CC BY
36
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
2022. Volume 59. Pp. 3-14
MSC2020: 53C25, 53D10, 53D15 © M.Y. Abass, Q. S. A. Al-Zamil
ON WEYL TENSOR OF ACR-MANIFOLDS OF CLASS C12 WITH APPLICATIONS
In this paper, we determine the components of the Weyl tensor of almost contact metric (ACR-) manifold of class C12 on associated G-structure (AG-structure) space. As an application, we prove that the con-formally flat ACR-manifold of class C12 with n > 2 is an n-Einstein manifold and conclude that it is an Einstein manifold such that the scalar curvature r has provided. Also, the case when n = 2 is discussed explicitly. Moreover, the relationships among conformally flat, conformally symmetric, {-conformally flat and ^-invariant Ricci tensor have been widely considered here and consequently we determine the value of scalar curvature r explicitly with other applications. Finally, we define new classes with identities analogously to Gray identities and discuss their connections with class C12 of ACR-manifold.
Keywords: almost contact metric manifold of class C12, n-Einstein manifold, Weyl tensor. DOI: 10.35634/2226-3594-2022-59-01
Introduction
Throughout this paper, we consider a Riemannian manifold with an odd dimension 2n +1 and furnished by an almost contact structure ($, n). Chinea and Gonzalez [9] obtained a complete classification for ACR-manifold through the study of the covariant derivative of fundamental 2-forms on the manifolds in question and consequently, these classifications imply a new class which is called class C12 with the following condition:
Vx(^)(Y, Z) = n(X){n(Z)V?(n)$Y - n(Y)V?(n)$Z} V X,Y,Z £ X(M),
where V is the Levi-Civita connection on M and Q(X, Y) = g(X, $Y). On the other hand, Bouzir et al. [6] studied the properties of the manifolds of class C12 when the dimension is 3, but the first author [2] determined the structure equations of Cartan for ACR-manifold of class C12 on the AG-structure space and determined the components of the Riemannian curvature tensor and Ricci tensor during this study; some of these results are given in the next section. Whereas, Candia and Falcitelli [7,8], generalized the class C12 into the class C5 © C12.
Moreover, as a quotation from the citation [19], we found that Weyl introduced a generalized curvature tensor which vanishes whenever the metric is conformally flat and this is why some times it is called conformal curvature tensor. The Weyl conformal curvature tensor field W is a tensor of type (3, 1) on ACR-manifold (M2n+1, g, n) and is defined to be (see [11])
W(Z,U)Y = R(Z,U)Y---—[S(U,Y)Z - S(Z,Y)U + g(U,Y)LZ - g(Z,Y)LU]
2n — 1
r
for all U,Y,Z £ X(M), where R is the Riemannian curvature tensor, S(U,Z) = g(LU,Z) is the Ricci tensor, L denotes the Ricci operator and r is a scalar curvature. Moreover, the Weyl tensor W of type (4, 0) is defined by W(X, Y, Z, U) = g(W(Z, U)Y, X) V X, Y, Z, U £ X(M). It is straightforward to show that the Weyl tensor possesses the same symmetries as the Rieman-nian tensor. However, Weyl tensor possesses very interesting property which is traceless tensor, in other words, it vanishes for any pair of contracted indices.
Geometrically, the Weyl tensor conveys information about the tidal force and shows how a body feels when moving along a geodesic. In fact, the major difference between the Weyl tensor and the Riemannian curvature tensor is that Weyl tensor does not decode information on how the volume of the body changes, but rather only how the shape (topology) of the body is distorted by the tidal force. More concretely, one could decompose the Riemannian curvature into trace and traceless parts which allows an easy proof that the Weyl curvature tensor is the conformally invariant part of the Riemannian curvature. So, without the slightest doubt, the Weyl tensor is no less important than the Riemannian curvature tensor from geometrical point of view.
In this light and as the trace part (Riemannian curvature) has been studied in [2], it would be interesting and reasonable to study the Weyl curvature tensor (conformally invariant part) of ACR-manifold of class C12 to complete the geometrical (trace and traceless parts) picture of ACR-manifold of class C12. On the other hand, Hwang and Yun [13] studied the Weyl curvature tensor that is weakly harmonic with some conditions. Whereas, Blair and Yildirim [5] discussed the conformally flat for another class.
The paper is structured as follows. In Section 1, we recall some definitions and theorems about ACR-manifold. In Section 2, we calculate the components of the Weyl tensor on AG-structure and its relation with n-Einstein manifold as an application. Also, we focus on £-conformally flat manifold of the class C12 and the result shows it is ^-invariant Ricci tensor; then the scalar curvature is calculated explicitly. In Section 3, interesting theorems are obtained on the discussions of the contact analog of Gray identities on Riemannian curvature tensor of the class C12 and their generalization to Weyl tensor.
