# On a Classification of AlmostCα-Manifolds

Authors: Tuğba Mert
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, pseudosymmetric and Ricci pseudosymmetric almostCα-manifold are studied. For an almost Cα-manifold, Riemann pseudosymmetric, Riemann Ricci pseudosymmetric, Ricci pseudosymmetric, projective pseudosymmetric, projective Ricci pseudosymmetric, concircular pseudosymmetric, and concircular Ricci pseudosymmetric cases are considered and new results are obtained.


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## Body

## 1. Introduction

In the period from the known past to the present, the place of geometry science in science and technology has always been preserved. Geometry was divided into various branches according to the needs of mankind over time and the studies were carried out more succinctly. One of these areas is differential geometry, where differential computation is applied to geometry. Differential geometry is one of the most popular fields of study in modern mathematics, as it finds applications in many disciplines. The beginning of differential geometry is based on Gauss’s work on the curvature of surfaces. These works of Gauss pioneered the concept of Riemannian manifold. Differential geometry is very closely concerned with the properties of Riemannian manifolds where the derivative is defined.In a Riemannian manifold, the Riemannian curvature tensor isR and for each U1,U2∈χM, if RU1,U2.R=0, then the manifold is said to be semisymmetric. Similarly, if RU1,U2.S=0, the manifold is called Ricci semisymmetric, if RU1,U2.P=0; the manifold is called projective semisymmetric, if RU1,U2.Z˜=0; and the manifold is called concircular semisymmetric, where S is the Ricci curvature tensor, P is the projective curvature tensor, and Z˜ is concircular curvature tensor. Studies on the symmetric Riemannian manifolds started with Cartan [1]. In the following periods, many authors have studied the symmetry cases of various manifolds ([2–12]).Again, interesting and important studies by many geometers continued to contribute to this field over time. In 2011, Dileo presented an important study on the geometry of the Kenmotsu type manifolds [13], while in 2019, a very important study was made for α− cosymplectic manifolds by Kupeli [14]. Many studies have also been made about almost Cα− manifolds in [15–18] and important characterizations of this type of manifolds have been obtained ([19, 20]).LetMφ,ξ,η,g be the 2n+1− dimensional almost contact metric manifold given by the φ,ξ,η,g contact structure, and let the Riemann curvature tensor of the manifold be R. If manifold M satisfies the following condition:(1)RU1,U2,U3,U4=RU1,U2,φU3,φU4+α−gU1,U3gU2,U4+gU1,U4gU2,U3+gU1,φU3gU2,φU4−gU1,φU4gU2,φU3.Then, manifold M is called almost Cα− manifold, where every U1,U2,U3,U4∈χM and at least one α∈ℝ [21].ACα− manifold is the general case of the co-Keahler, Sasakian, and Kenmotsu manifolds. That is, specially, if α=0, it is co-Keahler, if α=1, it is Sasakian, if α=−1, it is the Kenmotsu manifold [21]. Although many studies have been conducted on these manifolds, studies on Cα− manifold is quite limited.In this article, the pseudosymmetry and Ricci pseudosymmetry properties of almostCα− manifold, which are a subclass of almost contact metric manifolds and the general case of the co-Keahler, Sasakian, and Kenmotsu manifolds, have been studied geometrically.


