Some Curvature Properties of -Manifolds
Mehmet Atceken2013, Abstract and Applied Analysis
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
The object of the present paper is to study -manifolds with vanishing quasi-conformal curvature tensor. -manifolds satisfying Ricci-symmetric condition are also characterized.
Key takeaways
- Recently, in [1], Shaikh introduced and studied Lorentzian concircular structure manifolds (briefly (LCS)-manifold) which generalizes the notion of LP-Sasakian manifolds, introduced by Matsumoto [2].
- Also, Shaikh and his coauthors studied various types of (LCS) -manifolds by imposing the curvature restrictions (see [3][4][5][6]).
- A differentiable manifold of dimension is called (LCS)-manifold if it admits a (1, 1)-type tensor field , a covariant vector field , and a Lorentzian metric which satisfy
- Now let ( , ) be an -dimensional Riemannian manifold; then the concircular curvature tensor̃, the Weyl conformal curvature tensor , and the pseudo projective curvature tensor̃are, respectively, defined bỹ
- Thus we have the following theorem for (LCS) -conformally flat manifolds.
Related papers
On Curvature Tensors of an Almost C(α)-Manifold
The object of the present paper is to study the geometry of an almost C(alpha)-manifold when it is Ricci semisymmetric, projective semi-symmetric and quasi-conformal semi-symmetric. We show that the quasi-conformal curvature tensor is equivalent to conformal curvature tensor under the some conditions.
A Class of Riemannian Manifolds with Special Form of Curvature Tensor
Riemannian curvature is determined by the Ricci curvature by a special formula to introduce a special class of Riemannian manifolds which called Jawarneh manifold. Some geometric properties of Jawarneh manifold have been derived and a non-trivial example is obtained to prove the existence.
On the Pseudo-Quasi Conformal Curvature Tensor of a Normal Paracontact Metric Manifold
2020
In the present paper we have studied the curvature tensor of a normal paracontact metric manifold satisfying the conditions $ R(\xi,X) \widetilde{C}=0 $, $ \widetilde{C}(\xi,X)S=0 $, $ \widetilde{C}(\xi,X)P=0 $, $ \widetilde{C}(\xi,X)\widetilde{Z}=0 $ and pseudo quasi conformal flat, where $ R $, $ P $, $ S $, $ \widetilde{Z} $ and $ \widetilde{C} $ are the Riemannian curvature, projective curvature, Ricci, concircular curvature and pseudo-quasi conformal curvature tensors, respectively.
ON GENERALIZED QUASI-CONFORMAL N(k, μ)-MANIFOLDS
Communications of the Korean Mathematical Society, 2016
The object of the present paper is to introduce a new curvature tensor, named generalized quasi-conformal curvature tensor which bridges conformal curvature tensor, concircular curvature tensor, projective curvature tensor and conharmonic curvature tensor. Flatness and symmetric properties of generalized quasi-conformal curvature tensor are studied in the frame of (k, µ)-contact metric manifolds.
Locally conformal flat Riemannian manifolds with constant principal Ricci curvatures and locally conformal flat C-spaces
1997
It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given in dimensions 4,5,6,7 and 8. It is shown that any n-dimensional $(4\leq n \leq 8)$ locally conformal flat Riemannian manifold with constant principal Ricci curvatures is a Riemannian locally symmetric space.
On Quasi-Conformally Recurrent Manifolds with Harmonic Quasi-Conformal Curvature Tensor
Kyungpook mathematical journal, 2011
The main objective of the paper is to provide a full classification of quasiconformally recurrent Riemannian manifolds with harmonic quasi-conformal curvature tensor. Among others it is shown that a quasi-conformally recurrent manifold with harmonic quasi-conformal curvature tensor is any one of the following: (i) quasi-conformally symmetric, (ii) conformally flat, (iii) manifold of constant curvature, (iv) vanishing scalar curvature, (v) Ricci recurrent.
LP-Sasakian manifolds with quasi-conformal curvature tensor
SUT Journal of Mathematics
Ricci curvature of submanifolds in locally conformal almost cosymplectic manifolds
2003
In [Ch3], B.Y. Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. In [MMO], one dealt with similar problems for submanifolds in complex space forms. In this article, we obtain sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in locally conformal almost cosymplectic manifolds of pointwise constantsectional curvature.
Characterization on a Non-flat Riemannian Manifold
Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 2016
In this paper we study characterizations of odd and even dimensional pseudo generalized quasi Einstein manifold and we give three and four dimensional examples (both Riemannian and Lorentzian) of pseudo generalized quasi Einstein manifold to show the existence of such manifold. Also in the last section we give the examples of warped product on pseudo generalized quasi Einstein manifold. Keywords Einstein manifold Á Quasi-Einstein manifold Á Super quasi-Einstein manifold Á Mixed super quasi-Einstein manifold Á Pseudo generalized quasi Einstein manifold Á Warped product Mathematics Subject Classification 53C25 Á 53C15
On solutions of the Ricci curvature equation and the Einstein equation
Israel Journal of Mathematics, 2009
We consider the pseudo-Euclidean space (R n , g), with n ≥ 3 and g ij = δ ij i , i = ±1, where at least one i = 1 and nondiagonal tensors of the form T = ij f ij dx i dx j such that, for i = j, f ij (x i , x j ) depends on x i and x j . We provide necessary and sufficient conditions for such a tensor to admit a metricḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on R n , on the n-dimensional torus T n and on cylinders T k ×R n−k , that solve the Ricci equation or the Einstein equation.