As a future work, the authors can develop this work in the direction of the citations [4,10,12, 18,20].
§ 1. Preliminaries
We denote by M2n+1 and g, the smooth manifold M of dimension 2n +1 and the Riemannian metric respectively.
Definition 1.1 (see [3,15]). A Riemannian manifold (M2n+1, g) is called an ACR-man-ifold if it is supplied by a structure of triple (£, n, $), where $ is a (1, 1)-tensor over M, £ is a vector field on M and n is a 1-form of M, such that V U, V E X(M), the following hold:
$(£ ) = 0; n(£) = 1; n ◦ $ = 0; $2 + id = n 0 £;
g($U, $V) + n(U)n(V) = g(U, V).
Note that X(M) is the module of all vector fields on M. On the other hand, for the background of AG-structure space, the researchers can refer to the citation [15,17]. Moreover, on AG-structure space, the tensors g and $ of ACR-manifold M2n+1 are given by the following [15]:
1 0 0 \ /00 0 M= I o O In ; ($?)= o v^TIn o ]; (1.1)
0 In O \ 0 o -V^iin
where k, l = 0,1,..., 2n and In is n x n identity matrix.
Theorem 1.1 (see [2]). The components of Riemann curvature tensor R over AG-structure space of the class C12 with dimension 2n +1 are given by:
1) R0b0 + C Cb = Cb ;
2) Raobo + Ca Cb = Cab;
3) ^=
and the other components are 0 or can be obtained by the features of R or the conjugates (i. e., R%jkl = Rto the above components, where a,b,c,d = 1,2,... ,n, a = a + n, A= = AjbC] = C[bd] = C[bd] = 0 and Ca, Ca are the components of 6th structure tensor G (see [16]).
Theorem 1.2 (see [2]). The components of Ricci tensor S over AG-structure space that coming from the class C12 with dimension 2n + 1 are provided as follows:
1) Saa + 2Ca Ca = 2Caa;
2) Saa = 0;
3) Sab + Ca Cb = Cab;
4) Sab + Ca Cb = Cba + AC
and the remaining components are set by the symmetries or conjugates to the above components. Definition 1.2 (see [19]). An ACR-manifold (M2n+1, g, $, £, n) is called
(i) £-conformally flat if W(X, £, Y, Z) = 0;
(ii) conformally symmetric if W(X, Y, Z, $U) = 0;
(iii) $—conformally flat if W ($U, $X, $Y, $Z) = 0, for all U,X,Y,Z E X(M).
Definition 1.3 (see [17]). An ACR-manifold (M2n+1, $, £, n, g) is called
(i) of class CR1 if g (R($U, $X )$Y, $Z) = g (R($2U, $2 X )$Y, $Z);
(ii) of class CR2 if
g(R($X, $Y)$Z, $U) = g(R($2X, $2Y)$Z, $U) + g(R($2X, $Y)$2Z, $U)
+ g(R($2X, $Y)$Z, $2U);
(iii) of class CR3 if g(R($U, $X )$Y, $Z) = g (R($2U, $2 X )$2Y, $2 Z),
for all X, U, Y, Z e X(M). Moreover, over AG-structure space, the aforementioned classes are equivalent to the following:
CR1 ^^ Rabcd = Rabcd = Rabcd = 0; CR2 ^^ Rabcd = Rabcd = 0; CR3 ^^ Rabcd = 0.
Definition 1.4 (see [14]). An ACR-manifold (M2n+1, g, $, £, n) is said to be q-Einstein manifold if the Ricci tensor S of M attains the following equation:
S(U, V) = a g(U, V) + P n(U) n(V) V U, V E X(M),
where a,P E C^(M), (the set of all smooth functions on M). In particular, if P = 0 then M becomes Einstein manifold.
Definition 1.5 (see [15]). An ACR-manifold (M2n+1 ,g, $,£,n) has $-invariant Ricci tensor property if it satisfies the condition:
S($U,V) + S(U, $V) = 0 V U, V E X(M).
L e m m a 1.1 (see [1]). An ACR-manifold (M2n+1, g, $, £, n)possesses $-invariant Ricci tensor, if and only if, the components Sa0, Sab and their conjugates vanish, where a, b = 1, 2,..., n.
§ 2. The geometry of Weyl tensor on class C12
In this section, we discuss the geometry of Weyl tensor on class C12 as below.
Theorem 2.1. The components of Weyl tensor on AG-structure space of the class C12 are given by:
1) WaOcO = Q - CaCc ~ № ¿"CO + <$&<:} + 2ra(2^_i) ^c ,'
2) WaQ&d = CaC — CaC° — Sac,'
3) Wabcd = ~ Sbd
4) W^ = Afc - ^{Sb(i + S<ac + ^rry^
5) W^ = - S^ 5*};
6) W&hcd = ^{S-bc 5% - Sid 5« - S-ac 6bd + SM 6,b} + ^tt) {Sbd S* ~ 5%},
and the other components are 0 or obtained by the features of W or conjugates to the above components.