## 2. Preliminaries

LetM be a differentiable manifold with 2n+1 dimensional. If the condition(2)φ2U1=−U1+ηU1ξ,ηξ=1,satisfies on M, where φ is tensor field with type 1,1, ξ is a vector field and η is a 1− form, then we say that φ,ξ,η is an almost contact structure. Also, we say that M,φ,ξ,η is an almost contact manifold [22]. Let g be a metric with condition(3)gφU1,φU2=gU1,U2−ηU1ηU2,gU1,ξ=ηU1,for all U1,U2∈χM and ξ∈χM. In this case, we say that φ,ξ,η,g is an almost contact metric structure and M,φ,ξ,η,g is almost contact metric manifold [22]. Moreover, we have the property(4)gφU1,U2=−gU1,φU2,for all U1,U2∈χM on M manifold with 2n+1 dimensional. The fundamental 2-form of the φ,ξ,η,g almost contact metric structure is the Φ transformation such that(5)ΦU1,U2=gU1,φU2,for all U1,U2∈χM, where(6)η∧Φn≠0.We can write the Riemann curvature tensor of an almostCα-manifold which has a c− constant sectional curvature by(7)RU1,U2U3=c+3α4gU2,U3U1−gU1,U3U2+c−α4gU1,φU3φU2−gU2,φU3φU1+2gU1,φU2φU3+ηU2ηU3U1+gU1,U3ηU2ξ−gU2,U3ηU1ξ.If we takeU1=ξ in (4), then we obtain as follows:(8)Rξ,U2U3=αgU2,U3ξ−ηU3U2.If we takeU3=ξ in (4), then we have as follows:(9)RU1,U2ξ=αηU2U1−ηU1U2.Moreover, if we takeU2=ξ in (9), then we obtain as follows:(10)RU1,ξξ=αU1−ηU1ξ.Let us take an inner product of (7) by ξ∈χM. Then, we obtain as follows:(11)ηRU1,U2U3=αgU2,U3ηU1−gU1,U3ηU2.Let fork≥1,σ be a 0,k-type tensor field, a A be a 0,2-type tensor field and M is a Riemannian manifold. We define QA,σ-tensor field by(12)QA,σX1,X2,…,Xk;U1,U2=−σU1ΛAU2X1,X2,…,Xk…−σX1,X2,…,Xk−1,U1ΛAU2Xk,for all X1,X2,…,Xk,U1,U2∈ΓTM, where(13)U1∧AU2U3=AU2,U3U1−AU1,U3U2.Lemma 1.
LetM be an almost Cα− manifold with 2n+1− dimensional. Then, we have as follows:(14)SU1,U2=α3n−1+cn+12gU1,U2+α−cn+12ηU1ηU2,(15)SU1,ξ=2nαηU1,(16)QU1=α3n−1+cn+12U1+α−cn+12ηU1ξ,(17)Qξ=2nαξ,for each U1,U2,∈χM, where Q is the Ricci operator and S is the Ricci tensor of manifold M.IfS and g are linearly dependent, that is, there is a constant λ provided(18)SU1,U2=λgU1,U2,the almost Cα− manifold is called the Einstein manifold [23]. In particular, if the S Ricci tensor satisfies the relation(19)SU1,U2=agU1,U2+bηU1ηU2,for each U1,U2∈χM, M is called an η− Einstein manifold [23].Let (M, g) be a Riemannian manifold. For two-dimensional subspace Π of the tangent, let space TpM be(20)gU1,U1gU2,U2−gU1,U22≠0,for each U1,U2∈Π. In this case, KU1ΛU2 defined as follows(21)KU1ΛU2=gRU1,U2U2,U1gU1,U1gU2,U2−gU1,U22,is called the sectional curvature of the plane Π. If KU1pΛU2p is a constant for each U1,U2∈TpM,p∈M, M is called a c− constant section curvature space or real space form [22]. In this case, if the M Riemann manifold is a real space form and c− constant section curvature, the Riemann curvature tensor of M is as follows:(22)gRU1,U2U3,U4=cgU2,U3gU1,U4−gU1,U3gU2,U4,for each U1,U2,U3,U4∈χM.


## 3. Pseudosymmetric and Ricci Pseudosymmetric AlmostCα-Manifold

In this section, the cases of pseudosymmetry and Ricci pseudosymmetry of an almostCα− manifold are investigated. According to Riemann, Ricci, projective, and concircular curvature tensors, the pseudosymmetrical and Ricci pseudosymmetrical cases of the almost Cα− manifold can be given as follows.Definition 1.
LetM be an almost Cα− manifold with 2n+1− dimensional, R be the Riemann curvature tensor of M, and S be the Ricci curvature tensor of M.(i)
If the pairR⋅R and Qg,R are linearly dependent, that is, if a λ1 function can be found on the set M1=x∈M|gx≠Rx such that(23)R⋅R=λ1Qg,R,theM manifold is called a Riemann pseudosymmetric manifold.(ii)
If the pairR⋅R and QS,R are linearly dependent, that is, if a λ2 function can be found on the set M2=x∈M|Sx≠Rx such that(24)R⋅R=λ2QS,R,theM manifold is called a Riemann Ricci pseudosymmetric manifold.;Particularly, ifλ1=0, then this manifold is said to be semisymmetric.
