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Abstract
The purpose of the present paper is to discuss the geometrical properties of the Vaisman-Gray manifold (VG-manifold) of a pointwise holomorphic sectional conharmonic tensor (PHT-tensor). Furthermore, the necessary and sufficient conditions required for the VG-manifold to admit such a PHT-tensor have been determined. In particular, under certain conditions, we have established that the aforementioned manifold was an Einstein manifold.
The classification of the almost Hermitian structures was introduced by Gray and Hervella [4]. These structures have been categorized into sixteen different classes. Moreover, it has been found that the condition for each one of them depends on a Kozel’s operator method [15].
On the other hand, there is another significant classification method for the almost Hermitian structures that were introduced by Kirichenko. This method depends on the principle fibre bundle space of all complex frames of a smooth manifold M with the unitary structure group U(n). This space is called an adjoined G-structure space. For further information, refer to the following citations: [3], [8], [9], [10], and [11].
One of the interesting classes of almost Hermitian structures is a VG-manifold, which is denoted by W_{1} ⊕ W_{4}, where W_{1} and W_{4} denote the nearly Kähler manifold and the locally conformal Kähler manifold, respectively.
It is a well-known fact that the harmonic function is one whose Laplacian vanishes. In general, it is not a conformal transformation harmonic function. With regard to this fact, Ishi [6] introduced a tensor that remained invariant under a conharmonic transformation for an n-dimensional Riemannian manifold. In addition, Khan [7] determined the properties of the conharmonically flat Sasakian manifolds. Moreover, it has been proved that a special weakly Ricci symmetric Sasakian manifold is an Einstein manifold. Subsequently, Shihab [13] went on to determine the geometrical properties of the conharmonic curvature tensor belonging to the nearly Kähler manifold. Furthermore, it has been established that a Kähler manifold with a dimension greater than four is a conharmonic parakähler manifold if, and only if, it has a flat Ricci tensor. On the other hand, Zengin and Tasci [19] studied a pseudo conharmonically symmetric manifold. In particular, they proved that the aforementioned manifold with a non-zero scalar curvature has a closed associated 1-form. Lastly, Abood and Abdulameer [1] considered the conharmonically flat VG-manifold, exclusively and identified the necessary and sufficient conditions required for the VG-manifold to be an Einstein manifold.
In this article, we have employed the adjoined G-structure space to study the geometry of the VG-manifold that corresponds with a PHT-tensor.
2. Preliminaries
Let M be a smooth manifold of even dimension, C^{∞}(M) be an algebra of smooth functions on M, and X(M) be the module of smooth vector fields on M. An almost Hermitian manifold (AH-manifold) is a triple {M, J, g = 〈.,.〉}, where M is a smooth manifold, J is an almost complex structure, and g = 〈., .〉 is a Riemannian metric, such that the equality 〈JX, JY〉 = 〈X, Y〉 holds for X, Y ∈ X(M).
Suppose that ${T}_{p}^{c}(M)$ is the complexification of a tangent space T_{p}(M) at the point p ∈ M and {e_{1}, …, e_{n}, Je_{1}, …, Je_{n}} is a real adapted basis of AH-manifold. Then, in the module ${T}_{p}^{c}(M)$, there exists a basis given by {ɛ_{1}, …, ɛ_{n}, ε̂_{1}, …, ε̂_{n}} which is called as an adapted basis, where, ɛ_{a} = σ(e_{a}), ε̂_{a} = σ̄(e_{a}), and σ, σ̄ are two endomorphisms in the module X^{c}(M), which are given by $\sigma ={\scriptstyle \frac{1}{2}}(id-\sqrt{-1}{J}^{c})$ and $\overline{\sigma}=-{\scriptstyle \frac{1}{2}}(id+\sqrt{-1}{J}^{c})$, respectively, such that X^{c}(M) and J^{c} are the complexifications of X(M) and J, respectively. The corresponding frame of this basis is {p; ɛ_{1}, … ɛ_{n}, ε̂_{1}, …, ε̂_{n}}. Suppose that the indexes i, j, k, and l are in the range 1, 2, …, 2n and the indexes a, b, c, d and f are in the range 1, 2, …, n. Moreover, â = a + n.
For a manifoldM, it is a well known that the given AH-structure is equivalent to the given G-structure space in the principle fibre bundle of all complex frames of M with the unitary structure group U(n). Whereas, in the adjoined G-structure space, the components matrices of the almost complex structure J and the Riemannian metric g are given as follows:
where r, R and g are respectively the Ricci tensor, the Riemannian curvature tensor, and the Riemannian metric. Similar to the property of Riemannian curvature tensor, the conharmonic tensor has the following property:
In the adjoined G-structure space, an AH-manifold {M, J, g = 〈., .〉} is called a Vaisman-Gray manifold (VG-manifold) if B^{abc} = −B^{bac}, ${B}_{c}^{ab}={\alpha}^{[a}{\delta}_{c}^{b]}$; a locally conformal Kähler manifold (LCK-manifold) if B^{abc} = 0 and ${B}_{c}^{ab}={\alpha}^{[a}{\delta}_{c}^{b]}$; and a nearly Kähler manifold (NK-manifold) if B^{abc} = −B^{bac} and ${B}_{c}^{ab}=0$, where ${B}^{abc}=\frac{\sqrt{-1}}{2}{J}_{[\widehat{b},\widehat{c}]}^{a},{B}_{c}^{ab}=\frac{\sqrt{-1}}{2}{J}_{\widehat{b},c]}^{a},\alpha =\frac{1}{(n-1)}\delta F\circ J$ is a Lie form, F is a Kähler form which is given by F(X, Y) = 〈JX, Y〉, δ is a codrivative; X, Y ∈ X(M) and the bracket [ ] denotes the antisymmetric operation.
where, {${A}_{bcd}^{a}$ } are some functions on the adjoined G-structure space, {${A}_{bc}^{ad}$ } are a system of functions in the adjoined G-structure space that are symmetric by the lower and upper indices, which are called the components of the holomorphic sectional curvature tensor.