Proof. Suppose that (M2n+1, g, f, n) is an ACR-manifold of class C12, then according to equation (0.1) and for each X, Y, Z, U £ X(M), we get
W(X, Y, Z, U) = g(W(Z, U)Y, X) =
= R(X, Y, Z, U) - —}—{S(Y, U)g(X, Z) 2n — 1
— S(Y, Z)g(X, U) + g(Y, U)S(X, Z) — g(Y, Z)S(X, U)}
r
+ 2n(2n-l){9(Y> U)g{X' " 9(Y> Z)9{X> U)}* So, the components of Weyl tensor over AG-structure space are given by:
Wijki = Rijki — ^—^{Sji 9tk — Sjk gu + gji Sik — gjk Su}
r
+ 2n{2n - 1) ^9jl 9ik ~ 9jk 9il^
where i, j, k, l = 0,1,..., 2n. Now, we choose i = 0, a, a, j = 0, b, b , k = 0, c, c and l = 0, d, a!, where a, b, c, d = 1, 2,..., n and a, b, c , d = n + 1, n + 2,..., 2n. If we take all possible cases of indexes i, j, k, l, then we have only six cases and their conjugates or symmetries in which Wijk1 does not vanish and these six cases are
(i, j, k,l) £ {(a, 0,c, 0), (a, 0,c , 0), (a, b, c, d), (a, b, c, d), (a, b, c , d), (a, b, c, d)}.
So, using the components of R in Theorem 1.1, components of S in Theorem 1.2 and components of g in equation (1.1) and substituting them in equation (2.1), we arrive to the results by taking into account Rjfci = Rj.w.
Corollary 2.1. If the ACR-manifold (M2n+1, g, $, f, n) is of class C12, then for all X, Y, Z, U £ X(M), we have
W(U(X),U(Y),U(Z),U(U)) = W(n(X),Ti(Y),Ti(Z),Ti(U)) = 0,
where II = + and IT = |(-$2 +
Proof. Since on AG-structure space the following are equivalent:
W(n(X), n(Y), n(Z), n(U)) ^ Wa5Cd; a, b, c,d =1, 2,..., n, W(n(X),Ti(Y),Ti(Z),Ti(U)) d,b,c,d = n+l,n + 2,...,2n.
Then Theorem 2.1 gives the results.
Theorem 2.2. If the ACR-manifold (M2n+1,g, $,£,n) of class C12 having n > 2, is conformally flat then it is q-Einstein manifold with a = <¿4 (¿oo _ and ¡3 = S00 — a.
Proof. Suppose that M is conformally flat, then Wijk1 = 0 for all i, j, k, l = 0,1,..., 2n. Then we get from Theorem 2.1 that W„0c0 = 0, and Wabc(| = 0. Then replacing c by b in the first and contracting (c, d) in the second we get respectively the following:
Sat = (2n - 1 ){C6" - CaCb} - 5ab Soo + —Sab. (2.2)
b b 2n b
r
2S~ab = (2n - 1 + —5ab. (2.3)
Now, adding equations (2.2), (2.3), using the fact ACC = AC£ and then from Theorem 1.2, item 4, we obtain S~ab = ^(Soo - , if n > 2. But, Wa0eo = 0 gives = (2n - 1 ){Cac - CaCc} and from Theorem 1.2, item 3, we have S„c = 0. Also, we note that the remaining items of Theorem 2.1 satisfy the previous results. But, Definition 1.4 leads M to be an n-Einstein manifold having S0o = a + /3 and a = ¿4 (¿>00 -
Corollary 2.2. If the ACR-manifold (M2n+1 , g, $,£,n) of the class C12 with n > 2 is conformally flat then it is Einstein manifold with scalar curvature r = —n(2n — 5)S00.
Proof. If M is conformally flat, then M is n-Einstein manifold according to Theorem 2.2. But if ¡3 = 0, then M is Einstein manifold with S^o = a = ¿00 — and this implies that the value of r is given.
Corollary 2.3. If the ACR-manifold (M2n+1, g, $, n) of the class C12 is conformally flat then it has $-invariant Ricci tensor.
Proof. If M is conformally flat then Theorems 1.2 and 2.2 yields Sc0 = Sab = 0. But, Lemma 1. 1 attains the requirement.
Corollary 2.4. If the ACR-manifold (M5, g, $, n) of class C12 is conformally flat then
r = 2S00 and Aacca = 0.
