Ricci ϕ-invariance on almost cosymplectic three-manifolds

Quanxiang Pan

doi:10.1515/math-2023-0156

Article Open Access

Ricci ϕ-invariance on almost cosymplectic three-manifolds

Quanxiang Pan

Published/Copyright: November 28, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

Let M 3 be a strictly almost cosymplectic three-manifold whose Ricci operator is weakly ϕ -invariant. In this article, it is proved that Ricci curvatures of M 3 are invariant along the Reeb flow if and only if M 3 is locally isometric to the Lie group E ( 1 , 1 ) of rigid motions of the Minkowski 2-space equipped with a left-invariant almost cosymplectic structure.

Keywords: almost cosymplectic three-manifolds; Ricci ϕ-invariance; Lie group

MSC 2010: 53D15; 53C25

1 Introduction

The classification of an almost contact metric manifold under certain meaningful geometric condition is one of the core problems in almost contact Riemannian geometry. Let ( M , ϕ , ξ , η , g ) be an almost contact metric manifold of dimension 2 n + 1 , n ≥ 1 , and let Q be the Ricci operator of M related with the Ricci tensor Ric by g ( Q X , Y ) = Ric ( X , Y ) for any vector fields X and Y . The Ricci operator is said to be (strongly) ϕ -invariant (see [1]) if

(1.1) g ( Q ϕ X , ϕ Y ) = g ( Q X , Y )

for any vector fields X , Y . In view of the skew-symmetry of ϕ , it is easy to see that if equation (1.1) is valid, then we have

(1.2) Q ϕ = ϕ Q .

In literature, equation (1.2) was investigated by many authors from various points of view. It was proved in [2, Theorem 3.3] that if equation (1.2) on a contact metric three-manifold is valid, then the manifold must be either Sasakian, flat, or of constant Reeb sectional curvature k < 1 and constant ϕ -sectional curvature − k . In fact, from earlier literature [3], one knows that equation (1.1) is necessarily valid on any Sasakian manifold. Some recent results regarding equation (1.2) on a contact metric three-manifold can be seen in [4,5]. Moreover, it is known from [6–8] that equation (1.2) is also necessarily true in a Kenmotsu or cosymplectic manifold. Cho [9] studied almost cosymplectic three-manifolds satisfying equation (1.2).

There exists a natural almost contact metric structure on a real hypersurface in a complex space form. It was proved in [10, Main theorem] that on a three-dimensional real hypersurface in a nonflat complex space form, equation (1.2) is valid if and only if the real hypersurface is pseudo-Einstein. Real hypersurfaces of dimension > 3 in a complex projective and hyperbolic space satisfying equation (1.2) were classified in in the studies by Kimura [11] and Ki and Suh [12], respectively. We remak that the two kinds of classification theorem are different from each other. For example, a three-dimensional real hypersurface satisfying equation (1.2) may not be homogeneous, but a real hypersurface in C P n or C H n , n ≥ 3 , satisfying equation (1.2) must be Hopf and homogeneous.

The Ricci operator Q of an almost contact metric manifold is said to be weakly ϕ -invariant [1] if

(1.3) g ( Q ϕ X , ϕ Y ) = g ( Q X , Y )

for any vector fields X and Y orthogonal to the Reeb vector field ξ . Such an equality is much weaker than both equations (1.1) and (1.2), and it is more adapted to an almost contact metric structure. Equality (1.3) has received many attentions from some authors in differential geometry of real hypersurfaces. For example, type ( A ) and type ( B ) real hypersurfaces in a nonflat complex space form were characterized in [13,14] by means of equation (1.3). Here we remark that in geometry of real hypersurfaces, (1.3) often appeared as the form of

(1.4) g ( ( Q ϕ − ϕ Q ) X , Y ) = 0

for any vector fields X and Y orthogonal to the Reeb vector field (see [10,13–15]).

As introduced before, equation (1.3) has been studied by many authors from a viewpoint of real hypersurfaces, so it is very interesting to investigate equation (1.3) from a viewpoint of contact geometry. In this article, we investigate weak ϕ -invariance of Ricci operators on an almost cosymplectic three-manifold. We prove the equivalence of equations (1.2) and (1.3) under certain restriction. But, they are not equivalent in general situations. Finally, we present some concrete examples of almost cosymplectic three-manifolds to verify that the restriction we have employed is essential.