Let us now investigate the cases of Riemann pseudosymmetry and Riemann Ricci pseudosymmetry.Theorem 1.
If a2n+1− dimensional M almost Cα− manifold is a Riemann pseudo-symmetric manifold, then λ1=0.Proof.
Let us assume that the manifoldM is a Riemann pseudosymmetric manifold. Then, we can write(25)RU1,U2⋅RU3,U4,U5=λ1Qg,RU5,U4,U3;U1,U2,for each U1,U2,U3,U4,U5∈χM. In this case, we obtain as follows:(26)RU1,U2RU3,U4U5−RRU1,U2U3,U4U5−RU3,RU1,U2U4U5−RU3,U4RU1,U2U5,=−λ1RU1ΛgU2U5,U4U3+RU5,U1ΛgU2U4U3+RU5,U4U1ΛgU2U3.
If necessary arrangements are made in here, and we chooseU1=U3=ξ and using the expression (8) in (26), we obtain as follows:(27)αgU2,Rξ,U4U5ξ−αηRξ,U4U5U2−α2ηU2gU4,U5ξ+α2ηU5ηU2U4+αRU2,U4U5+α2ηU5gU2,U4ξ−α2ηU5gU2,U4ξ+αηU4Rξ,U2U5−α2ηU4gU2,U5ξ+α2gU2,U5U4+αηU5Rξ,U4U2=−λ1αgU2,U5ηU4ξ−αgU2,U5U4−ηU5RU2,U4ξ−αgU2,U4ηU5ξ+αgU2,U4U5−ηU4RU5,U2ξ+αηU4ηU2U5−αηU5ηU2U4−RU5,U4U2.
If we use the expression (8) again in (27) and make the necessary adjustments, then we obtain as follows:(28)αRU2,U4U5+α2gU2,U5U4−α2gU4,U5U2=−λ1αgU2,U5ηU4ξ−αgU2,U4ηU5ξ−αgU2,U5U4+αgU2,U4U5−RU5,U4U2.
If we applyξ∈χM to both sides of (28) and make use of (11), then we obtain as follows:(29)λ1ηRU5,U4U2=0.
SinceηRU5,U4U2≠0, it is clear that(30)λ1=0.
This completes our proof.Corollary 1.
If a2n+1− dimensional M almost Cα− manifold is a Riemann pseudosymmetric manifold, then M is a Riemann semisymmetric manifold.Corollary 2.
If a2n+1− dimensional M almost Cα− manifold is a Riemann pseudosymmetric manifold, then M is a real space form with a constant section curvature.Theorem 2.
If a2n+1− dimensional M almost Cα− manifold is a Riemann Ricci pseudosymmetric manifold, then M is either co-Keahler manifold or λ2=0.Proof.
Let us assume that the manifoldM is a Riemann Ricci pseudosymmetric manifold. Then, we can write as follows:(31)RU1,U2⋅RU3,U4,U5=λ2QS,RU5,U4,U3;U1,U2,for each U1,U2,U3,U4,U5∈χM. In this case, we obtain as follows:(32)RU1,U2RU3,U4U5−RRU1,U2U3,U4U5−RU3,RU1,U2U4U5−RU3,U4RU1,U2U5=−λ2RU1ΛSU2U5,U4U3+RU5,U1ΛSU2U4U3+RU5,U4U1ΛSU2U3.
If necessary arrangements are made in here, and we chooseU1=U3=ξ and using (8), (15) in (32), we obtain as follows:(33)αgU2,Rξ,U4U5ξ−αηRξ,U4U5U2−α2ηU2gU4,U5ξ+α2ηU5ηU2U4+αRU2,U4U5+α2ηU5gU2,U4ξ−α2ηU5gU2,U4ξ+αηU4Rξ,U2U5−α2ηU4gU2,U5ξ+α2gU2,U5U4+αηU5Rξ,U4U2=−λ2αSU2,U5ηU4ξ−αSU2,U5U4−2nαηU5RU2,U4ξ−αSU2,U4ηU5ξ+αSU2,U4U5−2nαηU4RU5,U2ξ+2nα2ηU4ηU2U5−2nα2ηU5ηU2U4−2nαRU5,U4U2.