Proof. Suppose M5 is conformally flat, then regarding the proof of Theorem 2.2, we have r = 25*00• But r = 2,S',,„ + ,S'()0 5aa = ¿.S'oo- Thus, contracting item 4, in Theorem 1.2, we get A^ = 0.
Theorem 2.3. The ACR-manifold (M2n+1, g, $, n) of class C12 is conformally symmetric, if and only if, it is conformally flat.
Proof. If M is conformally flat, then W(X, Y, Z, U) = 0 for all X, Y, U, Z G X(M). So, if we replace U by $U, then also we have W(X, Y, Z, $U) = 0 and this implies that M is conformally symmetric according to Definition 1.2.
Conversely, suppose that M is conformally symmetric. Then according to Definition 1.2, we have
W(X,Y,Z, $U) = 0 V X,Y,Z,U G X(M), Wijkt Xi Yj Zk U1 = 0 i, j, k, l, t = 0,1,..., 2n,
Wijkt = 0.
Considering the components of $ in equation (1.1), we get:
Wl3kd = Wijkd = 0; d =1, 2,..., n, d = d + n.
Now, ^om the symmetries of W, we obtain that all the components of W in Theorem 2.1 must be vanishing. Thus M is conformally flat.
Remark 2.1. Regarding Theorem 2.3, we discover the previous theorems and corollaries are stay valid if we replace the statement "conformally flat" by "conformally symmetric".
Theorem 2.4. If the ACR-manifold (M2n+1, g, $, £, q) of the class C12 is £-conformally flat with Afc = y and 7 G C^(M) then it is q-Einstein manifold, with
1 r
a = ^-^{5*00 + (2n - 1)7 - —}, (3 = S0o-a.
Proof. Suppose that M is £-conformally flat with A^ = 7 then we pay attention to Definition 1.2 and obtain W(X, £, Y, Z) = 0. Thus, on AG-structure space, we have Wi0jk = 0, where i, j, k = 0,1,..., 2n. Taking into account Theorem 2.1, we get Wa0c0 = Wa0a0 = 0, and this implies that S~ac = 0 and S~ac = (2n - 1 ){Cca - CaCc} - 5ac S00 + Moreover, from
Theorem 1.2, we have S&c = (2n - l)^ - Afc} - 5ac S00 + Therefore,
r
(2 n - 2) Sac = (2 n - 1 )A£ + {¿"oo - —K-
Since Aab = 7 , then M is q-Einstein manifold, with
1r
a = oo + (2n-1)7-—}, /3 = Soo-a-
Corollary 2.5. If the £-conformally flat ACR-manifold (M2n+1, $, £, q, g) of class C12 with Afb = y #a> is Einstein manifold, where 7 G C^ (M), then the scalar curvature
r = —2n{(2n - 3)S00 - (2n - 1)7}.
Proof. Suppose M is Einstein manifold, then from Theorem 2.4, we have P = 0 and this implies that S^o = a = -¿^{¿oo + (2n — 1)7 — So, the last equation gives:
r = —2n{(2n — 3)S00 — (2n — 1)7}.
Corollary 2.6. Every £-conformally flat ACR-manifold (M2n+1, g, $, £, q) of class C12 has $-invariant Ricci tensor.
Proof. Suppose that M is £-conformally flat, then Wa0c0 = Wi0i0i = 0. Thus, from Theorems 1.2 and 2.1, we deduce that Sa0 = Sab = 0. Then we establish the desired.
Theorem 2.5. If the ACR-manifold (M2n+1, $, £, q, g) of class C12 is $—conformally flat with A^ = 7 8%, then it is q-Einstein manifold with a = ^ + (2n — and ¡3 = Soo — ol.
Proof. Suppose that M is $-conformally flat, then from Definition 1.2, we have
W ($X, $Y, $Z, )=0 V X,Y,Z,U G X (M),
Wijki ($X )i ($Y )j ($Z )k ($U ) = 0, i, j, k, l = 0,1,..., 2n,
Wijki $j2 $k3 $t4 =0, t1, t2, is, t4 = 0,1,..., 2n.
According to the above equations and the components of $ in equation (1.1), we establish Wijk1 = 0 for i, j, k, / = 1, 2,..., 2n. Regarding Theorem 2.1, we acquire
W«6cd = Wabcd = Wabad = Wabcd = 0.
Regarding the proof of Theorem 2.2, we attain Sab = 0, and 2Sab = (2n — 1 )Aacl + 7^8%. Since Act, = 7 then M is q-Einstein manifold having a = + (2n — 1)^, and f3 = S0o — ol.
Corollary 2.7. If the Q-conformally flat ACR-manifold (M2n+1, g, Q, f, n) of class C12 with ACC = 7 ^C, then it is Einstein manifold with the scalar curvature r = 4nS00 — 2n(2n — 1)7.
