ111=−∂1φ, Γ222=∂2φ, for some function φ, Γjkl=0, unless j=k=l, then Σ,∇ is affine Szabó.Proof.
It easy to show that the Ricci tensor of∇ is skew-symmetric.A LagrangianL:U⟶ℝ in a manifold Σ is a function on a nonempty open set U⊂TΣ. A Lagrangian L:U⟶ℝ gives rise to equations of motion, which are the Euler–Lagrange equations, imposed on curves t⟶yt∈Σ and the velocity t⟶vt∈TΣ lies entirely in U. A fractional-linear function in a two-dimensional real vector space Π is a rational function of the form α/β, defined on a nonempty open subset of Π/kerβ, where α,β∈Π∗are linearly independent functionals. By using ([18], Theorem 11.1) and Theorem 3, we have.Theorem 5.
Let∇ be a torsion-free affine connection on a surface Σ. If every point in TΣ\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 TΣ, lie in U, then Σ,∇ is affine Szabó.Definition 1 (see [19]).
A tensor fieldT 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 6.
LetΣ,∇ be an affine Szabó surface. Then, Σ,∇ is recurrent if and only if, around each point, there exists a coordinate system U,uh with the nonzero components of ∇ which are(17)Γ111=−∂1φ,Γ222=∂2φ,for some scalar function φ such that ∂2∂1φ≠0. Moreover, Σ,∇ is not locally symmetric.Proof.
Consider the Ricci tensorρ=ρa+ρs, where ρa is the antisymmetric part of ρ and ρs is the symmetric part of ρ. Then, by using Theorem 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 ([19], Theorem 4.2) that one of the three possibilities for a nonflat recurrent affine surface is the one in which, around each point, there exists a coordinate system U,uh with the nonzero components of ∇ which are(18)Γ111=−∂1φ,Γ222=∂2φ,for some scalar function φ such that ∂2∂1φ≠0. Now, it is easy to calculate that ρ21=−ρ12=∂2∂1φ, which is never zero. So, Σ,∇ is not locally symmetric.By using the result of ([19], Theorem 2.2) and Theorem 3, we can say as follows.Theorem 7.
LetΣ,∇ be an affine Szabó recurrent surface. Then, the recurrence covector of a recurrence tensor is not locally a gradient.


## 4. The Deformed Riemannian Extensions of an Affine Szabó Manifold

A pseudo-Riemannian manifoldM,g is said to be Szabó if the Szabó operators SX=∇XR·,XX have constant eigenvalues on the unit pseudosphere bundles S±TM. Any Szabó manifold is locally symmetric in the Riemannian [15] and the Lorentzian [20] setting, but the higher signature case supports examples with nilpotent Szabó operators (cf. [21] and the references therein). Now, we will prove the following result.Theorem 8.
LetM,∇ be a 2-dimensional smooth torsion-free affine manifold. Then, the following assertions are equivalent:(1)
M,∇ is an affine Szabó manifold(2)
The deformed Riemannian extensionT∗M,g∇,ϕ of M,∇ is a pseudo-Riemannian nilpotent Szabó manifold of neutral signatureProof.
LetΓijk be the coefficients of the torsion-free affine connection ∇ and ϕij 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(19)g∇,ϕ=ϕ11u1,u2−2u3Γ111−2u4Γ112du1⊗du1+ϕ22u1,u2−2u3Γ221−2u4Γ222du2⊗du2+ϕ12u1,u2−2u3Γ121−2u4Γ122du1⊗du2+du1⊗du2+du1⊗du3+du3⊗du1+du2⊗du4+du4⊗du2.
A straightforward calculation shows that the nonzero Christoffel symbolsΓ˜αβγ of the Levi–Civita connection are given as follows:(20)Γ˜ijkΓijk;Γ˜i′jk′=−Γjki;Γ˜ij′k′=−Γijj,Γ˜ijk′∑r=12∂kΓijr−∂iΓjkr−∂jΓikr+2∑l=12ΓklrΓijl+12∂iϕjk+∂jϕik−∂kϕij−∑l=12ϕklΓijl,where i,j,k,l,r=1,2 and i′=i+2,j′=j+2,k′=k+2,r′=r+2. The nonzero components of the curvature tensor of T∗M,g∇,ϕ up to the usual symmetries are given as follows (we omit R˜kjih′, as it plays no role in our considerations):(21)R˜kjih=Rkjih,R˜kjih′,R˜kji′h′=−Rkjhi,R˜k′jih′=Rhijk,where Rkjih are the components of the curvature tensor of M,∇. For more details, see [14].
LetX˜=αi∂i+αi′∂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:(22)S˜X˜=S∇X0∗St∇X,where S∇X is the matrix of the affine Szabó operator on M relative to the basis ∂i. Note that the characteristic polynomial PλS˜X˜ of S˜X˜ and PλS∇X of S∇X are related by(23)PλS˜X˜=PλS∇X·PλSt∇X.
Now, if the affine manifoldM,∇ is assumed to be affine Szabó, then S∇X has zero eigenvalues for each vector field X on M. Therefore, it follows from (23) that the eigenvalues of S˜X˜ vanish for every vector field X˜ on T∗M. Thus, T∗M,g∇ is pseudo-Riemannian Szabó manifold.
Conversely, assume thatT∗M,g∇ is a pseudo-Riemannian Szabó manifold. If X=αi∂i is an arbitrary vector field on M, then X˜=αi∂i+1/2αi∂i′ is an unit vector field at every point of the zero section on T∗M. Then, from (23), we see that the characteristic polynomial PλS˜X˜ of S˜X˜ is the square of the characteristic polynomial PλS∇X of S∇X. Since for every unit vector field X˜ on T∗M, the characteristic polynomial PλS˜X˜ should be the same, and it follows that, for every vector field X on M, the characteristic polynomial PλS∇X is the same. Hence, M,∇ is affine Szabó.For an example, we have the following.Theorem 9.
(see [6]). Let M=ℝ2 and ∇ be the torsion-free connection defined by ∇∂1∂1=f1u1∂2 and ∇∂1∂2=f2u1∂2. Assume that f1 and f2 satisfies ∂1b=0 and ∂2b=0, where b=∂1f2+f22. Then, the pseudo-Riemannian metric g∇,ϕ on the cotangent bundle T∗M of neutral signature 2,2 defined by setting such that(24)g∇,ϕ=ϕ11−2u4f1du1⊗du1+ϕ22du2⊗du2+ϕ12−2u4f2du1⊗du2+du1⊗du2+du1⊗du3+du3⊗du1+du2⊗du4+du4⊗du2,is Szabó for any symmetric 0,2-tensor field ϕ.



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