If we use the expression (5) again in (33) and make the necessary adjustments, then we obtain as follows:(34)αRU2,U4U5+α2gU2,U5U4−α2gU4,U5U2=−λ2αSU2,U5ηU4ξ−αSU2,U4ηU5ξ−αSU2,U5U4+αSU2,U4U5−2nαRU5,U4U2.
If we applyξ∈χM to both sides of (34) and make use of (11), then we obtain as follows:(35)2nαλ2ηRU5,U4U2=0.
SinceηRU5,U4U2≠0, it is clear that(36)αλ2=0.
Thus, eitherα=0 or λ2=0 is obtained. This completes our proof.Corollary 3.
If a2n+1− dimensional M almost Cα− manifold is a Riemann Ricci pseudosymmetric manifold, then M is either co-Keahler manifold or a Riemann semisymmetric manifold.Corollary 4.
If a2n+1− dimensional M almost Cα− manifold is a Riemann Ricci pseudosymmetric manifold, then M is a real space form with a constant section curvature.Definition 2.
LetM be an almost Cα− manifold with 2n+1− dimensional, R be the Riemann curvature tensor of M and S be the Ricci curvature tensor of M. If the pair R⋅S and Qg,S are linearly dependent, that is, if a λ3 function can be found on the set M3=x∈M|gx≠Sx such that(37)R⋅S=λ3Qg,S,the M manifold is called a Ricci pseudosymmetric manifold.
Let us now investigate the case of Ricci pseudosymmetry.Theorem 3.
If a2n+1− dimensional M almost Cα− manifold is a Ricci pseudosymmetric manifold, then M is either an Einstein manifold or λ3=−α.Proof.
Let us assume that the manifoldM is a Ricci pseudosymmetric manifold. Then, we can write as follows:(38)RU1,U2⋅SU5,U4=λ3Qg,SU5,U4;U1,U2,for each U1,U2,U4,U5∈χM. In this case, we can write as follows:(39)SRU1,U2U5,U4+SU5,RU1,U2U4=−λ3SU1ΛgU2U5,U4+SU5,U1ΛgU2U4.
In equation (39), first U1=ξ is chosen and then (8) and (15) are used, we obtain as follows:(40)c2nα2gU2,U5ηU4−αηU5SU2,U4+2nα2gU2,U4ηU5−αηU4SU5,U2=−λ32nαηU4gU2,U5−ηU5SU2,U4+2nαηU5gU2,U4−ηU4SU5,U2.
If we chooseU5=ξ in (40) and make the necessary adjustments, we obtain as follows:(41)−αSU2,U4+2nα2gU2,U4=−λ3−SU2,U4+2nαgU2,U4,and from here, we can write that(42)α+λ3−SU2,U4+2nαgU2,U4=0.
It is clear from equation (42) that either(43)α+λ3=0,(44)orSU2,U4=2nαgU2,U4.
This completes the proof of the theorem.
The projective curvature tensorP is defined as follows:(45)PU1,U2U3=RU1,U2U3−12nSU2,U3U1−SU1,U3U2,for all each U1,U2,U3∈χM, by Yano and Sawaki [24]. If U1=ξ,U2=ξ, and U3=ξ are selected, respectively, in (45), then we obtain as follows:(46)Pξ,U2U3=αgU2,U3ξ−12nSU2,U3ξ,(47)PU1,ξU3=−αgU1,U3ξ+12nSU1,U3ξ,(48)PU1,U2ξ=0.
Let us take inner product of (45) by ξ∈χM, we get(49)ccηPU1,U2U3=ηU1αgU2,U3−12nSU2,U3−ηU2αgU1,U3−12nSU1,U3.Definition 3.