Proof. Suppose that M is Q-conformally flat. The consideration of Theorem 2.5 gives M to be Einstein manifold if /3 = 0, and then S^o = a = ^ + Thus, r = inSoo — 2n(2n—1)7.
Corollary 2.8. If the ACR-manifold (M2n+1, g, Q, f, n) of class C12 is Q-conformally flat, then it possesses Q-invariant Ricci tensor.
Proof. Suppose that M is Q-conformally flat. Then the proof of Theorem 2.5 gives Sab = 0, and from Theorem 1.2 we have Sc0 = 0. Then Lemma 1.1 produces the claim of this corollary.
Corollary 2.9. The ACR-manifold (M2n+1, g, Q, f, n) of class C12 is conformally flat, if and only if, it is f-conformally flat and Q-conformally flat.
Proof. The assertion of this corollary is achieved from Theorems 2.1, 2.2, 2.4, and 2.5.
§ 3. The contact analogs of Gray identities on class C12
In this section, we discuss the contact analogs of Gray identities on the Riemannian curvature tensor of the class C12 and their generalization to Weyl tensor.
Theorem 3.1. The classes CR1, CR2, and CR3 are equivalent on the ACR-manifold M of class C12.
Proof. Suppose that M is ACR-manifold of class C12. Then under Theorem 1.1, and Definition 1.3, we have
Rabcd = Rabcd = Rabcd = M G CR1;
RCbcd = Rabcd = 0. M G CR2;
Rabcd = 0. M G CR3. Then the classes CR1, CR2, and CR3 are equivalent on M.
D e f i n i t i o n 3.1. An ACR-manifold (M2n+1, g, Q, f, n) is called
(i) of class CW1 if g(W (QU, QX )QY, QZ) = g(W (Q2U, Q2X )QY, QZ);
(ii) of class CW2 if
g(W(QX, QY)QZ, QU) = g(W(Q2X, Q2Y)QZ, QU) + g(W(Q2X, QY)Q2Z, QU)
+ g(W(Q2X, QY)QZ, Q2U);
(iii) of class CW3 if g(W (QX, QY )QZ, QU) = g(W (Q2X, Q2Y )Q2Z, Q2U), for all X,Y,Z,U G X(M).
Now, since the Weyl tensor has the same properties as the Riemann curvature tensor, then from Definition 1.3, we get the following lemma.
L e m m a 3.1. On AG-structure space, the above classes are equivalent to the following:
CW1 ^ Wabcd = Wabcd = Wabcd = 0;
CW2 ^ Wabcd = Wabcd = 0;
CW3 ^ Wabcd = 0.
Interesting relations with n-Einstein manifolds and Q-invariant Ricci tensor are given in the following theorems.
Theorem 3.2. If the ACR-manifold (M2n+1, g, Q, f, n) of class C12 belongs to the class
1 r Q r •
2^41^00 ~ n-
n > 2.
CW\, then it is rj-Einstein manifold with a = t-zt{Soo — -} and ¡3 = Soo — ol, provided that
Proof. Suppose that M G C12 and M G CW1, then ^om Lemma 3.1, we have Wabcd = = Wabcd = Wabcd = 0. According to Theorem 2.1, we get:
0 = 2n _ $d ~ Sbd
1r 0 = 2n- l^Sbc ^ ~ S'bd ~ S&c ^ + S&d ^ + 2n{2n- 1)^ ~ ^
Contracting the above equations with respect to the indexes (a, d), we obtain:
Sbc = 0,
C _ 1 fr(n-l) b
Since r = 2Saa + Soo, then M is ^-Einstein manifold having a = oo — and f3 = Soo — a.
Corollary 3.1. If the ACR-manifold (M2n+1 ,g, Q,f,n), having n > 2 belongs to the classes C12 and CW1 then it is Einstein manifold with r = —n(2n — 5)S00.
Proof. Using Theorem 3.2, we conclude that M is Einstein manifold if P = 0, and then 5oo = ol = z^isoo — Thus, we obtain the result.
Corollary 3.2. If the ACR-manifold (M2n+1 ,g, Q,f,n), having n > 2 belongs to the classes C12 and CW1 then it possesses Q-invariant Ricci tensor.
Proof. According to the proof of Theorem 3.2, we attain the claim of this corollary.
Theorem 3.3. If the ACR-manifold (M5, g, Q, f, n) belongs to the classes C12 and CW1, then it possesses Q-invariant Ricci tensor and r = 2S00.
Proof. Consider M G C12 and M G CW1, then with the proof of Theorem 3.2, we have Sab = 0 and
r = 4Saa. r = 2r — 2S00. r = 2S00. So, this completes the proof.
Corollary 3.3. If the ACR-manifold (M5, g, Q, f, n) belongs to the classes C12 and CW1, then AcC = 0.
