Four-Dimensional Semi-Riemannian Szabó Manifolds

Diallo, Abdoul Salam; Gupta, Punam

doi:https://doi.org/10.1155/2020/6663361

Journal of Mathematics

On this page

Abstract Introduction Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 6663361 | https://doi.org/10.1155/2020/6663361

Four-Dimensional Semi-Riemannian Szabó Manifolds

Abdoul Salam Diallo¹and Punam Gupta²

Academic Editor: Ljubisa Kocinac

Received20 Oct 2020

Accepted14 Dec 2020

Published31 Dec 2020

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.

1. Introduction

Let be the cotangent bundle of -dimensional manifold 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 -dimensional cotangent bundle of any -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 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 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 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 be -differentiable.

2. Deformed Riemannian Extensions

Let be the cotangent bundle of -dimensional affine manifold with torsion-free affine connection and let be the natural projection defined by

A system of local coordinates around induces a system of local coordinates , around , where are components of covectors in each cotangent space , , with respect to the natural coframe . Let and , then, at each point ,is a basis for the cotangent space . For more details on the geometry of cotangent bundle, see [13].

The Riemannian extension is the pseudo-Riemannian metric on of neutral signature characterized by the identity [5].where is a complete lift of the vector field on , and the function is defined by

For more details, see [5]. In the locally induced coordinates on , the Riemannian extension [1] is expressed bywith respect to the basis , where are the coefficients of the torsion-free affine connection with respect to on .

Riemannian extensions provide a link between affine and pseudo-Riemannian geometries. Therefore, by using the properties of the Riemannian extension , we investigate the properties of the affine connection . Similarly, is locally symmetric if and only if is locally symmetric. In the same way, is projectively flat if and only if is locally conformally flat [14].

Let be a symmetric -tensor field on an affine manifold . In [14], the authors introduced a deformation of the Riemannian extension by means of a symmetric -tensor field on . They considered the cotangent bundle equipped with the metric , which is called the deformed Riemannian extension.

The deformed Riemannian extension denoted which is the metric of neutral signature on the cotangent bundle given by

In local coordinates, the deformed Riemannian extension is given bywith respect to the basis , where are the coefficients of the torsion-free affine connection and and are the local components of the symmetric -tensor field . Equivalently,

Note that the crucial terms 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 :

In the deformed Riemannian extension, the tensor 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

Let be an affine manifold and . The affine Szabó operator [15] with respect to is a function from to , , defined byfor any vector field and where is the curvature operator of the affine connection . The affine Szabó operator satisfies and , for . If , for , and , we havewhere .

Let be an affine manifold and . is said to affine Szabó at if the affine Szabó operator has the same characteristic polynomial for every vector field on . If is affine Szabó at each , then is known as affine Szabó. For more details, see [16].

Now, we give a known result for later use.

Theorem 1. (see [17]). Let be an -dimensional affine manifold and . Then, is affine Szabó at if and only if the characteristic polynomial of the affine Szabó operator is , for every .

We have a complete description of affine Szabó surfaces.

Theorem 2. (see [17]). Let be an affine surface. Then, is affine Szabó at if and only if the Ricci tensor of is cyclic parallel at .

Next, we investigate some particular case. The curvature of an affine surface is encoded by its Ricci tensor. We fix coordinates on and let , for , where . Then, a straightforward calculation shows that the components of the curvature tensor are given bywhere are the components of the Ricci tensor given

Let be a vector field on . It is easy to check that the affine Szabó operator expresses, with respect to the basis , aswhere the coefficients , , , and are given by

Its characteristic polynomial is given by

Here, we investigate affine surfaces whose Ricci tensor is skew-symmetric.

Theorem 3. Let be an torsion-free affine connection on a surface . Then, the Ricci tensor of is skew-symmetric and nonzero everywhere if and only if is affine Szabó.

Proof. If the Ricci tensor of is skew-symmetric, that is, and . Then, the Szabó operator is nilpotent.
Conversely, if is affine Szabó, then the trace and determinant of (14) will be zero, which is possible only if and .

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 surface has a neighborhood with coordinates in which the component functions of a torsion-free affine connection are , , for some function , , unless , then is affine Szabó.

Proof. It easy to show that the Ricci tensor of is skew-symmetric.