2 Preliminaries

An almost contact metric (Riemannian) manifold is defined as a Riemannian manifold ( M , g ) of dimension 2 n + 1 , n ≥ 1 , on which there is a quadruple ( ϕ , ξ , η , g ) satisfying

(2.1) ϕ 2 = − id + η ⊗ ξ , η ∘ ϕ = 0 , η ( ξ ) = 1 ,

(2.2) g ( ϕ X , ϕ Y ) = g ( X , Y ) − η ( X ) η ( Y ) , η ( X ) = g ( X , ξ )

for any vector fields X and Y , where ϕ is a ( 1 , 1 ) -type tensor field, ξ is a vector field called the Reeb vector field, η is a global 1-form called the almost contact 1-form, and g is a Riemannian metric [3]. By an almost cosymplectic manifold, we mean an almost contact metric manifold on which there holds d η = 0 and d Φ = 0 , where Φ is the fundamental 2-form defined by Φ ( X , Y ) = g ( X , ϕ Y ) [3,16].

On the product M 2 n + 1 × R of an almost contact metric manifold M 2 n + 1 and R , there is an almost complex structure J defined by

J X , f d d t = ϕ X − f ξ , η ( X ) d d t ,

where X denotes a vector field tangent to M 2 n + 1 , t is the coordinate of R , and f is a C ∞ -function on M 2 n + 1 × R . The almost contact metric manifold is said to be normal if J is integrable, or equivalently,

[ ϕ , ϕ ] = − 2 d η ⊗ ξ ,

where [ ϕ , ϕ ] is the Nijenhuis tensor of ϕ . A normal almost cosymplectic manifold is said to be a cosymplectic manifold (see [3]). An almost cosymplectic manifold is a cosymplectic manifold if and only if

(2.3) ∇ ϕ = 0 .

On an almost cosymplectic manifold, there exist three tensor fields, which are defined by l = R ( ⋅ , ξ ) ξ , h = 1 2 ℒ ξ ϕ , and h ′ = h ∘ ϕ , where R is the Riemannian curvature tensor of g and ℒ is the Lie differentiation. From [3,16], we know that the three ( 1 , 1 ) -type tensor fields l , h ′ , and h are symmetric and satisfy

(2.4) h ξ = 0 , l ξ = 0 , tr h = 0 , tr ( h ′ ) = 0 , h ϕ + ϕ h = 0 ,

and

(2.5) ∇ ξ = h ′ .

For some recent articles concerning almost cosymplectic three-manifolds we refer the reader to [17,18]. In this article, all manifolds are assumed to be smooth and connected.

3 Main Theorems and Proofs

As introduced before, condition (1.2) on almost cosymplectic manifolds has been studied by Cho [9]. Therefore, in this section, we consider only the Ricci weak ϕ -invariance on an almost cosymplectic three-manifold. From now on, let us suppose that M 3 is an almost cosymplectic three-manifold. It is known that an almost cosymplectic manifold of dimension three is cosymplectic if and only if h = 0 identically. Following Perrone [19], let U 1 be the open subset of M 3 on which h ≠ 0 and U 2 be the open subset defined by U 2 = { p ∈ M 3 : h = 0 in a neighborhood of p } . Therefore, U 1 ∪ U 2 is an open and dense subset of M 3 . For every point p ∈ U 1 ∪ U 2 , we find a local orthonormal basis { ξ , e 1 , e 2 = ϕ e 1 } of three distinct unit eigenvector fields of h in certain neighborhood of p . On U 1 , we assume that h e 1 = λ e 1 , and hence h e 2 = − λ e 2 , where λ is assumed to be a positive function. Note that λ is continuous on M 3 and smooth on U 1 ∪ U 2 . The following lemma is well known when studying almost cosymplectic three-manifolds (see, e.g., [19–22]).

Lemma 3.1

On U 1 , we have

∇ ξ e 1 = a e 2 , ∇ ξ e 2 = − a e 1 , ∇ e 1 ξ = − λ e 2 , ∇ e 2 ξ = − λ e 1 , ∇ e 1 e 1 = 1 2 λ ( e 2 ( λ ) + σ ( e 1 ) ) e 2 , ∇ e 2 e 2 = 1 2 λ ( e 1 ( λ ) + σ ( e 2 ) ) e 1 , ∇ e 2 e 1 = λ ξ − 1 2 λ ( e 1 ( λ ) + σ ( e 2 ) ) e 2 , ∇ e 1 e 2 = λ ξ − 1 2 λ ( e 2 ( λ ) + σ ( e 1 ) ) e 1 ,

where a is a smooth function, σ is the 1-form defined by σ ( ⋅ ) = S ( ⋅ , ξ ) , and S is the Ricci tensor.