The functions {${\alpha}_{b}^{a},{\alpha}_{a}^{b}$ } are the components of the covariant differential structure tensor of the first and second type, and {α_{ab}, α^{ab}} are the components of the Lie form on the adjoined G-structure space such that:
where, {ω^{a}, ω_{a}} are the components of mixture form and {${\omega}_{b}^{a}$ } are the components of the Riemannian connection of the metric g. Other components of the Riemannian curvature tensor R can be obtained by the property of symmetry for R.
The symbols ( ) and [ ] are usually used to denote the symmetric and antisymmetric respectively.
3. The main results
Definition 3.1
Let M be an AH-manifold. A holomorphic sectional conharmonic (HT-tensor) of a manifold M in the direction X ∈ X(M), X ≠ 0 is a function h(X), which is given by
If M is an AH-manifold of PHT- tensor, then the equation${\Vert X\Vert}^{4}=2{\delta}_{ad}^{\tilde{b}c}{X}^{a}{X}^{d}{X}_{b}{X}_{c}$holds, where${\tilde{\delta}}_{ad}^{bc}={\delta}_{a}^{b}{\delta}_{d}^{c}+{\delta}_{d}^{b}{\delta}_{a}^{c}$is a Kroneker delta of the second type.
The necessary condition for a VG-manifold to be a PHT-tensor is summarized in the following theorem.
Theorem 3.4
Suppose that M is a VG-manifold of the conharmonic tensor and the J-invariant Ricci tenor. Then, the necessary condition for a VG-manifold to be a PHT- tensor is for the components of the HT-curvature tensor to satisfy the following condition:
Therefore, according to the Definition 2.5, M is an NK-manifold.
Theorem 3.6
Let M be a VG-manifold of the PHT-tensor with a flat holomorphic sectional curvature tensor and J-invariant Ricci tensor. Then, M is an Einstein manifold.
Proof
Suppose thatM is a VG-manifold of the PHT-curvature tensor. According to the Theorem 3.4, we get
Since M has a J-invariant Ricci tensor, then M is an Einstein manifold.
Theorem 3.7
Let M be a VG-manifold of the PHT-tensor with a J-invariant Ricci tensor, then M is an Einstein manifold if, and only if, ${A}_{ac}^{bc}={c}_{1}{\delta}_{a}^{b}$, where c_{1}is a constant.
Proof
Suppose that M is a VG-manifold of the PHT-tensor. According to the Theorem 3.4, we have
Since M has a J-invariant Ricci tensor, then M is an Einstein manifold.
4. Conclusions
This article clearly aimed to study the geometrical properties of the VG-manifold of the pointwise holomorphic sectional curvature conharmonic tensor. We have found out the necessary conditions for the VG-manifold to be a manifold of the pointwise holomorphic sectional conharmonic tensor. Furthermore, we have formulated an interesting theoretical physical application. In particular, we have concluded the necessary and sufficient conditions for a VG-manifold to be an Einstein manifold.
References
Abood, HM, and Abdulameer, YA (2017). Conharmonically flat Vaisman-Gray manifold. Amer J Math Stat. 7, 38-43.
Ali, LK 2008. On almost Kähler manifold of a pointwise holomorphic sectional curvature tensor. Master’s thesis. University of Basra, College of Education.
Banaru, M . A new characterization of the Gray-Hervella classes of almost Hermitian manifolds., 8th International Conference on Differential Geometry and its Applications, August 27–31, 2001, Opava-Czech Republic.
Gray, A, and Hervella, LM (1980). The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann Math Pure Appl. 123, 35-58.
Ignatochkina, LA (2003). Vaisman-Gray manifolds with a J-nvariant conformal curvature tensor. Sb Math. 194, 225-235.
Ishi, Y (1957). On conharmonic transformations. Tensor (NS). 7, 73-80.
Khan, Q (2004). On conharmonically and special weakly Ricci symmetric Sasakian manifolds. Novi Sad J Math. 34, 71-77.
Kirichenko, VF (1975). Certain properties of tensors on K-spaces. Vestnik Moskov Univ Ser I Mat Meh. 30, 78-85.
Kirichenko, VF (1976). K-spaces of constant type. Sibirsk Mat Z. 17, 282-289.
Kirichenko, VF (1976). K-spaces of constant holomorphic sectional curvature. Mat Zametki. 19, 805-814.
Kirichenko, VF (1977). The differential geometry of K-spaces. Problems of Geometry. 8, 139-161.
Kirichenko, VF, and Arseneva, OE (1998). Self-dual geometry of generelized Hermitian surfaces. Sb Math. 189, 19-44.
Kirichenko, VF, and Shikhab, AA (2011). On the geometry of conharmonic curvature tensor of nearly Kähler manifolds. J Math Sci (NY). 177, 675-683.
Kobayashi, S, and Nomizu, K (1963). Foundations of differential geometry: John Wily and Sons
Kozal, JL (1957). Varicies Kahlerian-notes. Sao Paolo
Petrov, AZ (1961). Einstein space. Moscow: Phys-Math. Letr
Rachevski, PK (1964). Riemannian geometry and tensor analysis: M. Nauka
Tretiakova, EV (1999). Curvature identities for almost Kähler manifold. Moscow, No. 208-B99: VINITE
Zengin, FO, and Tasci, AY (2014). Pseudo Conharmonically Symmetric Manifolds. Eur J Pure Appl Math. 7, 246-255.