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Eight-Dimensional Walker Locally Symmetric Manifolds

Year 2024, Volume: 12 Issue: 1, 1 - 4, 30.04.2024

Silas Longwap Abdoul Salam Diallo

Abstract

A pseudo-Riemannian manifold which admits a field of parallel null $r$-planes, with $r\leq \frac{m}{2}$ is a Walker $m$-manifold. The even-dimensional Walker manifolds $(m=2r)$ with fields of parallel null planes of half dimension have some special interest. The main purpose of the present paper is to study a specifc Walker metric on a $8$-dimensional manifold and to give a theorem for the metric to be locally symmetric.

Keywords

Locally symmetric space Walker manifold. Pseudo-Riemannin

References

  • [1] R. Abounasr, A. Belhaj, J. Rasmussen and E. H. Saidi, Superstring theory on pp waves with ADE geometries, J. Phys., A 39 (2006), 2797-2841.
  • [2] M. Brozos-Vazquez, E. Garcia-Rio, P. Gilkey, S. Nikevic and R. Vazquez-Lorenzo. The Geometry of Walker Manifolds. Synthesis Lectures on Mathematics and Statistics, 5, (2009). (Morgan and Claypool Publishers, Williston, VT).
  • [3] M. Chaichi, E. Garcia-Rio and Y. Matsushita, Curvature properties of four-dimensional Walker metrics, Classical Quantum Gravity 22, (2005), 559-577.
  • [4] J. Davidov, J. C. Diaz-Ramos, E. Garc´ıa-R´ıo, Y. Matsushita, O. Muskarov and R. V´azquez-Lorenzo, Almost K¨ahler Walker 4-manifolds, J. Geom. Phys., 57, (2007), (3), 1075-1088.
  • [5] A. S. Diallo, S. Longwap and F. Massamba, Almost K¨ahler eight-dimensional Walker manifold, Novi Sad J. Math., 48, 2018, (1), 129-141.
  • [6] E. Garc´ıa-R´ıo E, S. Haze, N. Katayama, Y. Matsushita, Symplectic, Hermitian and K¨ahler structures on Walker 4-manifolds, J. Geom., 90 (2008), (1-2) 56-65.
  • [7] M. Iscan, A. Gezer and A. Salimov, Some properties concerning curvature tensors of eight-dimensional Walker manifolds, J. Math. Phys. Anal. Geom., 8, (2012), (1), 21-37.
  • [8] M. Iscan, Some notes concerning Norden-Walker 8-manifolds, Appl. Sci., 16, (2014), 23-31.
  • [9] S. Longwap and A. S. Diallo, Some geometric properties of a family Walker metric on an eight-dimensional manifolds, J. Nigerian Math. Soc. 41 (2022), (3), 223-234.
  • [10] Y. Matsushita, Four-dimensional Walker metrics and symplectic structure, J. Geom. Phys., 52, (2004), (1) 89-99.
  • [11] Y. Matsushita, S. Haze and P. R. Law, Almost Kahler Einstein structures on 8-dimensional Walker manifolds, Monatsh. Math., 150, (2007), 41-48.
  • [12] A. G. Walker, Canonical form for a Riemannian space with a parallel field of null planes, Quart J Math Oxford 1, (1950), (2), 69-79.