Proof. Suppose that M G C12 and M G CW1. Note that r = 2SdM + S00. Applying Theorem 3.3, we have Saa = |Soo- So, by using item 4 of Theorem 1.2, we obtain the result.
Theorem 3.4. The ACR-manifold (M2n+1, g, Q, f, n) of class C12 belongs to the class CW2, if and only if, it has Q-invariant Ricci tensor.
Proof. Consider M G C12 and M G CW2, then Wabcd = Wabcd = 0 and this implies that Sab = 0 according to Theorem 2.1. Thus M has Q-invariant Ricci tensor according to the combination of Theorem 1.2, Lemma 1.1, and the consequence obtained. Conversely, if M has Q-invariant Ricci tensor, then Sa0 = Sab = 0. Thus, combining the previous result with the Theorem 2.1, we have Wabcd = Wabcd = 0. So, M belongs to the class CW2.
Theorem 3.5. The ACR-manifold (M2n+1, g, n) of class C12 belongs to the class CW3, if and only if, it possesses $-invariant Ricci tensor.
Proof. Consider M G C12 and M G CW3, then Wâbcd = 0 and this implies that Sab = 0 under Theorem 2.1. Thus M has ^-invariant Ricci tensor according to the combination of Theorem 1.2, Lemma 1.1, and the resulting consequence. Conversely, if M has ^-invariant Ricci tensor, then we apply Lemma 1.1 and get Sa0 = Sab = 0. So, Theorem 2.1, item 3, yields Wabcd = 0. Thus, we conclude the implication of this theorem.
REFERENCES
1. Abood H. M., Abass M. Y. A study of new class of almost contact metric manifolds of Kenmotsu type, Tamkang Journal of Mathematics, 2021, vol. 52, no. 2, pp. 253-266. https://doi.org/10.5556/j.tkjm.52.2021.3276
2. Abass M.Y. Geometry of certain curvature tensors of almost contact metric manifold, PhD thesis, College of Education for Pure Sciences, University of Basrah, 2020. https://faculty.uobasrah.edu.iq/uploads/publications/1620305696.pdf
3. Abu-Saleem A., Rustanov A. R., Kharitonova S.V. Axiom of $-holomorphic (2r + 1)-planes for generalized Kenmotsu manifolds, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2020, no. 66, pp. 5-23 (in Russian). https://doi.org/10.17223/19988621/66/1
4. Alegre P., Fernandez L. M., Prieto-Martin A. A new class of metric f -manifolds, Carpathian Journal of Mathematics, 2018, vol. 34, no. 2, pp. 123-134. https://www.jstor.org/stable/26898721
5. Blair D. E., Yildirim H. On conformally flat almost contact metric manifolds, Mediterranean Journal of Mathematics, 2016, vol. 13, issue 5, pp. 2759-2770. https://doi.org/10.1007/s00009-015-0652-x
6. Bouzir H., Beldjilali G., Bayour B. On three dimensional C12-manifolds, Mediterranean Journal of Mathematics, 2021, vol. 18, issue 6, article number: 239. https://doi.org/10.1007/s00009-021-01921-3
7. de Candia S., Falcitelli M. Even-dimensional slant submanifolds of a C5 ® C12-manifold, Mediterranean Journal of Mathematics, 2017, vol. 14, issue 6, article number: 224.
https://doi. org/10.1007/s00009-017-1022-7
8. de Candia S., Falcitelli M. Curvature of C5 ® C12-manifold, Mediterranean Journal of Mathematics, 2019, vol. 16, issue 4, article number: 105. https://doi.org/10.1007/s00009-019-1382-2
9. Chinea D., Gonzalez C. A classification of almost contact metric manifolds, Annali di Matematica Pura edApplicata, 1990, vol. 156, no. 1, pp. 15-36. https://doi.org/10.1007/BF01766972
10. Chojnacka-Dulas J., Deszcz R., Glogowska M., Prvanovic M. On warped product manifolds satisfying some curvature conditions, Journal of Geometry and Physics, 2013, vol. 74, pp. 328-341. https://doi.org/10.1016/j.geomphys.2013.08.007
11. De U.C., Suh Y.J. On weakly semiconformally symmetric manifolds, Acta Mathematica Hungarica, 2019, vol. 157, issue 2, pp. 503-521. https://doi.org/10.1007/s10474-018-0879-7
12. Deszcz R., Glogowska M., Jelowicki J., Zafindratafa G. Curvature properties of some class of warped product manifolds, International Journal of Geometric Methods in Modern Physics, 2016, vol. 13, no. 01, 1550135. https://doi.org/10.1142/S0219887815501352
13. Hwang S., Yun G. Ridigity of Ricci solitons with weakly harmonic Weyl tensors, Mathematische Nachrichten, 2018, vol. 291, issue 5-6, pp. 897-907. https://doi.org/10.1002/mana.201600285
14. Kenmotsu K. A class of almost contact Riemannian manifolds, Tohoku Mathematical Journal, 1972, vol. 24, issue 1, pp. 93-103. https://doi.org/10.2748/tmj/1178241594
15. Kirichenko V. F. Differentsial'no-geometricheskie struktury na mnogoobraziyakh (Differential-geometric structures on manifolds), Odessa: Pechatnyi Dom, 2013.