On the Existence of Certain Metrics with Non Parallel Ricci Tensor
2001
A. Derdzinki [D] gave examples of Riemannian metrics with harmonic curvature and non parallel Ricci tensor on some compact manifolds $(M,g]$ . We examine their existence as well as their number wich naturally depends on the geometry of the manifolds.
Some complete manifolds with non-negative curvature operator
1987
where the scalar product at the left hand side is the one induced in the space of bi-vectors by the Riemannian metric. In this paper we will consider complete manifolds M with non-negative curvature operator (p>0) and its compact soul A (see [3]). The main result of this paper is: Theorem A. If nlM={O} and p>=O then M is a topological product of a soul by a Euclidean space. We will apply this result in two cases in which the non-negativity of the sectional curvatures of M(K>=O) implies the non-negativity of the curvature operator. The first of them is when the manifold is isometrically immersed in Euclidean spaces with the first normal space Nl(x) at most two-dimensional for each x~M (see [7]). We will prove the following: Theorem B. Let f: M" ~ R "+ 2 be an isometric immersion of an even-dimensional, complete, non-compact, oriented manifold with K>_O and Kp>O at.some point peM. Then M is either diffeomorphic to R" or to Akx R "-k, where A k is diffeomorphic to the product of two even dimensional spheres, or M has the homotopy type of RP 2 or CP 2. The second case is when p satisfies the following property: Definition. A Riemannian manifold M has pure curvature operator if for each xeM there exists an orthonormal frame {X 1 .... , X,} of TxM such that p(Xi/xXj)=KIjXi/xXj, where Kij is the sectional curvature of the plane spanned by X i and Xj.
On Pseudo Cyclic Ricci Symmetric Manifolds
Asian-European Journal of Mathematics, 2009
The object of the present paper is to introduce a type of non-flat Riemannian manifold called pseudo cyclic Ricci symmetric manifold and study its geometric properties. Among others it is shown that a pseudo cyclic Ricci symmetric manifold is a special type of quasi-Einstein manifold. In this paper we also study conformally flat pseudo cyclic Ricci symmetric manifolds and prove that such a manifold can be isometrically immersed in a Euclidean manifold as a hypersurface.
Almost Hermitian manifolds satisfying some curvature conditions
Kodai Mathematical Journal, 1979
On some type of curvature conditions
PDEs, Submanifolds and Affine Differential Geometry, 2002
In this paper we present a review of recent results on semi-Riemannian manifolds satisfying curvature conditions of pseudosymmetry type.
On the Ricci Tensor of Submanifolds in Conformal Kenmotsu Manifolds
Kyushu Journal of Mathematics, 2017
In this paper, we obtain several interesting results on submanifolds of conformal Kenmotsu manifolds. In addition to this we consider submanifolds of a conformal Kenmotsu manifold of which the Ricci tensor is parallel, Lie ξ-parallel or recurrent. We also present an illustration example of a three-dimensional conformal Kenmotsu manifold that is not Kenmotsu.
On Weakly Quasi-Conformally Symmetric Manifolds
SUT Journal of Mathematics
Four-dimensional, Ricci-flat manifolds which admit a metric
Filomat, 2015
This paper discusses the relationships between the metric, the connection and the curvature tensor of 4-dimensional, Ricci flat manifolds which admit a metric. It is shown that these metric and curvature objects are essentially equivalent conditions for such manifolds if one excludes certain very special cases and which occur when the signature is indefinite. In a similar vein, some relevant remarks are made regarding the Weyl conformal tensor.
On Ricci curvature of metric structures on g-manifolds
2020
We study the properties of Ricci curvature of ${\mathfrak{g}}$-manifolds with particular attention paid to higher dimensional abelian Lie algebra case. The relations between Ricci curvature of the manifold and the Ricci curvature of the transverse manifold of the characteristic foliation are investigated. In particular, sufficient conditions are found under which the ${\mathfrak{g}}$-manifold can be a Ricci soliton or a gradient Ricci soliton. Finally, we obtain a amazing (non-existence) higher dimensional generalization of the Boyer-Galicki theorem on Einstein K-manifolds for a special class of abelian ${\mathfrak{g}}$-manifolds.
Metric Ricci Curvature for Manifolds
Geometry, 2013
We introduce a metric notion of Ricci curvature for P L manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers Theorem, for surfaces as well as for a large class of higher dimensional manifolds.
Related topics
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.