Lie groups as 3-dimensional almost contact B-metric manifolds

Abstract

Abstract. The object of investigations are almost contact B-metric structures on 3-dimensional Lie groups considered as smooth manifolds. There are established the existence and some geometric characteristics of these manifolds in all basic classes. An example is given as a support of obtained results.

Lie groups as 3-dimensional almost paracontact almost paracomplex Riemannian manifolds

“…For the manifolds studied (M, φ, ξ, η, g), besides the metric g, we also have its associated metricg and their component η ⊗ η according to (3). Then, it is reasonable to consider the following more general case of the Einstein property, similarly to [13] for almost contact B-metric manifolds. An almost paracontact almost paracomplex Riemannian manifold is called η-paracomplex-Einstein when the following condition is valid…”

“…The same Lie group is equipped with an almost contact B-metric structure in [11] and then certain geometric characteristics for the obtained manifold are found. Further, in [13], the case of the lowest dimension is considered and some properties of the constructed manifold are determined.…”

“…A number of authors have studied Lie groups as manifolds equipped with various tensor structures and metrics that are compatible with the structures (including in the lowest-dimensional cases) -for example, [3] and [4] for almost contact metric manifolds, [13] and [10] for almost contact B-metric manifolds, [1] and [6] for almost complex manifolds with Hermitian metric, [8] and [24] for almost complex manifolds with Norden metric, [2] and [5] for hypercomplex hyper-Hermitian manifolds, [7] and [12] for almost hypercomplex Hermitian-Norden manifolds, [9] and [22] for Riemannian almost product manifolds, [16] and [25] for almost paracontact metric manifolds.…”

Geometric characteristics and properties of a two-parametric family of Lie groups with almost contact B-metric structure of the smallest dimension

“…Here, an object of special interest are the Lie groups considered as threedimensional almost contact B-metric manifolds. For example of such investigation see [19].…”

The decomposition of almost paracontact metric manifolds in eleven classes revisited

“…Moreover, we find explicit matrix representations of these Lie groups. Let us note that Lie groups as 3-dimensional almost contact B-metric manifolds were studied in [9] and their matrix representations were obtained in [8].…”