16. Kirichenko V. F., Dondukova N.N. Contactly geodesic transformations of almost-contact metric structures, Mathematical Notes, 2006, vol. 80, no. 1-2, pp. 204-213. https://doi.org/10.1007/s11006-006-0129-0
17. Kirichenko V. F., Kusova E. V. On geometry of weakly cosymplectic manifolds, Journal of Mathematical Sciences, 2011, vol. 177, no. 5, pp. 668-674. https://doi.org/10.1007/s10958-011-0494-4
18. Rustanov A. R., Polkina E.A., Kharitonova S.V. Projective invariants of almost C(A)-manifolds, Annals of Global Analysis and Geometry, 2022, vol. 61, issue 2, pp. 459-467. https://doi.org/10.1007/s10455-021-09818-w
19. Venkatesha, Naik D. M., Kumara H.A. Conformal curvature tensor on paracontact metric manifolds, Matematicki Vesnik, 2020, vol. 72, no. 3, pp. 215-225. http://www.vesnik.math.rs/vol/mv20304.pdf
20. Wang Y., Wang W. Curvature properties of almost Kenmotsu manifolds with generalized nullity conditions, Filomat, 2016, vol. 30, no. 14, pp. 3807-3816. https://doi.org/10.2298/FIL1614807W
Received 11.01.2022 Accepted 25.04.2022
Mohammed Yousif Abass, Doctor of Mathematics, Lecturer, Department of Mathematics, College of Science, University of Basrah, Basrah, Iraq. ORCID: https://orcid.org/0000-0003-1095-9963 E-mail: [email protected]
Qusay S. A. Al-Zamil, Doctor of Mathematics, Associate Professor, Department of Mathematics, College of Science, University of Basrah, Basrah, Iraq. ORCID: https://orcid.org/0000-0003-0888-638X E-mail: [email protected]
Citation: M. Y. Abass, Q. S. A. Al-Zamil. On Weyl tensor of ACR-manifolds of class C\2 with applications, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2022, vol. 59, pp. 3-14.
М. Ю. Абасс, К. С. А. Аль-Замиль
О тензоре Вейля ACR-многообразий класса Ci2 с приложениями
Ключевые слова: почти контактное метрическое многообразие класса C12, n-эйнштейновское многообразие, тензор Вейля.
УДК: 514.77
DOI: 10.35634/2226-3594-2022-59-01
В данной работе мы определяем компоненты тензора Вейля почти контактного метрического (ACR-) многообразия класса C12 на ассоциированном пространстве G-структуры (AG-структуры). В качестве приложения мы доказываем, что конформно плоское ACR-многообразие класса C12 с n > 2 является n-эйнштейновским многообразием и заключаем, что это эйнштейновское многообразие такое, что скалярная кривизна r обеспечена. Также в явном виде обсуждается случай, когда n = 2. Более того, здесь широко рассмотрены отношения между конформно плоским, конформно симметричным, £-конформно плоским и Ф-инвариантным тензором Риччи, и поэтому мы определяем значение скалярной кривизны r в явном виде с другими приложениями. Наконец, мы определяем новые классы с тождествами, аналогичными тождествам Грея, и обсуждаем их связь с классом C12 ACR-многообразий.