LetM be a 2n+1− dimensional almost Cα− manifold, R be the Riemann curvature tensor of M, S be the Ricci curvature tensor of M, and P be the projective curvature tensor of M.(i)
If the pairR⋅P and Qg,P are linearly dependent, that is, if a λ4 function can be found on the set M4=x∈M|gx≠Px such that(50)R⋅P=λ4Qg,P,theM manifold is called a projective pseudosymmetric manifold.(ii)
If the pairR⋅P and QS,P are linearly dependent, that is, if a λ5 function can be found on the set M5=x∈M|Sx≠Px such that(51)R⋅P=λ5QS,P.TheM manifold is called a projective Ricci pseudosymmetric manifold.
Let us now investigate the case of projective pseudosymmetry and projective Ricci pseudosymmetry.Theorem 4.
If a2n+1− dimensional M almost Cα− manifold is a projective pseudosymmetric manifold, then M is either an Einstein manifold or λ4=α.Proof.
Let us assume that the manifoldM is a projective pseudosymmetric manifold. Then, we can write as follows:(52)RU1,U2⋅PU5,U4,U3=λ4Qg,PU5,U4,U3;U1,U2,for each U1,U2,U3,U4,U5∈χM. In this case, we obtain as follows:(53)RU1,U2PU5,U4U3−PRU1,U2U5,U4U3−PU5,RU1,U2U4U3−PU5,U4RU1,U2U3=−λ4PU1ΛgU2U5,U4U3+PU5,U1ΛgU2U4U3+PU5,U4U1ΛgU2U3.
In the last equation, ifU1=ξ is chosen and necessary corrections are made, we can write(54)Rξ,U2PU5,U4U3−PRξ,U2U5,U4U3−PU5,Rξ,U2U4U3−PU5,U4Rξ,U2U3=−λ4gU2,U5Pξ,U4U3−gξ,U5PU2,U4U3+gU2,U4PU5,ξU3−gξ,U4PU5,U2U3+gU2,U3PU5,U4ξ−gξ,U3PU5,U4U2.
If (8), (46), and (47) are used in (54), we obtain follows:(55)αgU2,PU5,U4U3ξ−αηPU5,U4U3U2−αgU2,U5Pξ,U4U3+αηU5PU2,U4U3−αgU2,U4PU5,ξU3+αηU4PU,U2U3−αgU2,U3PU5,U4ξ+αηU3PU5,U4U2=−λ4αgU2,U5gU4,U3ξ−12ngU2,U5SU4,U3ξ−ηU5PU2,U4U3−αgU2,U4gU5,U3ξ+12ngU2,U4SU5,U3ξ−ηU4PU5,U2U3−ηU3PU5,U4U2.
IfU5=ξ is chosen in (55), equation (8) is used and necessary adjustments are made, we obtain as follows:(56)l−α2gU4,U3U2+αRU2,U4U3+α2nSU2,U3U4+α2gU2,U3ηU4ξ−α2nηU4SU2,U3ξ+α2ηU3gU4,U2ξ−12nηU2SU4,U3ξ−RU2,U4U3−α2nηU3SU4,U2ξ=−λ4αgU4,U3ηU2ξ+12nSU4,U3U2−12nSU2,U3U4−αηU4gU2,U3ξ+12nηU4SU2,U3ξ−αgU4,U2ηU3ξ+12nηU3SU4,U2ξ.
If we applyξ∈χM to both sides of equation (54) and make use of equation (11), we obtain as follows:(57)α−λ4αηU3gU4,U2−12nα−λ4ηU3SU4,U2=0.
IfU3=ξ is chosen in the last equation, we obtain as follows:(58)α−λ4−SU4,U2+2nαgU4,U2=0,that is,(59)α=λ4,SU4,U2=2nαgU4,U2.
Thus, the proof of the theorem is complete.Theorem 5.
If a2n+1− dimensional M almost Cα− manifold is a projective Ricci pseudosymmetric manifold, then M is either an Einstein manifold, co-Keahler or λ5=1/2n.Proof.
Let us assume that the manifoldM is a projective Ricci pseudosymmetric manifold. Then, we can write(60)RU1,U2⋅PU5,U4,U3=λ5QS,PU5,U4,U3;U1,U2,for each U1,U2,U3,U4,U5∈χM. In this case, we obtain as follows:(61)RU1,U2PU5,U4U3−PRU1,U2U5,U4U3−PU5,RU1,U2U4U3−PU5,U4RU1,U2U3=−λ5PU1ΛSU2U5,U4U3+PU5,U1ΛSU2U4U3+PU5,U4U1ΛSU2U3.
