Almost-para-Ricci-like solitons with torse-forming vertical potential of constant length on para-Sasaki-like Riemannian Π-manifolds

In 1976, Sato introduced the concept of an almost paracontact structure. On such a manifold, two types of compatible metrics can be considered—the induced transformations, which are isometries, or the antiisometries on the paracontact distribution of the tangent space. The main difference between them lies in the type of associated tensor (0,2) of the metric on the structure. In compatible manifolds, it is a metric, while in metric manifolds, it is a 2-form.

Furthermore, different geometers contributed to the development of the study of the first case manifolds—Riemannian almost paracontact manifolds. In 1980, Sasaki introduced the notion of a Riemannian almost paracontact manifold of type (p,q). By p and q he meant the multiplicities of the eigenvalues ​​+1 and -1 of the structural endomorphism φ.In addition to these eigenvalues, the tensor field φ also has a simple eigenvalue 0.

In the other possible case, i.e., the case of almost paracontact metric manifolds, the metric is pseudo-Riemannian with a signature (n+1, n). These geometric objects have been well studied by a number of authors.

An almost paracontact structure (φ, ξ, η) that possesses a traceless endomorphism φ plays the role of an almost paracomplex structure on the paracontact distribution of a (2n+1)-dimensional smooth manifold and is called a Π-structure. Such a manifold, equipped with a Π-structure, is called a Π-manifold, and when it is also equipped with a Riemannian metric g, it is called a Riemannian Π-manifold.

In this work, a generalization of para-Ricci-like solitons with torse-forming potential on para-Sasaki-like Riemannian Π-manifolds, which is constant multiple of the Reeb vector field, is investigated. Necessary and sufficient conditions are established for such solitons to be equivalent to almost-para-Einstein-like metrics. Further results are derived concerning parallel symmetric covariant tensors of second order. An explicit example in an arbitrary dimension is constructed to illustrate the obtained results.