Filomat 2022 Volume 36, Issue 19, Pages: 6699-6711
https://doi.org/10.2298/FIL2219699C
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Riemannian concircular structure manifolds
Chaubey Sudhakar Kumar (Section of Mathematics, Department of Information Technology, University of Technology and Applied Sciences-Shinas, Postal Code, Oman), sudhakar.chaubey@shct.edu.om
Suh Young Jin (Department of Mathematics and RIRCM, Kyungpook National University, Daegu, South Korea), yjsuh@knu.ac.kr
In this manuscript, we give the definition of Riemannian concircular
structure manifolds. Some basic properties and integrability condition of
such manifolds are established. It is proved that a Riemannian concircular
structure manifold is semisymmetric if and only if it is concircularly flat.
We also prove that the Riemannian metric of a semisymmetric Riemannian
concircular structure manifold is a generalized soliton. In this sequel, we
show that a conformally flat Riemannian concircular structure manifold is a
quasi-Einstein manifold and its scalar curvature satisfies the partial
differential equation △r = ∂2r/∂t2 + α(n−1)∂r/∂t. To validate the
existence of Riemannian concircular structure manifolds, we present some
non-trivial examples. In this series, we show that a quasi-Einstein manifold
with a divergence free concircular curvature tensor is a Riemannian
concircular structure manifold.
Keywords: Riemannian manifolds, (RCS)n-manifolds, curvature tensors, symmetric spaces, torse-forming vector field, concircular vector field, generalized soliton
Show references
O. Bahadir, S. K. Chaubey, Some notes on LP-Sasakian manifolds with generalized symmetric metric connection, Honam Math. J. 42(3) (2020) 461-476.
J. Berndt, H. Tamaru, Homogeneous codimension one foliations on noncompact symmetric spaces, J. Differential Geom. 63(1) (2003) 1-40.
J. Berndt, H. Tamaru, Cohomogeneity one actions on noncompact symmetric spaces of rank one, Trans. Amer. Math. Soc. 359(7) (2007) 3425-3438.
A. M. Blaga, B.-Y. Chen, Harmonic forms and generalized solitons, arXiv:2107.04223v1 [math.DG], 2021.
D. E. Blair, J.-S. Kim, M. M. Tripathi, On the concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc. 42(5) (2005) 883-892.
S. Capozziello, C. A. Mantica, L. G. Molinari, Cosmological perfect-fluids in f (R) gravity, Int. J. Geom. Methods Mod. Phys. 16(1) (2019) 1950008.
M. C. Chaki, R. K. Maity, On quasi Einstein manifolds, Publ. Math. Debrecen 57(3-4) (2000) 297-306.
S. K. Chaubey, J. W. Lee, S. K. Yadav, Riemannian manifolds with a semi-symmetric metric P-connection, J. Korean Math. Soc. 56(4) (2019) 1113-1129.
S. K. Chaubey, U. C. De, Three-Dimensional Riemannian Manifolds and Ricci solitons, Quaest. Math. 45(5) (2022) 765-778.
S. K. Chaubey, U. C. De, Characterization of three-dimensional Riemannian manifolds with a type of semi-symmetric metric connection admitting Yamabe soliton. J. Geom. Phys. 157 (2020) 103846.
S. K. Chaubey, Y. J. Suh, U. C. De, Characterizations of the Lorentzian manifolds admitting a type of semi-symmetric metric connection, Anal. Math. Phys. 10(4) (2020) Paper No. 61 15 pp.
T. Chave, G. Valent, Quasi-Einstein metrics and their renormalizability properties, Journées Relativistes 96, Part I (Ascona, 1996). Helv. Phys. Acta 69(3) (1996) 344-347.
B.-Y. Chen, Some results on concircular vector fields and their applications to Ricci solitons, Bull. Korean Math. Soc. 52(5) (2015) 1535-1547.
B.-Y. Chen, Classification of torqued vector fields and its applications to Ricci solitons, Kragujevac J. Math. 41(2) (2017) 239-250.
B.-Y. Chen, K. Yano, On submanifolds of submanifolds of a Riemannian manifold, J. Math. Soc. Japan 23(3) (1971) 548-554.
A. Fialkow, Conformal geodesics, Trans. Amer. Math. Soc. 45(3) (1939) 443-473.
E. T. Kobayashi, A remark on the Nijenhuis tensor, Pacific J. Math. 12 (1962) 963-977.
N. Koiso, On Rotationally symmetric Hamilton’s equations for Kähler-Einstein metrics, Max-Planck-Institute preprint series (1987) 16-87.
J. H. Kwon, Y. S. Pyo, Y. J. Suh, On semi-Riemannian manifolds satisfying the second Bianchi identity, J. Korean Math. Soc. 40(1) (2003) 129-167.
C. A. Mantica, L. G. Molinari, Y. J. Suh, S. Shenawy, Perfect-fluid, generalized Robertson-Walker space-times, and Gray’s decomposition, J. Math. Phys. 60(5) (2019) 052506.
C. A. Mantica, L. G. Molinari, Generalized Robertson-Walker spacetimes-a survey, Int. J. Geom. Methods Mod. Phys. 14(3) (2017) 1730001.
A. Mihai, I. Mihai, Torse forming vector fields and exterior concurrent vector fields on Riemannian manifolds and applications, J. Geom. Phys. 73 (2013) 200-208.
J. Mikeš, L. Rachůnek, Torse-forming vector fields in T-semisymmetric Riemannian spaces, Steps in differential geometry (Debrecen, 2000), 219-229, Inst. Math. Inform., Debrecen, 2001.
M. S. Najdanović, M. Lj. Zlatanović, I. Hinterleinter, Conformal and geodesic mappings of generalized equidistant spaces, PUBLICATIONS DE L’INSTITUT MATH´EMATIQUE Nouvelle s´erie, tome 98(112) (2015) 71-84.
A. Nijenhuis, Xn−1-forming sets of eigenvectors. Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math. 13, (1951), 200-212.
J. D. Pérez, On Ricci tensor of real hypersurfaces of quaternionic projective spaces, Internat. J. Math. & Math. Sci. 19 (1996), 193-198.
J. A. Schouten, Ricci-Calculus: An introduction to tensor analysis and its geometrical applications, 2nd. ed., Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1954.
Y. J. Suh, J.-H. Kwon, H. Y. Yang, Conformally symmetric semi-Riemannian manifolds, J. Geom. Phys. 56(5) (2006) 875-901.
Z. I. Szabó, Structure theorems on Riemannian spaces satisfying R(X, Y)・R = 0. I. The local version, J. Differential Geometry 17(4) (1982) 531-582.
Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965) 251-275.
H. Weyl, Reine Infinitesimalgeometrie (German), Math. Z. 2(3-4) (1918) 384-411.
K. Yano, Recent topics in differential geometry, Bull. Korean Math. Soc. 13(2) 1976 113-120.
K. Yano, On semi-symmetric metric connections, Rev. Roumaine Math Pures Appl. 15 1970 1579-1586.
K. Yano, Concircular geometry I. Concircular transformations, Proc. Imp. Acad. Tokyo 16 (1940) 195-200.
K. Yano, On torse forming direction in a Riemannian space, Proc. Imp. Acad. Tokyo 20 (1944) 340-345.
M. Zlatanović, I. Hinterleitner, M. Najdanović, On Equitorsion Concircular Tensors of Generalized Riemannian Spaces, Filomat 28(3) (2014) 463-471.