Almost Riemann Solitons with Vertical Potential on Conformal Cosymplectic Contact Complex Riemannian Manifolds

Abstract

Almost-Riemann solitons are introduced and studied on an almost contact complex Riemannian manifold, i.e., an almost-contact B-metric manifold, which is obtained from a cosymplectic manifold of the considered type by means of a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. The potential of the studied soliton is assumed to be in the vertical distribution, i.e., it is collinear to the Reeb vector field. In this way, manifolds from the four main classes of the studied manifolds are obtained. The curvature properties of the resulting manifolds are derived. An explicit example of dimension five is constructed. The Bochner curvature tensor is used (for a dimension of at least seven) as a conformal invariant to obtain these properties and to construct an explicit example in relation to the obtained results.

Ricci–Bourguignon Almost Solitons with Special Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds

“…Therefore, the identity in (15) holds, as do the formulas in (16). Thus, we find that no different results are obtained for β = 0 compared to the case β ̸ = − 1 2n .…”

“…Similarly, since for a Sasaki-like accR manifold ∇x ξ = −φx is true [16], we have (L ϑ g)(x, y) = h(x, y) − 2kg(φx, φy) and considering the last equality of (2), the last expression takes the following form…”

“…The identity in ( 17) is a generalization of the corresponding result in the β = 0 case. In this way, we obtain the formulas in (16). Due to (17) the sum of τ and τ is a constant that depends only on the dimension of the manifold as in (15).…”