Curvature-like Tensor Research Articles

Covariant Derivative of the Curvature Tensor of Kenmotsu Manifolds

In this paper, we define a (1, 3)-tensor field T(X, Y)Z on Kenmotsu manifolds and give a necessary and sufficient condition for T to be a curvature-like tensor. Next, we present some properties related to the curvature-like tensor T and prove that $$M^{2m+1}$$ is an $$\eta $$ -Einstein–Kenmotsu manifold if and only if $$\sum ^{m}_{j=1}T( \varphi (e_j), e_j) X = 0$$ . Besides, we define a (1, 4)-tensor field t on the Kenmotsu manifold M which determines when M is a Chaki T-pseudo-symmetric manifold. Then, we obtain a formula for the covariant derivative of the curvature tensor of Kenmotsu manifold M. We also find some conditions under which an $$\eta $$ -Einstein–Kenmotsu manifold is a Chaki T-pseudo-symmetric. Finally, we give an example to verify our results and prove that every three-dimensional Kenmotsu manifold is a generalized pseudo-symmetric manifold.

A (CHR)3-flat trans-Sasakian manifold

In [4] M. Prvanovic considered several curvaturelike tensors defined for Hermitian manifolds. Developing her ideas in [3], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly (CHR)3-curvature tensor. Then, we mainly researched (CHR)3-curvature tensor in a Sasakian manifold. Also we proved, that a conformally (CHR)3-flat Sasakian manifold does not exist. In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the (CHR)3-curvature tensor in a trans-Sasakian manifold. Also, the (CHR)3-Ricci tensor ρ3 and the (CHR)3-scalar curvature τ3 in a trans-Sasakian manifold have been obtained. Moreover, we define the notion of the (CHR)3-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We proved that a (CHR)3-flat trans-Sasakian manifold is a generalized ɳ-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric g of a trans-Sasakian manifold, if the latter is (CHR)3-flat.

A new curvaturelike tensor field in an almost contact Riemannian manifold II

In the last paper, we introduced a new curvaturlike tensor field in an almost contact Riemannian manifold and we showed some geometrical properties of this tensor field in a Kenmotsu and a Sasakian manifold. In this paper, we define another new curvaturelike tensor field, named (CHR)3-curvature tensor in an almost contact Riemannian manifold which is called a contact holomorphic Riemannian curvature tensor of the second type. Then, using this tensor, we mainly research (CHR)3-curvature tensor in a Sasakian manifold. Then we define the notion of the flatness of a (CHR)3-curvature tensor and we show that a Sasakian manifold with a flat (CHR)3-curvature tensor is flat. Next, we introduce the notion of (CHR)3-?-Einstein in an almost contact Riemannian manifold. In particular, we show that Sasakian (CHR)3- ?-Einstein manifold is ?-Einstain. Moreover, we define the notion of (CHR)3- space form and consider this in a Sasakian manifold. Finally, we consider a conformal transformation of an almost contact Riemannian manifold and we get new invariant tensor fields (not the conformal curvature tensor) under this transformation. Finally, we prove that a conformally (CHR)3-flat Sasakian manifold does not exist.

A new curvature-like tensor in an almost contact Riemannian manifold

In a M. Prvanović’s paper [5], we can find a new curvature-like tensor in an almost Hermitian manifold.In this paper, we define a new curvature-like tensor, named contact holomorphic Riemannian, briefly (CHR), curvature tensor in an almost contactRiemannian manifold. Then, using this tensor, we mainly research (CHR)-curvature tensor in a Kenmotsu and a Sasakian manifold. We introducethe flatness of a (CHR)-curvature tensor and show that a Kenmotsu anda Sasakian manifold with a flat (CHR)-curvature tensor is flat, see Theorems3.1 and 4.1. Next, we introduce the notion of an (CHR)-n-Einstein inan almost contact Riemannian manifold. In particular, in a Sasakian or aKenmotsu manifold, a (CHR)-n-Einstein manifold is n-Einstein, see Theorem5.3. Finally, from this tensor, we introduce a notion of a (CHR)-spaceform in an almost contact Riemannian manifold. In particular, if a Kenmotsuand a Sasakian manifold are (CHR)-space form, then the (CHR)-curvaturetensor satisfies a special equation, see Theorems 6.2 and 7.1.