Year 2024, Volume: 12 Issue: 1, 1 - 4, 30.04.2024

Silas Longwap Abdoul Salam Diallo

Abstract

References

  • [1] R. Abounasr, A. Belhaj, J. Rasmussen and E. H. Saidi, Superstring theory on pp waves with ADE geometries, J. Phys., A 39 (2006), 2797-2841.
  • [2] M. Brozos-Vazquez, E. Garcia-Rio, P. Gilkey, S. Nikevic and R. Vazquez-Lorenzo. The Geometry of Walker Manifolds. Synthesis Lectures on Mathematics and Statistics, 5, (2009). (Morgan and Claypool Publishers, Williston, VT).
  • [3] M. Chaichi, E. Garcia-Rio and Y. Matsushita, Curvature properties of four-dimensional Walker metrics, Classical Quantum Gravity 22, (2005), 559-577.
  • [4] J. Davidov, J. C. Diaz-Ramos, E. Garc´ıa-R´ıo, Y. Matsushita, O. Muskarov and R. V´azquez-Lorenzo, Almost K¨ahler Walker 4-manifolds, J. Geom. Phys., 57, (2007), (3), 1075-1088.
  • [5] A. S. Diallo, S. Longwap and F. Massamba, Almost K¨ahler eight-dimensional Walker manifold, Novi Sad J. Math., 48, 2018, (1), 129-141.
  • [6] E. Garc´ıa-R´ıo E, S. Haze, N. Katayama, Y. Matsushita, Symplectic, Hermitian and K¨ahler structures on Walker 4-manifolds, J. Geom., 90 (2008), (1-2) 56-65.
  • [7] M. Iscan, A. Gezer and A. Salimov, Some properties concerning curvature tensors of eight-dimensional Walker manifolds, J. Math. Phys. Anal. Geom., 8, (2012), (1), 21-37.
  • [8] M. Iscan, Some notes concerning Norden-Walker 8-manifolds, Appl. Sci., 16, (2014), 23-31.
  • [9] S. Longwap and A. S. Diallo, Some geometric properties of a family Walker metric on an eight-dimensional manifolds, J. Nigerian Math. Soc. 41 (2022), (3), 223-234.
  • [10] Y. Matsushita, Four-dimensional Walker metrics and symplectic structure, J. Geom. Phys., 52, (2004), (1) 89-99.
  • [11] Y. Matsushita, S. Haze and P. R. Law, Almost Kahler Einstein structures on 8-dimensional Walker manifolds, Monatsh. Math., 150, (2007), 41-48.
  • [12] A. G. Walker, Canonical form for a Riemannian space with a parallel field of null planes, Quart J Math Oxford 1, (1950), (2), 69-79.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Silas Longwap

Abdoul Salam Diallo

Early Pub Date April 29, 2024
Publication Date April 30, 2024
Submission Date September 12, 2022
Acceptance Date January 10, 2024
Published in Issue Year 2024 Volume: 12 Issue: 1

Cite

APA Longwap, S., & Diallo, A. S. (2024). Eight-Dimensional Walker Locally Symmetric Manifolds. Konuralp Journal of Mathematics, 12(1), 1-4.
AMA Longwap S, Diallo AS. Eight-Dimensional Walker Locally Symmetric Manifolds. Konuralp J. Math. April 2024;12(1):1-4.
Chicago Longwap, Silas, and Abdoul Salam Diallo. “Eight-Dimensional Walker Locally Symmetric Manifolds”. Konuralp Journal of Mathematics 12, no. 1 (April 2024): 1-4.
EndNote Longwap S, Diallo AS (April 1, 2024) Eight-Dimensional Walker Locally Symmetric Manifolds. Konuralp Journal of Mathematics 12 1 1–4.
IEEE S. Longwap and A. S. Diallo, “Eight-Dimensional Walker Locally Symmetric Manifolds”, Konuralp J. Math., vol. 12, no. 1, pp. 1–4, 2024.
ISNAD Longwap, Silas - Diallo, Abdoul Salam. “Eight-Dimensional Walker Locally Symmetric Manifolds”. Konuralp Journal of Mathematics 12/1 (April 2024), 1-4.
JAMA Longwap S, Diallo AS. Eight-Dimensional Walker Locally Symmetric Manifolds. Konuralp J. Math. 2024;12:1–4.
MLA Longwap, Silas and Abdoul Salam Diallo. “Eight-Dimensional Walker Locally Symmetric Manifolds”. Konuralp Journal of Mathematics, vol. 12, no. 1, 2024, pp. 1-4.
Vancouver Longwap S, Diallo AS. Eight-Dimensional Walker Locally Symmetric Manifolds. Konuralp J. Math. 2024;12(1):1-4.
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