In the last equation, ifU1=ξ is chosen and necessary corrections are made, we can write(62)Rξ,U2PU5,U4U3−PRξ,U2U5,U4U3−PU5,Rξ,U2U4U3−PU5,U4Rξ,U2U3=−λ5SU2,U5Pξ,U4U3−Sξ,U5PU2,U4U3+SU2,U4PU5,ξU3−Sξ,U4PU5,U2U3+SU2,U3PU5,U4ξ−Sξ,U3PU5,U4U2.
If (8), (46), and (47) are used in (62), we obtain as follows:(63)αgU2,PU5,U4U3ξ−αηPU5,U4U3U2−αgU2,U5Pξ,U4U3+αηU5PU2,U4U3−αgU2,U4PU5,ξU3+αηU4PU5,U2U3−αgU2,U3PU5,U4ξ+αηU3PU5,U4U2=−λ5αSU2,U5gU4,U3ξ−12nSU2,U5SU4,U3ξ−2nαηU5PU2,U4U3−αSU2,U4gU5,U3ξ+12nSU2,U4SU5,U3ξ−2nαηU4PU5,U2U3−2nαηU3PU5,U4U2.
IfU5=ξ is chosen in (63), equation (8) is used and necessary adjustments are made, we obtain as follows:(64)l−α2gU4,U3U2+αRU2,U4U3+α2nSU2,U3U4+α2gU2,U3ηU4ξ−α2nSU2,U3U4+α2ηU3gU4,U2ξ−α2nηU3SU4,U2ξ=−λ52nα2gU4,U3ηU2ξ−αηU2SU4,U3ξ−2nαRU2,U4U3+αSU4,U3U2−αSU2,U3U4−2nα2ηU4gU2,U3ξ+αηU4SU2,U3ξ−2nα2gU4,U2ηU3ξ+αηU3SU4,U2ξ.
If we applyξ∈χM to both sides of equation (65) and make use of equation (11), we obtain as follows:(65)2nα21−2nλ5ηU3gU4,U2−α1−2nλ5ηU3SU4,U2=0.
IfU3=ξ is chosen in the last equation, we yield as follows:(66)α1−2nλ5−SU4,U2+2nαgU4,U2=0,that is,(67)α=0,1−2nλ5=0,orSU4,U2=2nαgU4,U2.
Thus, the proof of the theorem is complete.
LetM be a Riemannian manifold with 2n+1− dimensional. Then, the concircular curvature tensor Z˜ is defined as(68)Z˜U1,U2U3=RU1,U2U3−r2n2n+1gU2,U3U1−gU1,U3U2,for each U1,U2,U3∈χM [25]. If we choose, respectively, U1=ξ,U2=ξ, and U3=ξ in (68), then we obtain as follows:(69)Z˜ξ,U2U3=α−r2n2n+1gU2,U3ξ−ηU3U2,(70)Z˜U1,ξU3=α−r2n2n+1−gU1,U3ξ+ηU3U1,(71)Z˜U1,U2ξ=α−r2n2n+1ηU2U1−ηU1U2.
In addition, we chooseU3=ξ in (69), we obtain as follows:(72)Z˜ξ,U2ξ=α−r2n2n+1ηU2ξ−U2.and finally, we take inner product of (68) by ξ∈χM, we have(73)ηZ˜U1,U2U3=ηRU1,U2U3−r2n2n+1gU2,U3ηU1−gU1,U3ηU2.Definition 4.
LetM be a 2n+1− dimensional almost Cα− manifold, R be the Riemann curvature tensor of M, S be the Ricci curvature tensor of M, and Z˜ be the concircular curvature tensor of M.(i)
If the pairR⋅Z˜ and Qg,Z˜ are linearly dependent, that is, if a λ6 function can be found on the set M6=x∈M|gx≠Z˜x such that(74)R⋅Z˜=λ6Qg,Z˜,theM manifold is called a concircular pseudo-symmetric manifold.(ii)
If the pairR⋅Z˜ and QS,Z˜ are linearly dependent, that is, if a λ7 function can be found on the set M7=x∈M|Sx≠Z˜x such that(75)R⋅Z˜=λ7QS,Z˜,theM manifold is called a concircular Ricci pseudosymmetric manifold.
