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Journal ArticleDOI

On a class of three-dimensional trans-sasakian manifolds

31 Oct 2012-Communications of The Korean Mathematical Society (The Korean Mathematical Society)-Vol. 27, Iss: 4, pp 795-808
TL;DR: In this paper, a 3D trans-Sasakian manifold with conservative curvature tensor and 3D conformally flat trans-sakian manifolds are studied.
Abstract: The object of the present paper is to study 3-dimensional trans-Sasakian manifolds with conservative curvature tensor and also 3-dimensional conformally flat trans-Sasakian manifolds. Next we consider compact connected -Einstein 3-dimensional trans-Sasakian manifolds. Finally, an example of a 3-dimensional trans-Sasakian manifold is given, which verifies our results.
Citations
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Book
01 Jan 1970

329 citations

Journal ArticleDOI
01 Jan 2018-Filomat
TL;DR: In this article, it was shown that a trans-Sakian 3-manifold is locally symmetric if and only if it is locally isometric to the sphere space S3(c2), the hyperbolic space H3(-c 2), the Euclidean space R3, the product space R x H2(-c2) or R x S2(c 2) where c is a nonzero constant.
Abstract: In this paper, it is proved that a trans-Sasakian 3-manifold is locally symmetric if and only if it is locally isometric to the sphere space S3(c2), the hyperbolic space H3(-c2), the Euclidean space R3, the product space R x S2(c2) or R x H2(-c2), where c is a nonzero constant. Some examples are constructed to illustrate main results. We also give some new conditions for a compact trans-Sasakian 3-manifold to be proper.

9 citations

Journal ArticleDOI
15 Jul 2013
TL;DR: In this article, the Ricci soliton in 3-dimensional -Sasakian manifolds is shown to be a shrinking or expanding soliton and the manifold is a conformal Killilng vector field.
Abstract: We study Ricci solitons in -Sasakian manifolds and show that it is a shrinking or expanding soliton and the manifold is Einstein with Killing vector field Further, we prove that if is conformal Killilng vector field, then the Ricci soliton in 3-dimensional -Sasakian manifolds is shrinking or expanding but cannot be steady

7 citations


Cites background from "On a class of three-dimensional tra..."

  • ...De [16], De and Tripathi [17], Shaikh et al....

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Journal ArticleDOI
TL;DR: In this article, the Ricci solitons in trans-Sasakian manifolds satisfying,,, and, where,, and are quasiconformal, projective, and conharmonic curvature tensors.
Abstract: We study and obtain results on Ricci solitons in trans-Sasakian manifolds satisfying , , , and , where , , and are quasiconformal, projective, and conharmonic curvature tensors.

7 citations


Cites background from "On a class of three-dimensional tra..."

  • ...De [16], De and Tripathi [17], Shaikh et al....

    [...]

Journal ArticleDOI
08 Aug 2021
TL;DR: In this paper, the authors study 3D compact and connected trans-Sasakian manifold and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds.
Abstract: In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. First, four results in this paper deal with finding necessary and sufficient conditions on a compact and connected trans-Sasakian manifold to be homothetic to a compact and connected Sasakian manifold, and the fifth result deals with finding necessary and sufficient condition on a connected trans-Sasakian manifold to be homothetic to a connected Sasakian manifold. Finally, we find necessary and sufficient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to a compact and simply connected Einstein Sasakian manifold.

5 citations

References
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Book
01 Jan 1976
TL;DR: In this paper, the tangent sphere bundle is shown to be a contact manifold, and the contact condition is interpreted in terms of contact condition and k-contact and sasakian structures.
Abstract: Contact manifolds.- Almost contact manifolds.- Geometric interpretation of the contact condition.- K-contact and sasakian structures.- Sasakian space forms.- Non-existence of flat contact metric structures.- The tangent sphere bundle.

1,259 citations


Additional excerpts

  • ...3) g(X,φY ) = −g(φX, Y ), g(X, ξ) = η(X) for all X and Y tangent to M ([2], [3])....

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Journal ArticleDOI
TL;DR: In this paper, it was shown that sixteen classes of almost Hermitian manifolds can be found in the Euclidean space, and that they are Hermitians in a natural way.
Abstract: It is shown that in a natural way there are precisely sixteen classes of almost Hermitian manifolds.

823 citations


"On a class of three-dimensional tra..." refers background in this paper

  • ...An almost contact metric structure (φ,ξ,η,g) on a connected manifold M is called trans-Sasakian structure [17] if (M × R, J , G) belongs to the class W4 [10], where J is the almost complex structure on M ×R defined by J (...

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  • ...Again in the Gray-Hervella classification of almost Hermite manifolds [10], there appears a class W4 of Hermitian manifolds which are closely related to locally conformally Kähler manifolds....

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Book
01 Jan 1970

329 citations


"On a class of three-dimensional tra..." refers background in this paper

  • ...It is classical that on a 3-dimensional conformally flat Riemannian manifold [19], we have (5....

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