𝜁 (Ric)-vector fields on doubly warped product manifolds

Yıl 2023, Cilt: 1 Sayı: 2, 67 - 73, 17.12.2023

Sibel Gerdan Aydın Moctar Traore Hakan Mete Taştan

https://doi.org/10.26650/ijmath.2023.00008

Öz

We investigate 𝜁 (Ric)-vector fields on doubly warped product manifolds. We obtain some results when the vector field is also 𝜁 (Ric) on factor manifolds.We prove that if a vector field is a 𝜁 (Ric)-vector field on a doubly warped product manifold, it is also a 𝜁 (Ric)-vector field on the factor manifolds under certain conditions. Also, we show that a vector field on a doubly warped product manifold can be a 𝜁 (Ric)-vector field with some conditions. Moreover we give two important applications of this concept in the Lorentzian settings, which are the doubly warped product generalized Robertson-Walker space-time and doubly warped product standard static space-time.

Anahtar Kelimeler

𝜁(Ric)-vector field, warped product manifold, standard static space-times, generalized Robertson-Walker space-times

Kaynakça

• Allison, D.E., 1988, Geodesic completeness in static space-times, Geom. Dedic., 26, 85-97. google scholar
• Allison, D.E., 1998, Energy conditions in standard static space-times, Gen. Relativ. Gravit., 20(2), 115-122. google scholar
• Allison, D.E., Ünal, B., 2003, Geodesic structure of standard static space-times, J. Geom. Phys. 46(2), 193–200. google scholar
• Besse, A.L., 2007, Einstein Manifolds, Classics in Mathematics, Springer: Berlin, Germany. google scholar
• Bishop, R. L., O’Neill, B., 1969, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145(1), 1-49. google scholar
• Blaga, A.M., Taştan, H.M., 2022, Gradient solitons on doubly warped product manifolds, Rep. Math. Phys., 89(3), 319-333. google scholar
• Chen, B.Y., 2017, Differential geometry of warped product manifolds and submanifolds, World Scientific. google scholar
• Chen, B.Y., 2015, Some results on concircular vector fields and their applications to Ricci solitons, Bull. Korean Math. Soc., 52(5), 1535–1547. google scholar
• De, U.C., Shenaway, S., Ünal, B., 2021, 𝜑(Ric)-vector fields on warped product manifolds and applications, Afr. Mat, 32, 1709-1716. google scholar
• Ehrlich, P.E., 1974, Metric deformations of Ricci and sectional curvature on compact Riemannian manifolds, P.h.D. Dissertation, SUNNY Stony Brook, New York. google scholar
• El-Sayied, H. K., Mantica, C. A., Shenawy, S. Syied, N., 2020, Gray’s Decomposition on Doubly Warped Product Manifolds and Applications, Filomat, 34(11), 3767-3776. google scholar
• Flores, J.L., Sánchez, M., 2000, Geodesic connectedness and conjugate points in GRW spacetimes, J. Geom. Phys., 36(3-4), 285-314. google scholar
• Gutierrez, M., Olea, B., 2012, Semi-Riemannian manifolds with a doubly warped structure, Rev. Mat. Iberoam., 28(1), 1-24. google scholar
• Hinterleitner, I., Kiosak, V.A, 2008, 𝜑(Ric)-vector fields in Riemannian spaces, Arch. Math., 44(5), 385–390. google scholar
• Hinterleitner, I., Kiosak, V.A, 2009, 𝜑(Ric)-vector fields on conformally flat spaces, AIP. Conf. Prof., 1191(1), 98-103. google scholar
• Kırık, B., Özen Zengin, F., 2015, Conformal mappings of quasi-Einstein manifolds admitting special vector fields, Filomat, 29(3), 525–534. google scholar
• Kırık, B., Özen Zengin, F., 2015, Generalized quasi-Einstein manifolds admitting special vector fields, Acta Math. Acad. Paedagog. Nyregyhziensis, 31(1), 61–69. google scholar
• Kırık, B., Özen Zengin, F., 2019, Applications of a special generalized quasi-Einstein manifold, Bull.Iran.Math. Soc., 45, 89–102. google scholar
• Kobayashi, S., 1995, Transformations groups in differential geometry, Classic in Mathematics, Springer: Berlin, Germany. google scholar
• O’Neill, B., 1983, Semi-Riemannian Geometry with Applications to Relativity, Academic Press Limited: London, England. google scholar
• Sánchez, M., 1998, On the geometry of generalized Robertson–Walker spacetimes: geodesics, Gen. Relativ. Gravit., 30(6), 915-93. google scholar
• Sánchez, M., 1999, On the geometry of generalized Robertson–Walker space times: curvature and killing fields, J. Geom. Phys., 31(1), 1-15. google scholar
• Özen Zengin, F., Kırık, B., Conformal mappings of nearly quasi-Einstein manifolds, Miskolc Math. Notes, 14(2), 629–636. google scholar