Second order parallel tensors and Ricci solitons in $$3$$ 3 -dimensional normal paracontact geometry

Ricci solitons in three-dimensional paracontact geometry

“…Since the basic paper [24], in the last years several authors studied paracontact metric structures and their further generalization, namely, almost paracontact metric structures, emphasizing similarities and differences with respect to the most investigated contact case. Some recent results on paracontact and almost paracontact metric structures may be found in [1], [6], [8], [9], [13], [15], [17], [22], [23] and references therein. Using the relationship between Ricci solitons and infinitesimal harmonic transformations pointed out in [21], it was proved in [11] that a paracontact Ricci soliton, that is, a paracontact metric manifold for which the Reeb vector field ξ satisfies (1.1), is necessarily expanding and must satisfy Qξ = −2nξ, where Q denotes the Ricci operator.…”

Ricci almost solitons and gradient Ricci almost solitons in $(k,\mu)$-paracontact geometry

“…Let g be the Lie algebra of a Lie group G admits a basis {e 1 , e 2 , e 3 , e 4 , e 5 } such that [6] [e 1 We consider the metric such that g(e 1 , e 1 ) = g(e 4 , e 4 ) = g(e 5 , e 5 ) = 1, g(e 2 , e 2 ) = g(e 3 , e 3 ) = −1 and g(e i , e j ) = 0, for i = j.…”

“…Published April 18, 2017 [10], Bejan and Crasmareanu [1] about Ricci solitons and gradient Ricci solitons. Ricci solitons and gradient Ricci solitons on several types of (almost) contact metric manifolds were studied by several authors such as Cho [8,9], De and Matsuyama [11], Deshmukh [12], Hamilton [14], Turan et al [21], Wang [22], Wang et al [23], Yildiz et al [24] and many others.…”

Invariant distributions and holomorphic vector fields in paracontact geometry

“…The second subject of the present paper is the class of paracontact-holomorphic vector fields that form a Lie subalgebra on a normal almost paracontact manifold; recently this type of vector fields was studied as providing the potential vector field of Ricci solitons in (3 -dimensional) almost paracontact geometries in [1]. These vector fields vanish a∂ -operator expressed in terms of Levi-Civita as well as the canonical paracontact connection from [14].…”