A Lagrangian in a manifold is a function on a nonempty open set . A Lagrangian gives rise to equations of motion, which are the Euler–Lagrange equations, imposed on curves and the velocity lies entirely in . A fractional-linear function in a two-dimensional real vector space is a rational function of the form , defined on a nonempty open subset of , where 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 . If every point in has a neighborhood with a fractional-linear Lagrangian such that the solutions of the Euler–Lagrange equations for coincide with those geodesics of which, lifted to , lie in , then is affine Szabó.

Definition 1 (see [19]). A tensor field is said to be recurrent if there exists a 1-form such that = , where is an affine connection. In particular, an affine surface is said to be recurrent if its Ricci tensor is recurrent.

Theorem 6. Let be an affine Szabó surface. Then, is recurrent if and only if, around each point, there exists a coordinate system with the nonzero components of which arefor some scalar function such that . Moreover, is not locally symmetric.

Proof. Consider the Ricci tensor , where is the antisymmetric part of and is the symmetric part of . Then, by using Theorem 3, we can say that 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 with the nonzero components of which arefor some scalar function such that . Now, it is easy to calculate that , which is never zero. So, is not locally symmetric.

By using the result of ([19], Theorem 2.2) and Theorem 3, we can say as follows.

Theorem 7. Let 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 manifold is said to be Szabó if the Szabó operators have constant eigenvalues on the unit pseudosphere bundles . 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. Let be a 2-dimensional smooth torsion-free affine manifold. Then, the following assertions are equivalent:(1) is an affine Szabó manifold(2)The deformed Riemannian extension of is a pseudo-Riemannian nilpotent Szabó manifold of neutral signature

Proof. Let be the coefficients of the torsion-free affine connection and denote the local components of . Then, the deformed Riemannian extension of the torsion-free affine connection is the pseudo-Riemannian metric tensor on of signature given byA straightforward calculation shows that the nonzero Christoffel symbols of the Levi–Civita connection are given as follows:where and . The nonzero components of the curvature tensor of up to the usual symmetries are given as follows (we omit , as it plays no role in our considerations):where are the components of the curvature tensor of . For more details, see [14].
Let be a vector field on . Then, the matrix of the Szabó operator with respect to the basis is of the form:where is the matrix of the affine Szabó operator on relative to the basis . Note that the characteristic polynomial of and of are related byNow, if the affine manifold is assumed to be affine Szabó, then has zero eigenvalues for each vector field on . Therefore, it follows from (23) that the eigenvalues of vanish for every vector field on . Thus, is pseudo-Riemannian Szabó manifold.
Conversely, assume that is a pseudo-Riemannian Szabó manifold. If is an arbitrary vector field on , then is an unit vector field at every point of the zero section on . Then, from (23), we see that the characteristic polynomial of is the square of the characteristic polynomial of . Since for every unit vector field on , the characteristic polynomial should be the same, and it follows that, for every vector field on , the characteristic polynomial is the same. Hence, is affine Szabó.

For an example, we have the following.

Theorem 9. (see [6]). Let and be the torsion-free connection defined by and . Assume that and satisfies and , where . Then, the pseudo-Riemannian metric on the cotangent bundle of neutral signature defined by setting such thatis Szabó for any symmetric -tensor field .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

