On the structure tensors of almost contact B-metric manifolds

Abstract

Abstract. The space of the structure (0,3)-tensors of the covariant derivatives of the structure endomorphism and the metric on almost contact B-metric manifolds is considered. A known decomposition of this space in orthogonal and invariant subspaces with respect to the action of the structure group is used. We determine the corresponding components of the structure tensor and consider the case of the lowest dimension 3 of the studied manifolds. Some examples are commented.

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

“…Proof. By a similar way, using (6), (9), (11) and (12), we establish that the definition condition for a bi-invariant endomorphism φ is satisfied if and only if we have…”

“…Vice versa, let the latter equalities be satisfied. Then, applying (6) and (11) to them, we deduce (14) and (15). Therefore, g is Killing.…”

“…According to (8), (9) and (13), we establish that g is Killing if and only if the equalities (14) C 0 12 = −C 1 02 = C 2 01 , C i ij = 0 are valid. Then, using (11) and (12), we obtain the following equalities (15) − F 011 = F 022 = 2F 110 = 2F 101 = −2F 220 = −2F 202 = C 0 12 and θ = θ * = ω = 0. Therefore, by virtue of (6), we get…”

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

“…In [8] is introduced a real connected Lie group L as a manifold from the class F 9 ⊕ F 10 . We consider now this example for dim L = 3 and give geometrical characteristics in relation with some previous results in this work.…”

“…. , 11) are the components of F in the corresponding basic classes F s and x = x i e i , y = y j e j , z = z k e k are arbitrary vectors we have [8]:…”