Geometric structures associated to a contact metric (κ,μ)-space

Abstract

We prove that any contact metric (κ, µ)-space (M, ϕ, ξ, η, g) admits a canonical paracontact metric structure that is compatible with the contact form η. We study this canonical paracontact structure, proving that it satisfies a nullity condition and induces on the underlying contact manifold (M, η) a sequence of compatible contact and paracontact metric structures satisfying nullity conditions. We then study the behavior of that sequence, which is related to the Boeckx invariant I M and to the bi-Legendrian structure of (M, ϕ, ξ, η, g). Finally we are able to define a canonical Sasakian structure on any contact metric (κ, µ)-space whose Boeckx invariant satisfies |I M | > 1.

Sasaki–Einstein and paraSasaki–Einstein metrics from (κ,μ)-structures

“…There is a strict relationship between the theory of contact metric (κ, µ)-spaces and of paracontact geometry, as shown in [15] and [17]. In fact, given a non-Sasakian contact metric (κ, µ)-space (M, ϕ, ξ, η, g), one can define canonically two integrable paracontact metric structures on M , ( ϕ 1 , ξ, η, g 1 ) and ( ϕ 2 , ξ, η, g 2 ), which are compatible with the same underlying contact form and Reeb vector field as the (κ, µ)-space M .…”

“…We conclude the subsection by recalling the following formula for the Lie derivative of the operator h in any non-Sasakian (κ, µ)-space (cf. [17,Lemma 4.5])…”

Paracontact Metric κ , μ -Manifold Satisfying the Miao-Tam Equation

“…In this section, we recall some basic definitions and facts on paracontact metric manifolds which we will use later. For more details and some examples, we refer to [15][16][17][18][19][20][21][22][23][24][25][26].…”

“…It follows from ( 6) that RðφX, φYÞξ = 0. Then, replacing X by φX and Y by φY in (17), respectively, we obtain…”

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

“…Every paraSasakian manifold is a K-paracontact manifold, but the converse is not always true, as it is shown in three dimensional case [3]. Paracontact metric manifolds have been studied by Cappelletti-Montano et al [6,7], Martin-Molina [16,17] and many others. According to Cappelletti-Montano et al [6] we have the following definition.…”