Abstract
The framework of the paper is the phase universe, described by the total space of the cotangent bundle of a manifold M, which is of interest for both mathematics and theoretical physics. When M carries a symmetric linear connection, then \(T^*M\) is endowed with a semi-Riemannian metric, namely the classical Riemann extension, introduced by Patterson and Walker and then by Willmore. We consider here a generalization provided by Sekizawa and Kowalski of this metric, called the natural Riemann extension, which is also a metric of signature (n, n). We give the complete classification of conformal and Killing vector fields with respect to an arbitrary natural Riemann extension. Ricci soliton is a topic that has been increasingly studied lately. Necessary and sufficient conditions for the phase space to become a Ricci soliton (or Einstein) are given at the end.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bejan, C.L., Kowalski, O.: On some differential operators on natural Riemann extensions. Ann. Glob. Anal. Geom. 48, 171–180 (2015)
Bejan, C.L., Eken, S.: Conformality on semi-Riemannian manifolds. Mediterr. J. Math. 13(4), 2185–2198 (2016)
Caratheodory, C.: Conformal Representation. Cambridge University Press, Cambridge (1932)
Deshmukh, S.: Conformal vector fields and eigenvectors of Laplacian operator, mathematical Physics. Anal. Geometry 15(2), 163–172 (2012)
Dunajski, M., Mettler, T.: Gauge theory on projective surfaces and anti-self-dual Einstein metrics in dimension four. J. Geom. Anal. 28(3), 2780–2811 (2018)
Dunajski, M., Tod, P.: Four dimensional metrics conformal to Kähler. Math. Proc. Camb. Philos. Soc. 148, 485–504 (2010)
Faleitelli, M., Ianus, S., Pastore, A.M.: Riemannian Submersions and Related Topics. World Scientific, Singapore (2004)
Kowalski, O., Sekizawa, M.: On natural Riemann extensions. Publ. Math. Debrecen 78(3-4), 709–721 (2011)
Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, New York (1993)
Konno, T.: Killing vector field on tangent sphere bundles. Kodai Math. J. 21, 61–72 (1998)
Manev, M., Ivanova, M.: Canonical-type connection on almost contact manifolds with B-metric. Ann. Glob. Anal. Geom. 43(4), 397–408 (2013)
Patterson, E.M., Walker, A.G.: Riemannian extensions. Q. J. Math. Oxford Ser. 2(3), 19–28 (1952)
Sahin, B.: Conformal Riemannian maps between Riemannian manifolds, their harmonicity and decomposition theorems. Acta Appl. Math. 109, 829–847 (2010)
Sekizawa, M.: Natural transformations of affine connections on manifolds to metrics on cotangent bundles. In: Proceedings of 14th Winter School on Abstract Analysis (Srni, 1986), Rend. Circ. Mat. Palermo, Vol 14, pp. 129–142 (1987)
Willmore, T.J.: An Introduction to Differential Geometry. Clarendon Press, Oxford (1959)
Yano, K., Patterson, E.M.: Vertical and complete lifts from a manifold to its cotangent bundle. J. Math. Soc. Jpn. 19, 91–113 (1967)
Yano, K., Ishihara, S.: Tangent and Cotangent Bundles. Marcel Dekker, New York (1973)
Acknowledgements
The authors would like to express their sincere gratitude to Professor M. Sekizawa for helpful comments and suggestions, and to the unknown referee for his/her careful reading of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Professor Vasile Oproiu (1941–2020).
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Abbassi, M.T.K., Amri, N. & Bejan, CL. Conformal Vector Fields and Ricci Soliton Structures on Natural Riemann Extensions. Mediterr. J. Math. 18, 55 (2021). https://doi.org/10.1007/s00009-020-01690-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-020-01690-5