Current Developments in Differential Geometry and Its Related Fields by Adachi Toshiaki Hashimoto Hideya Hristov Milen J

Author:Adachi, Toshiaki,Hashimoto, Hideya,Hristov, Milen J.
Language: eng
Format: epub
ISBN: 9789814719797
Publisher: World Scientific Publishing Company
Published: 2016-07-24T16:00:00+00:00

2.Almost contact B-metric manifolds

We shall start by recalling some properties and notations concerned with almost contact manifolds. Let (M, φ, ξ, η, g) be a (2n + 1)-dimensional almost contact B-metric manifold. Here, (φ, ξ, η) is a triplet of a tensor field φ of type (1,1), a vector field ξ and its dual 1-form η satisfying

φξ = 0, φ2 = −Id + η ⊗ ξ, η ∘ φ = 0, η(ξ) = 1,

where Id is the identity. Moreover, g is a pseudo-Riemannian metric, called a B-metric, satisfying

g(φx, φy) = −g(x, y) + η(x)η(y)

for arbitrary tangent vectors x, y ∈ TpM at an arbitrary point p ∈ M ([4]). The triplet (φ, ξ, η) is called an almost contact structure on M. We note that the restriction of a B-metric on the contact distribution H = ker(η) coincides with the corresponding Norden metric with respect to the almost complex structure, the restriction of φ on H, acting as an anti-isometry on the metric on H, the restriction of g to H. Thus, we obtain a correlation between a (2n + 1)-dimensional almost contact B-metric manifold and a 2n-dimensional almost complex manifold with Norden metric (or an n-dimensional complex Riemannian manifold).

The associated metric of g on M is given by (x, y) = g(x, φy) + η(x)η(y). It is also a B-metric, hence (M, φ, ξ, η, ) is also an almost contact B-metric manifold. Both metrics g and are indefinite of signature (n + 1, n). The structure group of (M, φ, ξ, η, g) is × , where is the identity on span(ξ) and = (n; ℂ) ∩ (n, n).

By using the Levi-Civita connection ∇ of g, we define a tensor F of type (0,3) on M by F(x, y, z) = g((∇xφ) y, z). We then have

F(x, y, z) = F(x, z, y) = F(x, φy, φz) + η(y)F (x, ξ, z) + η(z)F (x, y, ξ)

(see [4]). Almost contact B-metric manifolds were classified into eleven basic classes i (i = 1, 2, …, 11) with respect to F in [4]. The special class 0 is determined by the condition F(x, y, z) = 0. Hence 0 is the class of almost contact B-metric manifolds having ∇-parallel structures, that is, ∇φ = ∇ξ = ∇η = ∇g = ∇ = 0. We should note that this special class 0 is contained in each class i (i = 1, …, 11). In this paper, we study almost contact manifolds with B-metric which belong to the class 7. This basic class is characterized by the following conditions:

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