Applying Lemma 3.1, we obtain the Ricci operator Q associated with the Ricci tensor given by

(3.1) Q = α id + β η ⊗ ξ + ϕ ∇ ξ h − σ ( ϕ 2 ) ⊗ ξ + σ ( e 1 ) η ⊗ e 1 + σ ( e 2 ) η ⊗ e 2 ,

where α = 1 2 ( r + tr ( h 2 ) ) , β = − 1 2 ( r + 3 tr ( h 2 ) ) , and r is the scalar curvature. From equation (3.1), we write the Ricci operator as follows:

(3.2) Q ξ = − 2 λ 2 ξ + σ ( e 1 ) e 1 + σ ( e 2 ) e 2 , Q e 1 = σ ( e 1 ) ξ + 1 2 ( r + 2 λ 2 − 4 λ a ) e 1 + ξ ( λ ) e 2 , Q e 2 = σ ( e 2 ) ξ + ξ ( λ ) e 1 + 1 2 ( r + 2 λ 2 + 4 λ a ) e 2 ,

with respect to the local orthonormal basis { ξ , e 1 , e 2 } .

Lemma 3.2

On U 1 , if the Ricci operator is weakly ϕ -invariant, then we have

(3.3) e 1 ( λ 2 ) + ξ ( σ ( e 1 ) ) − λ σ ( e 2 ) = 0 ,

(3.4) e 2 ( λ 2 ) + ξ ( σ ( e 2 ) ) − λ σ ( e 1 ) = 0 .

Proof

If the Ricci operator is weakly ϕ -invariant, by the skew-symmetry of ϕ and equation (2.1), then from equation (1.3), we have

ϕ Q ϕ X + Q X − η ( Q X ) ξ = 0

for any vector field X orthogonal to ξ . With the help of equation (2.1), the action of ϕ on the above equality gives

(3.5) Q ϕ X − ϕ Q X − η ( Q ϕ X ) ξ = 0

for any vector field X orthogonal to ξ . With the help of equation (3.2), we obtain

Q e 2 − ϕ Q e 1 − η ( Q e 2 ) ξ = 2 ξ ( λ ) e 1 + 4 a λ e 2 ,

Q e 1 + ϕ Q e 2 − η ( Q e 1 ) ξ = 4 a λ e 1 + 2 ξ ( λ ) e 2 .

So, if the Ricci operator is weakly ϕ -invariant, then from equation (3.5) and the above two equalities, we obtain

(3.6) ξ ( λ ) = 0 and a = 0 ,

where we remark that λ is assumed to be a positive function on U 1 . With the help of equation (3.6), according to Lemma 3.1, we have

(3.7) [ ξ , e 1 ] = λ e 2 ,

(3.8) [ ξ , e 2 ] = λ e 1 ,

and

(3.9) [ e 1 , e 2 ] = − 1 2 λ ( e 2 ( λ ) + σ ( e 1 ) ) e 1 + 1 2 λ ( e 1 ( λ ) + σ ( e 2 ) ) e 2 .

Consequently, substituting equations (3.7)–(3.9) into the well-known Jacobi identity

[ ξ , [ e 1 , e 2 ] ] + [ e 1 , [ e 2 , ξ ] ] + [ e 2 , [ ξ , e 1 ] ] = 0 ,

we obtain

(3.10) e 1 ( λ ) + 1 2 λ ξ ( e 2 ( λ ) + σ ( e 1 ) ) − 1 2 ( e 1 ( λ ) + σ ( e 2 ) ) = 0

and

(3.11) e 2 ( λ ) + 1 2 λ ξ ( e 1 ( λ ) + σ ( e 2 ) ) − 1 2 ( e 2 ( λ ) + σ ( e 1 ) ) = 0 ,

where we used the first term of equation (3.6). In view of this, from equations (3.7) and (3.8), we obtain

(3.12) ξ ( e 1 ( λ ) ) = λ e 2 ( λ )

and

(3.13) ξ ( e 2 ( λ ) ) = λ e 1 ( λ ) ,

respectively. Substituting the above two equalities into equations (3.10) and (3.11), we obtain equations (3.3) and (3.4), respectively.□

From the proof of Lemma 3.2, the following corollary is valid.