СПИСОК ЛИТЕРАТУРЫ
1. Abood H. M., Abass M. Y. A study of new class of almost contact metric manifolds of Kenmotsu type, Tamkang Journal of Mathematics, 2021, vol. 52, no. 2, pp. 253-266. https://doi.org/10.5556/j.tkjm.52.2021.3276
2. Abass M. Y. Geometry of certain curvature tensors of almost contact metric manifold, PhD thesis, College of Education for Pure Sciences, University of Basrah, 2020. https://faculty.uobasrah.edu.iq/uploads/publications/1620305696.pdf
3. Абу-Салеем А., Рустанов А. Р., Харитонова С. В. Аксиома Ф-голоморфных (2r + 1)-плоскостей для обобщенных многообразий Кенмоцу // Вестник Томского государственного университета. Математика и механика. 2020. № 66. С. 5-23. https://doi.org/10.17223/19988621/66/1
4. Alegre P., Fernandez L. M., Prieto-Martin A. A new class of metric f -manifolds, Carpathian Journal of Mathematics, 2018, vol. 34, no. 2, pp. 123-134. https://www.jstor.org/stable/26898721
5. Blair D. E., Yildirim H. On conformally flat almost contact metric manifolds, Mediterranean Journal of Mathematics, 2016, vol. 13, issue 5, pp. 2759-2770. https://doi.org/10.1007/s00009-015-0652-x
6. Bouzir H., Beldjilali G., Bayour B. On three dimensional C12-manifolds, Mediterranean Journal of Mathematics, 2021, vol. 18, issue 6, article number: 239. https://doi.org/10.1007/s00009-021-01921-3
7. de Candia S., Falcitelli M. Even-dimensional slant submanifolds of a C5 ф C12-manifold, Mediterranean Journal of Mathematics, 2017, vol. 14, issue 6, article number: 224. https://doi.org/10.1007/s00009-017-1022-7
8. de Candia S., Falcitelli M. Curvature of C5 ф C12-manifold, Mediterranean Journal of Mathematics, 2019, vol. 16, issue 4, article number: 105. https://doi.org/10.1007/s00009-019-1382-2
9. Chinea D., Gonzalez C. A classification of almost contact metric manifolds, Annali di Matematica Pura edApplicata, 1990, vol. 156, no. 1, pp. 15-36. https://doi.org/10.1007/BF01766972
10. Chojnacka-Dulas J., Deszcz R., Glogowska M., Prvanovic M. On warped product manifolds satisfying some curvature conditions, Journal of Geometry and Physics, 2013, vol. 74, pp. 328-341. https://doi.org/10.1016/j.geomphys.2013.08.007
11. De U.C., Suh Y.J. On weakly semiconformally symmetric manifolds, Acta Mathematica Hungarica, 2019, vol. 157, issue 2, pp. 503-521. https://doi.org/10.1007/s10474-018-0879-7
12. Deszcz R., Glogowska M., Jelowicki J., Zafindratafa G. Curvature properties of some class of warped product manifolds, International Journal of Geometric Methods in Modern Physics, 2016, vol. 13, no. 01, 1550135. https://doi.org/10.1142/S0219887815501352
13. Hwang S., Yun G. Ridigity of Ricci solitons with weakly harmonic Weyl tensors, Mathematische Nachrichten, 2018, vol. 291, issue 5-6, pp. 897-907. https://doi.org/10.1002/mana.201600285
14. Kenmotsu K. A class of almost contact Riemannian manifolds, Tohoku Mathematical Journal, 1972, vol. 24, issue 1, pp. 93-103. https://doi.org/10.2748/tmj/1178241594
15. Кириченко В.Ф. Дифференциально-геометрические структуры на многообразиях. Одесса: Печатный дом, 2013.
16. Кириченко В.Ф., Дондукова Н. Н. Контактно геодезические преобразования почти контактных метрических структур // Математические заметки. 2006. Т. 80. Вып 2. С. 209-219. https://doi.org/10.4213/mzm2802
17. Кириченко В.Ф., Кусова Е.В. О геометрии слабо косимплектических многообразий // Фундаментальная и прикладная математика. 2010. Т. 16. Вып. 2. С. 33-42. http://mi.mathnet.ru/fpm1304
18. Rustanov A. R., Polkina E.A., Kharitonova S.V. Projective invariants of almost C(A)-manifolds, Annals of Global Analysis and Geometry, 2022, vol. 61, issue 2, pp. 459-467. https://doi.org/10.1007/s10455-021-09818-w
19. Venkatesha, Naik D. M., Kumara H.A. Conformal curvature tensor on paracontact metric manifolds, Matematicki Vesnik, 2020, vol. 72, no. 3, pp. 215-225. http://www.vesnik.math.rs/vol/mv20304.pdf
20. Wang Y., Wang W. Curvature properties of almost Kenmotsu manifolds with generalized nullity conditions, Filomat, 2016, vol. 30, no. 14, pp. 3807-3816. https://doi.org/10.2298/FIL1614807W
Поступила в редакцию 11.01.2022 Принята в печать 25.04.2022
Абасс Мохаммед Юсиф, д. м. н., преподаватель, кафедра математики, Научный колледж, Университет Басры, Басра, Ирак.
ORCID: https://orcid.org/0000-0003-1095-9963 E-mail: [email protected]
Аль-Замиль Кусай С. А., д. м. н., доцент, кафедра математики, Научный колледж, Университет Басры, Басра, Ирак.
ORCID: https://orcid.org/0000-0003-0888-638X E-mail: [email protected]
Цитирование: М. Ю. Абасс, К. С. А. Аль-Замиль. О тензоре Вейля ACR-многообразий класса C\2 с приложениями // Известия Института математики и информатики Удмуртского государственного университета. 2022. Т. 59. С. 3-14.