Let us now investigate the cases of concircular pseudosymmetry and concircular Ricci pseudosymmetry.Theorem 6.
If a2n+1− dimensional M almost Cα− manifold is a concircular pseudosymmetric manifold, then we can see either λ6=0 or α=r/2n2n+1.Proof.
Let us assume that the manifoldM is a concircular pseudosymmetric manifold. Then, we can write(76)RU1,U2⋅Z˜U5,U4,U3=λ6Qg,Z˜U5,U4,U3;U1,U2,for each U1,U2,U3,U4,U5∈χM. In this case, we yield as follows:(77)RU1,U2Z˜U5,U4U3−Z˜RU1,U2U5,U4U3−Z˜U5,RU1,U2U4U3−Z˜U5,U4RU1,U2U3=−λ6Z˜U1ΛgU2U5,U4U3+Z˜U5,U1ΛgU2U4U3+Z˜U5,U4U1ΛgU2U3.
If necessary arrangements are made in here and we chooseU1=U5=ξ and using (8), (69) in (77), we obtain as follows:(78)αgU2,Z˜ξ,U4U3ξ−αηZ˜ξ,U4U3U2−αα−r2n2n+1ηU2gU4,U3ξ+αα−r2n2n+1ηU3ηU2U4+αZ˜U2,U4U3+αα−r2n2n+1ηU3gU2,U4ξ−αα−r2n2n+1ηU3gU2,U4ξ+αηU4Z˜ξ,U2U3−αα−r2n2n+1ηU4gU2,U3ξ+αα−r2n2n+1gU2,U3U4+αηU3Z˜ξ,U4U2=−λ6α−r2n2n+1gU4,U3ηU2ξ−α−r2n2n+1ηU2ηU3U4−Z˜U2,U4U3−α−r2n2n+1gU2,U4ηU3ξ+α−r2n2n+1gU2,U4ηU3ξ−ηU4Z˜ξ,U2U3+α−r2n2n+1gU2,U3ηU4ξ−α−r2n2n+1gU2,U3U4−ηU3Z˜ξ,U4U2.
If we use the expression (69) again in (78) and make the necessary adjustments, then we obtain as follows:(79)−αα−r2n2n+1gU4,U3U2+αZ˜U2,U4U3−αα−r2n2n+1gU2,U3U4=−λ6α−r2n2n+1gU4,U3ηU2ξ−Z˜U2,U4U3+α−r2n2n+1ηU4ηU3U2−α−r2n2n+1gU2,U3U4−α−r2n2n+1gU4,U2ηU3ξ.
If we replace the expression (68) in (79) and make the necessary adjustments, we have the following:(80)−α2gU4,U3U2+α2gU2,U3U4+αRU2,U4U3=−λ6α−r2n2n+1gU4,U3ηU2ξ−RU2,U4U3+αr2n2n+1gU4,U3U2−αr2n2n+1gU2,U3U4+α−r2n2n+1ηU4ηU3U2−α−r2n2n+1gU2,U3U4−α−r2n2n+1gU4,U2ηU3ξ.
If we applyξ∈χM to both sides of (80) and make use of (11), then we obtain as follows:(81)λ6α−r2n2n+1−gU4,U2+ηU4ηU2=0.
On the other hand, we know that there is ag metric such that(82)gφU4,φU2=gU4,U2−ηU4ηU2,on a 2n+1 dimensional almost contact manifold [26]. Therefore, equation (81) can be written as follows:(83)λ6α−r2n2n+1gφU4,φU2=0.
It is clear from the last equation that(84)λ6=0,orα=r2n2n+1.
This completes our proof.Corollary 5.
If a2n+1− dimensional M almost Cα− manifold is a concircular pseudosymmetric manifold, then M is either α=r/2n2n+1 or a concircular semisymmetric manifold.Corollary 6.
If a2n+1− dimensional M almost Cα− manifold is a concircular pseudosymmetric manifold, then M is a real space form with a constant section curvature.Theorem 7.