Corollary 3.3

On an almost cosymplectic three-manifold whose Reeb vector field is an eigenvector field of the Ricci operator with eigenvalue zero, the Ricci operator is weakly ϕ -invariant if and only if it is (strongly) ϕ -invariant.

In order to give a solution of equations (3.3) and (3.4), we next have to employ a restriction.

Lemma 3.4

On U 1 , if the Ricci operator is weakly ϕ -invariant and Ricci curvatures are invariant along the Reeb flow, then the manifold is locally isometric to the group E ( 1 , 1 ) of rigid motions of the Minkowski 2-space endowed with a left-invariant almost cosymplectic structure. In this case, the scalar curvature is a negative constant − 2 λ 2 .

Proof

If the Ricci curvatures are invariant along the Reeb flow, equalities (3.3) and (3.4) transform into

(3.14) 2 e 1 ( λ ) = σ ( e 2 )

and

(3.15) 2 e 2 ( λ ) = σ ( e 1 ) ,

respectively, where we used λ > 0 on U 1 . The action of equation (3.14) along the Reeb vector field ξ implies ξ ( e 1 ( λ ) ) = 0 . This, combined with equation (3.12), gives that e 2 ( λ ) = 0 , where we used again λ > 0 on U 1 . Similarly, the action of equation (3.15) along the Reeb vector field ξ implies ξ ( e 2 ( λ ) ) = 0 . This, combined with equation (3.13), gives that e 1 ( λ ) = 0 , where we used again λ > 0 on U 1 . Recalling the first term of equation (3.6), now λ is a constant on U 1 . From equations (3.7)–(3.9), we obtain

(3.16) [ ξ , e 1 ] = λ e 2 , [ e 1 , e 2 ] = 0 , [ e 2 , ξ ] = − λ e 1 .

According to Milnor [23] and the above relations, it is easily seen that the manifold is locally isometric to the Lie group E ( 1 , 1 ) of rigid motions of the Minkowski 2-space equipped with a left-invariant almost cosymplectic structure.

Applying Lemma 3.1, the curvature tensor R of an almost cosymplectic three-manifold on U 1 can be given as follows:

R ( e 1 , ξ ) ξ = − λ 2 e 1 .

R ( e 2 , ξ ) ξ = − λ 2 e 2 .

R ( e 1 , e 2 ) e 2 = λ 2 e 1 .

The scalar curvature is given by − 2 λ 2 . From equation (3.2), the Ricci tensor is of the following form:

Ric = diag { − 2 λ 2 , 0 , 0 }

with respect to the orthonormal ϕ -basis { ξ , e 1 , e 2 } of the tangent space. Hence, the scalar curvature is a negative constant − 2 λ 2 .□

Remark 3.5

In fact, following Lemmas 3.2 and 3.4, it is easy to see that the Ricci operator of a group whose Lie algebra is given by equation (3.16) is always weakly ϕ -invariant, where we refer the reader to [24] for the definitions of almost contact metric structures on Lie groups of dimension three.

It is easily seen from Lemma 3.4 and from the study by Cho [9] that the Ricci weak ϕ -invariance and equation (1.2) are mutually equivalent if the Ricci curvatures are invariant along the Reeb flow on a strictly almost cosymplectic three-manifold. Here, when referring to a strictly almost cosymplectic manifold, we mean an almost cosymplectic manifold, which is not cosymplectic. Next, we study Ricci ϕ -invariance on a cosymplectic three-manifold.

Lemma 3.6

The Ricci operator of a cosymplectic three-manifold is weakly ϕ -invariant.

Proof

On a cosymplectic three-manifold we always have h = 0 . Using this in equation (2.5), it is easily seen that the Reeb vector field ξ is parallel, and hence, we obtain R ( X , Y ) ξ = 0 and

(3.17) Q ξ = 0 .

