Search Results (11)

The subject of this study is almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds. The considerations are restricted to a special class of these manifolds, namely those of the Sasaki-like type, because of their geometric construction and the explicit expression of their classification tensor by the pair of B-metrics. Here, each of the two B-metrics is considered as an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form. The soliton potential is chosen to be a conformal Killing vector field (in particular, concircular or concurrent) and then a generalization of the notion of conformality using contact conformal transformations of B-metrics. The resulting manifolds, equipped with the introduced almost solitons, are geometrically characterized. In the five-dimensional case, an explicit example on a Lie group depending on two real parameters is constructed, and the properties obtained in the theoretical part are confirmed. Full article

The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as an anti-isometry for these metrics, called B-metrics, if its action is restricted to the contact distribution of the manifold. In this paper, some curvature properties of a special class of these manifolds, called Sasaki-like, are studied. Such a manifold is defined by the condition that its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). Each of the two B-metrics on the considered manifold is specialized here as an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form, i.e., has an additional curvature property such that the metric is a self-similar solution of a special intrinsic geometric flow. Almost solitons are generalizations of solitons because their defining condition uses functions rather than constants as coefficients. The introduced (almost) solitons are a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein). The soliton potential is chosen to be collinear with the Reeb vector field and is therefore called vertical. The special case of the soliton potential being solenoidal (i.e., divergence-free) with respect to each of the B-metrics is also considered. The resulting manifolds equipped with the pair of associated $\eta$-Ricci–Bourguignon almost solitons are characterized geometrically. An example of arbitrary dimension is constructed and the properties obtained in the theoretical part are confirmed. Full article

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are equipped with a pair of pseudo-Riemannian metrics that are mutually associated with each other using the tensor structure. Here, we consider a special class of these manifolds, namely those of the Sasaki-like type. They have an interesting geometric interpretation: the complex cone of such a manifold is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). The basic metric on the considered manifold is specialized here as a soliton, i.e., has an additional curvature property such that the metric is a self-similar solution to an intrinsic geometric flow. Almost solitons are more general objects than solitons because they use functions rather than constants as coefficients in the defining condition. A $\beta$-Ricci–Bourguignon-like almost soliton ($\beta$ is a real constant) is defined using the pair of metrics. The introduced soliton is a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein) which, in principle, arise from a single metric rather than a pair of metrics. The soliton potential is chosen to be pointwise collinear to the Reeb vector field, or the Lie derivative of any B-metric along the potential to be the same metric multiplied by a function. The resulting manifolds equipped with the introduced almost solitons are characterized geometrically. Suitable examples for both types of almost solitons are constructed, and the properties obtained in the theoretical part are confirmed. Full article

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are, in principle, equipped with a pair of mutually associated pseudo-Riemannian metrics. Each of these metrics is specialized as a Yamabe almost soliton with a potential collinear to the Reeb vector field. The resulting manifolds are then investigated in two important cases with geometric significance. The first is when the manifold is of Sasaki-like type, i.e., its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). The second case is when the soliton potential is torse-forming, i.e., it satisfies a certain recurrence condition for its covariant derivative with respect to the Levi–Civita connection of the corresponding metric. The studied solitons are characterized. In the three-dimensional case, an explicit example is constructed, and the properties obtained in the theoretical part are confirmed. Full article

In this paper, we establish some inequalities between the normalized $\delta$-Casorati curvatures and the scalar curvature (i.e., between extrinsic and intrinsic invariants) of spacelike statistical submanifolds in Sasaki-like statistical manifolds, endowed with a semi-symmetric metric connection. Moreover, we study the submanifolds satisfying the equality cases of these inequalities. We also present an appropriate example. Full article

A Yamabe soliton is defined on an arbitrary almost-contact B-metric manifold, which is obtained by a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. The cases when the given manifold is cosymplectic or Sasaki-like are studied. In this manner, manifolds are obtained that belong to one of the main classes of the studied manifolds. The same class contains the conformally equivalent manifolds of cosymplectic manifolds by the usual conformal transformation of the B-metric on contact distribution. In both cases, explicit five-dimensional examples are given, which are characterized in relation to the results obtained. Full article

Para-Ricci-like solitons with arbitrary potential on para-Sasaki-like Riemannian $\Pi$-manifolds are introduced and studied. For the studied soliton, it is proved that its Ricci tensor is a constant multiple of the vertical component of both metrics. Thus, the corresponding scalar curvatures of both considered metrics are equal and constant. An explicit example of the Lie group as the manifold under study is presented. Full article

The objects of study are para-Ricci-like solitons on para-Sasaki-like, almost paracontact, almost paracomplex Riemannian manifolds, namely, Riemannian Π-manifolds. Different cases when the potential of the soliton is the Reeb vector field or pointwise collinear to it are considered. Some additional geometric properties of the constructed objects are proven. Results for a parallel symmetric second-order covariant tensor on the considered manifolds are obtained. An explicit example of dimension 5 in support of the given assertions is provided. Full article

We introduce and study a new type of soliton with a potential Reeb vector field on Riemannian manifolds with an almost paracontact structure corresponding to an almost paracomplex structure. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field were considered. It was proved a necessary and sufficient condition for the manifold to admit a para-Ricci-like soliton, which is the structure that is para-Einstein-like. Explicit examples are provided in support of the proven statements. Full article

In this paper, we obtain the upper bounds for the normalized $\delta$ -Casorati curvatures and generalized normalized $\delta$ -Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the necessary and sufficient condition for a Sasaki-like statistical manifold to be $\eta$ -Einstein. Finally, we provide the condition under which the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations. Full article