If a2n+1− dimensional M almost Cα− manifold is a concircular Ricci pseudosymmetric manifold, then M is either a co-Keahler, α=r/2n2n+1 or λ7=0.Proof.
Let us assume that the manifoldM is a concircular Ricci pseudosymmetric manifold. Then, we can write as follows:(85)RU1,U2⋅Z˜U5,U4,U3=λ7QS,Z˜U5,U4,U3;U1,U2,for each U1,U2,U3,U4,U5∈χM. In this case, we yield the following:(86)RU1,U2Z˜U5,U4U3−Z˜RU1,U2U5,U4U3−Z˜U5,RU1,U2U4U3−Z˜U5,U4RU1,U2U3=−λ7Z˜U1ΛSU2U5,U4U3+Z˜U5,U1ΛSU2U4U3+Z˜U5,U4U1ΛSU2U3.
If necessary arrangements are made in here and we chooseU1=U5=ξ and using (8), (69), and (70) in (86), then we obtain as follows:(87)αgU2,Z˜ξ,U4U3ξ−αηZ˜ξ,U4U3U2−αα−r2n2n+1ηU2gU4,U3ξ+αα−r2n2n+1ηU3ηU2U4+αZ˜U2,U4U3+αα−r2n2n+1ηU3gU2,U4ξ−αα−r2n2n+1ηU3gU2,U4ξ+αηU4Z˜ξ,U2U3−αα−r2n2n+1ηU4gU2,U3ξ+αα−r2n2n+1gU2,U3U4+αηU3Z˜ξ,U4U2=−λ7α−r2n2n+1gU4,U3SU2,ξξ−α−r2n2n+1SU2,ξηU3U4−2nαZ˜U2,U4U3−α−r2n2n+1SU2,U4ηU3ξ+α−r2n2n+1SU2,U4ηU3ξ−2nαηU4Z˜ξ,U2U3+α−r2n2n+1SU2,U3ηU4ξ−α−r2n2n+1SU2,U3U4−2nαηU3Z˜ξ,U4U2.
If we use the expression (69) again in the last equation and make the necessary adjustments, then we obtain as follows:(88)−αα−r2n2n+1gU4,U3U2+αZ˜U2,U4U3+αα−r2n2n+1gU2,U3U4=−λ72nαα−r2n2n+1gU4,U3ηU2ξ−2nαZ˜U2,U4U3−2nαα−r2n2n+1gU2,U3ηU4ξ+2nαα−r2n2n+1ηU4ηU3U2+α−r2n2n+1SU2,U3ηU4ξ−α−r2n2n+1SU2,U3U4−2nαα−r2n2n+1gU4,U2ηU3ξ.
If we replace the expression (68) in (88) and make the necessary adjustments, we have as follows:(89)−α2gU4,U3U2+α2gU2,U3U4+αRU2,U4U3=−λ72nαα−r2n2n+1gU4,U3ηU2ξ−2nαRU2,U4U3+2nαr2n2n+1gU4,U3U2−2nαr2n2n+1gU2,U3U4−2nαα−r2n2n+1gU2,U3ηU4ξ+2nαα−r2n2n+1ηU4ηU3U2+α−r2n2n+1SU2,U3ηU4ξ−α−r2n2n+1SU2,U3U4−2nαα−r2n2n+1gU4,U2ηU3ξ.
If we applyξ∈χM to both sides of (89) and make use of (11), then we get(90)2nαα−r2n2n+1λ7−gU4,U2+ηU4ηU2=0.
On the other hand, we know that there is ag metric such that(91)gφU4,φU2=gU4,U2−ηU4ηU2,on a 2n+1 dimensional almost contact manifold [26]. Therefore, equation (90) can be written as follows:(92)2nαα−r2n2n+1λ7gφU4,φU2=0.
It is clear from the last equation that(93)α=0,λ7=0,orα=r2n2n+1.
This completes our proof.Corollary 7.
If a2n+1− dimensional M almost Cα− manifold is a concircular Ricci pseudosymmetric manifold, then M is either a co-Keahler, α=r/2n2n+1 or a concircular semisymmetric manifold.Corollary 8.
If a2n+1− dimensional M almost Cα− manifold is a concircular Ricci pseudosymmetric manifold, then M is a real space form with a constant section curvature.



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