Gravity and induced matter on nearly Kähler manifolds

We show that the conservation of energy–momentum tensor of a gravitational model with Einstein–Hilbert like action on a nearly Kähler manifold with the scalar curvature of a curvature-like tensor, is consistent with the nearly Kähler properties. In this way, the nearly Kähler structure is automatically manifested in the action as a induced matter field. As an example of nearly Kähler manifold, we consider the group manifold of R×II ×S3×S3 on which we identify a nearly Kähler structure and solve the Einstein equations to interpret the model. It is shown that the nearly Kähler structure in this example is capable of alleviating the well known fine tuning problem of the cosmological constant. Moreover, this structure may be considered as a potential candidate for dark energy.

A DECOMPOSITION OF THE CURVATURE TENSOR FIELD ON THE ABBENA-THURSTON MANIFOLD

Abstract. In this paper, we decompose the curvature tensor ( eld)on the Abbena-Thurston manifold into three curvature-like tensor elds, and investigate geometric properties. 1. IntroductionLet (V; ) be an n-dimensional real inner product space. Insection 2, we introduce the notion of a curvature-like tensor of type(1;3) on (V; ). We putL(V) := fLjLis a curvature like tensor on (V; )g;L 1 (V) := fL2L(V) jL(u;v) = cu^vfor u;v2V and some c2Rg;L ! (V) := fL2L(V) jthe Ricci tensor Ric L of Lis 0g;L 2 (V) := fL2L 1 (V) ? j = 0 for all L 0 2L ! (V)g:Then L(V) is decomposed into the orthogonal direct sum L 1 (V) L ! (V) L 2 (V). Let L= L 1 + L ! + L 2 (L2L(V)) be the decom-position corresponding to L 1 (V) L ! (V) L 2 (V). The component L ! of L2L(V) is said to be the Weyl tensor of L. The curvature-liketensors L 1 ;L ! ;L 2 of L= L 1 +L ! +L 2 2L(V) are given in terms of theRicci tensor Ric L and the scalar curvature S L of L(cf. Lemma 2.1).In this paper, using Lemma 2.1 we decompose the curvature tensor( eld) on the Abbena-Thurston manifold (cf. section 3.1) into threecurvature-like tensor elds. Geometric properties on low-dimensionalmanifolds (Heisenberg group, Abbena-Thurston manifold etcetera) have

A best possible inequality for curvature-like tensor fields

A best possible inequality for curvature-like tensor fields

A CURVATURE-LIKE TENSOR FIELD ON A SASAKIAN MANIFOLD

We investigate a curvature-like tensor deflned by (3.1) in Sasakian manifold of dimension‚ 5, and show that this ten- sor satisfles some properties. Especially, we determine compact Sasakian manifolds with vanishing this tensor and improve some theorems concerning contact conformal curvature tensor and spec- trum of Laplacian acting on p(0 • p • 2)-forms on the manifold by using this tensor component.

Conformally symmetric semi-Riemannian manifolds

In this paper we introduce the concept of conformal curvature-like tensor on a semi-Riemannian manifold, which is weaker than the notion of conformal curvature tensor defined on a Riemannian manifold. By such kind of conformal curvature-like tensor we give a complete classification of conformally symmetric semi-Riemannian manifolds with generalized non-null stress energy tensor.

δ-invariants and their applications to centroaffine geometry

We introduce the notion of δ-invariant for curvature-like tensor fields and establish optimal general inequalities in case the curvature-like tensor field satisfies some algebraic Gauss equation. We then study the situation when the equality case of one of the inequalities is satisfied and prove a dimension and decomposition theorem. In the second part of the paper, we apply these results to definite centroaffine hypersurfaces in R n + 1 . The inequality is specified into an inequality involving the affine δ-invariants and the Tchebychev vector field. We show that if a centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is a proper affine hypersphere. Furthermore, we prove that if a positive definite centroaffine hypersurface in R n + 1 , n ⩾ 3 , satisfies the equality case of one of the inequalities, it is foliated by ellipsoids. And if a negative definite centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is foliated by two-sheeted hyperboloids. Some further applications of the inequalities are also provided in this article.

On the decomposition of generalized ${\cal S}$-curvature-like tensor fields

On the decomposition of generalized ${\cal S}$-curvature-like tensor fields