Canonical connections on paracontact manifolds

Abstract

Abstract. The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A Dhomothetic transformation is determined as a special gauge transformation. The η-Einstein manifold are defined, it is prove that their scalar curvature is a constant and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with a D-homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.MSC: 53C15, 5350, 53C25, 53C26, 53B30

Mixed 3-Sasakian structures and curvature

“…To this end, we shall need some formulas for the curvature tensor of a manifold with parasasakian structure and of a manifold with indefinite Sasakian structure. Some results recently proved in [23] will also be recovered.…”

“…Moreover, one may find similar definitions in [14] and [23], where the further condition that the restriction ϕ| Im(ϕ) is an almost paracomplex structure on the distribution Im(ϕ) is required. The notion of normality for an almost paracontact structure is defined, as in the classical almost contact case (cf.…”

“…The notion of normality for an almost paracontact structure is defined, as in the classical almost contact case (cf. [3]), through the integrability of the almost product structure F canonically induced on the manifold M × R, defined by [14], [23]). Other properties of almost paracontact manifolds (M, ϕ, ξ, η), which are immediate consequences of the above definition, are ϕ(ξ) = 0, η • ϕ = 0, ker(ϕ) = Span(ξ), ker(η) = Im(ϕ) and T M = Im(ϕ) ⊕ Span(ξ).…”

Para-contact product semi-Riemannian submersions

“…(ii) the tensor …eld ' induces an almost para-complex structure on the distribution D = ker ; that is, the eigendistributions D + ; D corresponding to the eigenvalues 1, -1 of '; respectively, have equal dimension m: M is said to be almost para-contact manifold if it is endowed with an almost paracontact structure( [7], [16], [19], [26]). …”

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

“…An almost paracontact structure is said to be normal [25] if and only if the (1, 2)-type torsion tensor N φ = [φ, φ] − 2dη ⊗ ξ vanishes identically, where [φ, φ] denotes the Nijenhuis tensor of φ. If an almost paracontact manifold M equipped with a pseudo-Riemannian metric g of signature (n + 1, n) such that…”