E. M. Patterson and A. G. Walker, “Riemann extensions,” The Quarterly Journal of Mathematics, vol. 3, no. 1, pp. 19–28, 1952.
View at: Publisher Site | Google Scholar
Z. Afifi, “Riemann extensions of affine connected spaces,” The Quarterly Journal of Mathematics, vol. 5, no. 1, pp. 312–320, 1954.
View at: Publisher Site | Google Scholar
E. García-Río, D. N. Kupeli, M. E. Vázquez-Abal, and R. Vázquez-Lorenzo, “Affine Osserman connections and their Riemann extensions,” Differential Geometry and Its Applications, vol. 11, no. 2, pp. 145–153, 1999.
View at: Publisher Site | Google Scholar
A. S. Diallo, “The Riemann extension of an affine Osserman connection on 3-dimensional manifolds,” Global Journal of Advanced Research on Clasical and Modern Geometry, vol. 2, no. 2, pp. 69–75, 2013.
View at: Google Scholar
E. García-Río, P. Gilkey, S. Nikcević, and R. Vázquez-Lorenzo, “Applications of affine and Weyl geometry,” in Synthesis Lectures on Mathematics and Statistics, Morgan & Claypool Publishers, Williston, VT, USA, 2013.
View at: Google Scholar
A. S. Diallo and P. Gupta, “On a deformed Riemannian extension of affine Szabó connections,” Balkan Journal of Geometry and its Applications, vol. 25, no. 2, pp. 30–38, 2020.
View at: Google Scholar
O. Kowalski and M. Sekizawa, “On natural Riemann extensions,” Publicationes Mathematicae Debrecen, vol. 78, no. 3-4, pp. 709–721, 2011.
View at: Publisher Site | Google Scholar
O. Kowalski and M. Sekizawa, “The Riemann extensions with cyclic parallel Ricci tensor,” Mathematische Nachrichten, vol. 287, no. 8-9, pp. 955–961, 2014.
View at: Publisher Site | Google Scholar
L. Vanhecke and T. J. Willmore, “Riemann extensions of D’Atri spaces,” Tensor (N.S.), vol. 38, pp. 154–158, 1982.
View at: Google Scholar
A. Bonome, R. Castro, E. García-Río, L. Hervella, and R. Vázquez-Lorenzo, “Nonsymmetric Osserman indefinite Kähler manifolds,” Proceedings of the American Mathematical Society, vol. 126, no. 9, pp. 2763–2769, 1998.
View at: Publisher Site | Google Scholar
M. Brozos-Vázquez, E. García-Río, P. Gilkey, S. Nikcević, and R. Vézquez-Lorenzo, The Geometry of Walker Manifolds, Synthesis Lectures on Mathematics and Statistics 5, Morgan & Claypool Publ., Williston, VT, USA, 2009.
A. S. Diallo, M. Hassirou, and I. Katambe, “The twisted Riemannian extension of an affine Osserman connection on 2-dimensional manifolds,” Mathematical Sciences Research Journal, vol. 17, no. 8, pp. 201–207, 2013.
View at: Google Scholar
K. Yano and S. Ishihara, “Tangent and cotangent bundles: differential geometry,” in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1973.
View at: Google Scholar
E. Calviño-Louzao, E. García-Río, and R. Vázquez-Lorenzo, “Riemann extensions of torsion-free connections with degenerate Ricci tensor,” Canadian Journal of Mathematics, vol. 62, no. 5, pp. 1037–1057, 2010.
View at: Publisher Site | Google Scholar
Z. I. Szabó, “A short topological proof for the symmetry of 2 point homogeneous spaces,” Inventiones Mathematicae, vol. 106, pp. 61–64, 1991.
View at: Google Scholar
A. S. Diallo, S. Longwap, and F. Massamba, “On three dimensional affine Szabó manifolds,” Balkan Journal of Geometry and Its Applications, vol. 22, no. 2, pp. 21–36, 2017.
View at: Google Scholar
A. S. Diallo and F. Massamba, “Affine Szab ó connections on smooth manifolds,” Revista de la Unión Matemática Argentina, vol. 58, no. 1, pp. 37–52, 2017.
View at: Google Scholar
A. Derdzinski, “Connections with skew-symmetric Ricci tensor on surfaces,” Results in Mathematics, vol. 52, no. 3-4, pp. 223–245, 2008.
View at: Publisher Site | Google Scholar
Y.-C. Wong, “Two dimensional linear connexions with zero torsion and recurrent curvature,” Monatshefte für Mathematik, vol. 68, no. 2, pp. 175–184, 1964.
View at: Publisher Site | Google Scholar
P. Gilkey and I. Stavrov, “Curvature tensors whose Jacobi or Szabó operator is nilpotent on null vectors,” Bulletin of the London Mathematical Society, vol. 34, no. 06, pp. 650–658, 2002.
View at: Publisher Site | Google Scholar
P. B. Gilkey, R. Ivanova, and T. Zhang, “Szabó osserman IP pseudo-riemannian manifolds,” Publicationes Mathematicae Debrecen, vol. 62, no. 3-4, pp. 387–401, 2003.
View at: Google Scholar

Copyright

Copyright © 2020 Abdoul Salam Diallo and Punam Gupta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

402

Downloads

670

Citations