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2024 | OriginalPaper | Chapter

Weak \(\beta \)-Kenmotsu Manifolds and \(\eta \)-Ricci Solitons

Authors : Dhriti Sundar Patra, Vladimir Rovenski

Published in: Differential Geometric Structures and Applications

Publisher: Springer Nature Switzerland

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Abstract

Weak contact metric structures on a smooth manifold have been recently introduced by V. Rovenski and R. Wolak in 2022. In this paper, we define a new structure of this kind called a weak \(\beta \)-Kenmotsu structure (that generalizes the notion by K. Kenmotsu with \(\beta =1\) and its extension for \(\beta \ne 0\) by Z. Olszak). We show that a weak \(\beta \)-Kenmotsu manifold is locally the warped product \((-\varepsilon ,\varepsilon )\times _\sigma \bar{M}\), where \((\partial _t\,\sigma )/\sigma =\beta \ne 0\), and \((\bar{M},\bar{g})\) is equipped with a parallel skew-symmetric (1,1)-tensor \(\bar{\phi }\) such that \(\bar{\phi }^{\,2}\) is negative definite. Then, we show that an \(\eta \)-Einstein weak \(\beta \)-Kenmotsu manifold with \(\beta =const\ne 0\) admitting an \(\eta \)-Ricci soliton structure is an Einstein manifold. Finally, we prove that a weak \(\beta \)-Kenmotsu manifold and admitting an \(\eta \)-Ricci soliton structure, whose non-zero potential vector field is weak contact or is collinear to \(\xi \), is an Einstein manifold.

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previous chapter Einstein-Type Metrics and Ricci-Type Solitons on Weak f-K-Contact Manifolds

next chapter Twistor Bundles of Foliated Riemannian Manifolds

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Title: Weak -Kenmotsu Manifolds and -Ricci Solitons
Authors: Dhriti Sundar Patra
Vladimir Rovenski
Publisher: Springer Nature Switzerland
Book: Differential Geometric Structures and Applications
Print ISBN: 978-3-031-50585-0

Electronic ISBN: 978-3-031-50586-7

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-50586-7_3

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