On the other hand, on a Riemannian three-manifold, the curvature tensor R is given as follows:

(3.18) R ( X , Y ) Z = g ( Y , Z ) Q X − g ( X , Z ) Q Y + g ( Q Y , Z ) X − g ( Q X , Z ) Y − r 2 ( g ( Y , Z ) X − g ( X , Z ) Y )

for any vector fields X , Y , and Z , where r is the scalar curvature. Replacing Z by ξ in equation (3.18), with the help of equation (3.17) and R ( X , Y ) ξ = 0 , we see that the Ricci operator of a cosymplectic three-manifold is given as follows:

(3.19) Q = r 2 ( I − η ⊗ ξ ) .

Obviously, the Reeb vector field is an eigenvector field of the Ricci operator, then according to equation (1.3), we see that the Ricci operator of a cosymplectic three-manifold is weakly ϕ -invariant.□

Finally, based on the above several lemmas and remark, the main theorem of this article is given as the follows.

Theorem 3.7

Let M 3 be an almost cosymplectic three-manifold, then the following two statements are valid:

If the manifold is cosymplectic, then the Ricci operator is weakly ϕ -invariant.
If the manifold is non-cosymplectic, then the Ricci operator is weakly ϕ -invariant, and the Ricci curvatures are invariant along Reeb flow if and only if the manifold is locally isometric to Lie group E ( 1 , 1 ) of rigid motions of the Minkowski 2-space equipped with a left-invariant almost cosymplectic structure.

An almost cosymplectic manifold ( M 2 n + 1 , ϕ , ξ , η , g ) is said to be a ( κ , μ ) -almost cosymplectic manifold [25] if the following equality is valid:

R ( X , Y ) ξ = κ ( η ( Y ) X − η ( X ) Y ) + μ ( η ( Y ) h X − η ( X ) h Y )

for any vector fields X and Y , where κ and μ denote two constants. This kind of almost cosymplectic manifolds play a fundamental role in almost cosymplectic geometry [26]. The Ricci operator of a ( κ , μ ) -almost cosymplectic manifold is given by [25]:

Q = μ h + 2 n κ η ⊗ ξ ,

where h = 1 2 ℒ ξ ϕ . By the anti-commutation of ϕ and h , the above equality gives

Q ϕ − ϕ Q = 2 μ h ϕ .

We also have h 2 = κ ϕ 2 . From this, the following fact is immediate.

Remark 3.8

Let M 2 n + 1 be a ( κ , μ ) -almost cosymplectic manifold of dimension 2 n + 1 . The following three statements are valid:

M 2 n + 1 satisfies equation (1.1) if and only if κ = 0 and μ is arbitrary.
M 2 n + 1 satisfies equation (1.2) if and only if either κ is arbitrary and μ = 0 or κ = 0 and μ is arbitrary.
M 2 n + 1 satisfies equation (1.3) if and only if either κ = 0 or μ = 0 .

Remark 3.9

Hopf hypersurfaces in nonflat complex space forms satisfying equation (1.3) were classified completely by Okumura in [1, Theorem 1]. But, without Hopf restriction, the classification for non-ruled real hypersurfaces satisfying equation (1.3) is still open (see [1, Remark 3]). This article deals with equation (1.3) on a special almost contact metric manifold of dimension three. However, the study of equation (1.3) on a general contact metric manifold, even on an almost cosymplectic manifold of dimension ≥ 5 (not the type of almost cosymplectic manifolds in Remark 3.8), is also open.

Before closing this note, we now present some concrete examples (expect for Remark 3.8) to verify that the two kinds of ϕ -invariance of Ricci operators in a strictly almost cosymplectic three-manifold are not equivalent in general cases, and the invariance of the Ricci curvatures under Reeb flow in Theorem 3.7 is essential.

Example 3.10

Let G be a non-unimodular Lie group of dimension three with a left-invariant metric g whose Lie algebra is given as follows:

(3.20) [ e 1 , e 2 ] = α e 2 , [ e 2 , e 3 ] = 0 , [ e 1 , e 3 ] = β e 2 ,

where { e 1 , e 2 , e 3 } is an orthonormal basis with respect to g and α , β ≠ 0 ∈ R . Let us define ξ ≔ e 3 and one-form η by η = g ( ξ , ⋅ ) . We define a ( 1 , 1 ) -type tensor field ϕ by ϕ ξ = 0 , ϕ e 1 = e 2 , and ϕ e 2 = − e 1 . So, ( G , ϕ , ξ , η , g ) forms a left-invariant cosymplectic three-manifold. By using the Koszul formula and equation (3.20), we have

