# ∗ -Ricci Tensor on α -Cosymplectic Manifolds

TL;DR: In this paper , the authors studied α-cosymplectic manifold and showed that the Ricci tensor tensor is a semisymmetric manifold, which is an extension of the RICCI tensor.

Abstract: In this paper, we study α-cosymplectic manifold
$M$
admitting
$\ast $
-Ricci tensor. First, it is shown that a
$\ast $
-Ricci semisymmetric manifold
$M$
is
$\ast $
-Ricci flat and a
$\varphi $
-conformally flat manifold
$M$
is an
$\eta $
-Einstein manifold. Furthermore, the
$\ast $
-Weyl curvature tensor
${\mathcal{W}}^{\ast}$
on
$M$
has been considered. Particularly, we show that a manifold
$M$
with vanishing
$\ast $
-Weyl curvature tensor is a weak
$\varphi $
-Einstein and a manifold
$M$
fulfilling the condition
$R\left({E}_{1},{E}_{2}\right)\cdot {\mathcal{W}}^{\ast}=0$
is
$\eta $
-Einstein manifold. Finally, we give a characterization for α-cosymplectic manifold
$M$
admitting
$\ast $
-Ricci soliton given as to be nearly quasi-Einstein. Also, some consequences for three-dimensional cosymplectic manifolds admitting
$\ast $
-Ricci soliton and almost
$\ast $
-Ricci soliton are drawn.

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TL;DR: In this paper , the authors investigated the properties of a 3-dimensional Kenmotsu manifold satisfying certain curvature conditions endowed with Ricci solitons and showed that such a manifold is φ-Einstein.

Abstract: The present paper deals with the investigations of a Kenmotsu manifold satisfying certain curvature conditions endowed with 🟉-η-Ricci solitons. First we find some necessary conditions for such a manifold to be φ-Einstein. Then, we study the notion of 🟉-η-Ricci soliton on this manifold and prove some significant results related to this notion. Finally, we construct a nontrivial example of three-dimensional Kenmotsu manifolds to verify some of our results.

1 citations

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TL;DR: In this paper , the authors concentrate on hyper generalized and quasi-generalized φ-varphi-recurrent π-cosymplectic manifolds and obtain some significant characterizations which classify such manifolds.

Abstract: In this paper, we concentrate on hyper generalized $\varphi-$recurrent $\alpha-$cosymplectic manifolds and quasi generalized $\varphi-$recurrent $\alpha-$cosymplectic manifolds and obtain some significant characterizations which classify such manifolds.

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TL;DR: In this article , the Schouten-van Kampen connection on α-cosymplectic manifolds admits pseudo-projective and W8-curvature tensors.

TL;DR: In this paper , the authors concentrate on hyper generalized φ -recurrent α -cosymplectic manifolds and quasi generalized ε-generalized φ-recurrent ε -recurrence α -co-symmetric manifold and obtain some significant characterizations which classify such manifolds.

Abstract: A bstract . In this paper, we concentrate on hyper generalized φ -recurrent α -cosymplectic manifolds and quasi generalized φ -recurrent α -cosymplectic manifolds and obtain some significant characterizations which classify such manifolds.

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08 Jan 2002

TL;DR: In this article, the authors describe a complex geometry model of Symplectic Manifolds with principal S1-bundles and Tangent Sphere Bundles, as well as a negative Xi-sectional Curvature.

Abstract: Preface * 1. Symplectic Manifolds * 2. Principal S1-bundles * 3. Contact Manifolds * 4. Associated Metrics * 5. Integral Submanifolds and Contact Transformations * 6. Sasakian and Cosymplectic Manifolds * 7. Curvature of Contact Metric Manifolds * 8. Submanifolds of Kahler and Sasakian Manifolds * 9. Tangent Bundles and Tangent Sphere Bundles * 10. Curvature Functionals and Spaces of Associated Metrics * 11. Negative Xi-sectional Curvature * 12. Complex Contact Manifolds * 13. Additional Topics in Complex Geometry * 14. 3-Sasakian Manifolds * Bibliography * Subject Index * Author Index

1,822 citations

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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

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TL;DR: In this paper, Tanno et al. showed that the curvature tensor R of a locally symmetric Riemannian space satisfies R(X, Y) R − 0 for all tangent vectors X and 7, where the linear endomorphism R(x, y) acts on R as a derivation.

Abstract: Introduction The curvature tensor R of a locally symmetric Riemannian space satisfies R(X, Y) R — 0 for all tangent vectors X and 7, where the linear endomorphism R(X, Y) acts on R as a derivation. This identity holds in a space of recurrent curvature also. The spaces with R(X9 Y) R = 0 have been investigated first by E. Cartan [2] as these spaces can be considered as a direct generalization of the notion of symmetric spaces. Further on remarkable results were obtained by the authors A. Lichnerowicz [13], R. S. Couty [3], [4] and N. S. Sinjukov [19], [20], [21]. In one of his papers K. Nomizu [15] conjectered that an irreducible, complete Riemannian space with dim > 3 and with the above symmetric property of the curvature tensor is always a locally symmetric space. But this conjecture was refuted by H. Takagi [22] who constructed 3-dimensional complete irreducible nonlocally-symmetric hypersurfaces with R(X, Y) R — 0. These two papers were very stimulating for the further investigations. We also have to mention the following authors in this field: S. Tanno [23], [24], [25], K. Sekigawa [16], [17] and P. I. Kovaljev [9], [10], [11]. In the following we call a space satisfying R(X, Y) R = 0 a semi-symmetric space. The main purpose of this paper is to determine all semi-symmetric spaces in a structure theorem. In §1 we give local decomposition theorems using the infinitesimal holonomy group, and in §2 we give some basic formulas. We would like to make it perfectly clear that the results of these chapters are concerning general Riemannian spaces, and not only semi-symmetric spaces. In §3 we construct several nonsymmetric semi-symmetric spaces and in §4 we show that every semi-symmetric space can be decomposed locally on an everywhere dense open subset into the direct product of locally symmetric spaces and of the spaces constructed in §3.

266 citations

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129 citations