Eight-Dimensional Walker Locally Symmetric Manifolds

Yıl 2024, Cilt: 12 Sayı: 1, 1 - 4, 30.04.2024

Silas Longwap Abdoul Salam Diallo

Öz

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.

Anahtar Kelimeler

Locally symmetric space, Walker manifold., Pseudo-Riemannin

Kaynakça

• [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.