( ∇ e i e j ) = 0 − β 2 ξ β 2 e 2 − α e 2 − β 2 ξ α e 1 β 2 e 1 − β 2 e 2 β 2 e 1 0

for any i , j ∈ { 1 , 2 , 3 } . Using this the Ricci tensor of G is given by

Ric i j = − α 2 − β 2 2 0 0 0 − α 2 + β 2 2 − α β 0 − α β − β 2 2

with respect to the ϕ -basis { e 1 , e 2 , e 3 } . According to this, we see that the Ricci operator is neither (strongly) ϕ -invariant nor weakly ϕ -invariant although the Ricci curvatures are invariant along the Reeb flow.

The above example has been appeared in [21,24]. Next, we continue to check the example shown in [21].

Example 3.11

Let M 3 be an open subset of R 3 , which is defined by M 3 ≔ { ( x , y , z ) ∈ R 3 : z > 0 } . On M 3 , there exists an almost cosymplectic structure of dimension three defined by

ξ = ∂ ∂ z , η = d z ,

ϕ ∂ ∂ x = z 2 ∂ ∂ y , ϕ ∂ ∂ y = − 1 z 2 ∂ ∂ x , ϕ ∂ ∂ z = 0 ,

g = z 2 d x ⊗ d x + 1 z 2 d y ⊗ d y + d z ⊗ d z .

The fundamental two-form Φ of M 3 is given by Φ = 2 d x ∧ d y . We define e 1 ≔ 1 z ∂ ∂ x , and hence e 2 ≔ ϕ e 1 = z ∂ ∂ y and e 3 ≔ ξ . Therefore, { e 1 , e 2 , e 3 } forms an orthonormal global frame on M 3 . The Levi-Civita connection ∇ of M 3 is given by

( ∇ e i e j ) = − 1 z e 3 0 1 z e 1 0 1 z e 3 − 1 z e 2 0 0 0

for any i , j ∈ { 1 , 2 , 3 } . From this, the Ricci tensor of M 3 is

Ric i j = 1 z 2 0 0 0 − 1 z 2 0 0 0 − 2 z 2

with respect to the orthonormal ϕ -basis { e 1 , e 2 , e 3 } . According to this, the Ricci operator is not weakly ϕ -invariant, and the scalar curvature of the manifold is not invariant along the Reeb flow.

The following example is shown in [27].

Example 3.12

Let M = R 2 × R + ⊂ R 3 and denote by { x , y , z } the Cartesian coordinates of R 3 restricted to M . There exists an almost cosymplectic structure defined by

ξ = ∂ ∂ z , η = d z ,

ϕ ∂ ∂ x = z 2 e x ∂ ∂ y , ϕ ∂ ∂ y = − e x z 2 ∂ ∂ x , ϕ ∂ ∂ z = 0 ,

g = z 2 d x ⊗ d x + e 2 x z 2 d y ⊗ d y + d z ⊗ d z .

The orthonormal ϕ -basis of the manifold is given as follows:

e 1 = 1 z ∂ ∂ x , e 2 = z e x ∂ ∂ y , e 3 = ∂ ∂ z .

The Levi-Civita connection of the metric g is

( ∇ e i e j ) = − 1 z e 3 0 1 z e 1 1 z e 2 − 1 z e 1 + 1 z e 3 − 1 z e 2 0 0 0

for any i , j ∈ { 1 , 2 , 3 } . The Ricci tensor of the manifold is

Ric i j = 0 0 2 z 2 0 − 2 z 2 0 2 z 2 0 − 2 z 2

with respect to the orthonormal ϕ -basis { e 1 , e 2 , e 3 } . From this, we see that neither the Ricci operator is weakly ϕ -invariant nor the Ricci curvatures are invariant along Reeb flow.

Acknowledgement

The author would like to thank the anonymous reviewers for their useful comments and suggestions.

Funding information: This article was supported by the Doctoral Foundation of the Henan Institute of Technology (Grant No. KQ1828).
Conflict of interest: The author states that there is no conflict of interest.
Data availability statement: This paper has no associated data.

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Received: 2023-01-25

Revised: 2023-08-21

Accepted: 2023-11-03

Published Online: 2023-11-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2023-0156

Keywords for this article

almost cosymplectic three-manifolds; Ricci ϕ-invariance; Lie group

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