A class of almost contact riemannian manifolds
01 Jan 1972-Tohoku Mathematical Journal (Mathematical Institute, Tohoku University)-Vol. 24, Iss: 1, pp 93-103
TL;DR: In this article, Tanno has classified connected almost contact Riemannian manifolds whose automorphism groups have themaximum dimension into three classes: (1) homogeneous normal contact manifolds with constant 0-holomorphic sec-tional curvature if the sectional curvature for 2-planes which contain
Abstract: Recently S. Tanno has classified connected almostcontact Riemannian manifolds whose automorphism groups have themaximum dimension [9]. In his classification table the almost contactRiemannian manifolds are divided into three classes: (1) homogeneousnormal contact Riemannian manifolds with constant 0-holomorphic sec-tional curvature if the sectional curvature for 2-planes which contain
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TL;DR: In this paper, generalized Sasakian-space-forms are introduced and studied, by using some different geometric techniques such as Riemannian submersions, warped products or conformal and related transformations.
Abstract: Generalized Sasakian-space-forms are introduced and studied. Many examples of these manifolds are presented, by using some different geometric techniques such as Riemannian submersions, warped products or conformal and related transformations. New results on generalized complex-space-forms are also obtained.
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TL;DR: In this article, the local structure of trans-Sasakian manifolds of dimension ⩾ 5 has been characterized by giving suitable examples, and the authors give suitable examples for trans-sakian manifold decomposition.
Abstract: In this paper, we completely characterize the local structure of trans-Sasakian manifolds of dimension ⩾ 5 by giving suitable examples.
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TL;DR: In this paper, the authors consider locally symmetric almost Kenmotsu manifold and show that the manifold is locally isometric to the Riemannian product of an n+1-dimensional manifold of constant curvature.
Abstract: We consider locally symmetric almost Kenmotsu manifolds showing that such a manifold is a Kenmotsu manifold if and only if the Lie derivative of the structure, with respect to the Reeb vector field $\xi$, vanishes. Furthermore, assuming that for a $(2n+1)$-dimensional locally symmetric almost Kenmotsu manifold such Lie derivative does not vanish and the curvature satisfies $R_{XY}\xi =0$ for any $X, Y$ orthogonal to $\xi$, we prove that the manifold is locally isometric to the Riemannian product of an $(n+1)$-dimensional manifold of constant curvature $-4$ and a flat $n$-dimensional manifold. We give an example of such a manifold.
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Cites background from "A class of almost contact riemannia..."
...Kenmotsu manifolds appear for the first time in [9], where they have been locally classified....
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...As proved in [9], a Kenmotsu manifold is locally symmetric if and only if it is a space of constant sectional curvature K = −1....
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...([9]) Let (M, φ, ξ, η, g) be a Kenmotsu manifold....
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TL;DR: In this article, a complete classification for almost contact metric manifolds through the study of the covariant derivative of the fundamental 2-form on those manifolds was obtained, and the classification was extended to almost contact manifold.
Abstract: It is obtained a complete classification for almost contact metric manifolds through the study of the covariant derivative of the fundamental 2- form on those manifolds.
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"A class of almost contact riemannia..." refers background in this paper
...We define a (1,1)-tensor field R1 as follows: g(R1(X), Y) = R1(X, Y)....
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...In his classification table the almost contact Riemannian manifolds are divided into three classes: (1) homogeneous normal contact Riemannian manifolds with constant ƒÓ-holomorphic sectional curvature if the sectional curvature for 2-planes which contain e, say K (X,ƒÌ ), > 0, (2) global Riemannian products of a line or a circle and a Kaehlerian manifold with constant holomorphic sectional curvature, if K (X, ƒÌ) = 0 and (3) a warped product space L x f CE, if K (X, ƒÌ) <0....
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...It is known that the manifold of the class (1) in the above statement is characterized by some tensor equations; it has a Sasakian structure....
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...12) A (1,1)-tensor field ƒÓ is defined ƒÓby...
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TL;DR: In this article, the authors studied the problem of finding a (φ, ξ, η, g)-connection in a space with a normal contact structure and proved that the space with such a contact structure is an Einstein space.
Abstract: Introduction. Recently S. Sasaki [3]° defined the notion of (φ, ξ, η, g) structure of a differentiable manifold. Further, S. Sasaki and Y. Hatakeyama [ 4 ] [ 5 ] showed that the structure is closely related to contact structure. By means of this notion, it is shown that a space with a contact structure can be dealt with as we deal with an almost complex space. So, by similar manner, some problems discussed in the latter space may be considered in the former. On the other hand, S. Tachibana [6] [7] proved many interesting theorems in an almost complex space. In this paper, the present author tries to study, in the space with a certain contact structure, the problem corresponding to S. Tachibana's results. We shall devote § 1 to preliminaries and in this section introduce a normal contact structure. In §2, we ennumerate identities which will be useful in the later sections. We shall prove in § 3 that a space with a normal contact structure satisfying VkRjt = 0 be necessarily an Einstein one and that a symmetric space with a normal contact structure reduces to the space of constant curvature respectively. The differential form R is dealt with in § 4, and in this section, we shall show a necessary and sufficient condition that the space be an Einstein space by means of the form R. Finally in § 5, we introduce a certain type of (φ, η,
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