Conformal Vector Fields and Ricci Soliton Structures on Natural Riemann Extensions

• Published in: Mediterranean journal of mathematics Vol. 18; no. 2
• Main Authors: Abbassi, Mohamed Tahar Kadaoui, Amri, Noura, Bejan, Cornelia-Livia
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2021; Springer Nature B.V
• Subjects: Fields (mathematics); Mathematics; Mathematics and Statistics; Solitary waves; Theoretical physics; Ricci soliton; 53B20; 53C24; Cotangent bundle; Killing vector field; conformal vector field; 53B05; 53C25; natural Riemann extension; 53C07
• ISSN: 1660-5446; 1660-5454
• DOI: 10.1007/s00009-020-01690-5

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.