#### Abstract

In this paper, pseudosymmetric and Ricci pseudosymmetric almost -manifold are studied. For an almost -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.

#### 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 is and for each , if , then the manifold is said to be semisymmetric. Similarly, if , the manifold is called Ricci semisymmetric, if ; the manifold is called projective semisymmetric, if ; and the manifold is called concircular semisymmetric, where is the Ricci curvature tensor, is the projective curvature tensor, and is concircular curvature tensor. Studies on the symmetric Riemannian manifolds started with Cartan . In the following periods, many authors have studied the symmetry cases of various manifolds ().

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 , while in 2019, a very important study was made for cosymplectic manifolds by Kupeli . Many studies have also been made about almost manifolds in  and important characterizations of this type of manifolds have been obtained ([19, 20]).

Let be the dimensional almost contact metric manifold given by the contact structure, and let the Riemann curvature tensor of the manifold be . If manifold satisfies the following condition:Then, manifold is called almost manifold, where every and at least one .

A manifold is the general case of the co-Keahler, Sasakian, and Kenmotsu manifolds. That is, specially, if , it is co-Keahler, if , it is Sasakian, if , it is the Kenmotsu manifold . Although many studies have been conducted on these manifolds, studies on manifold is quite limited.

In this article, the pseudosymmetry and Ricci pseudosymmetry properties of almost 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

Let be a differentiable manifold with dimensional. If the conditionsatisfies on , where is tensor field with type , is a vector field and is a form, then we say that is an almost contact structure. Also, we say that is an almost contact manifold . Let be a metric with conditionfor all and . In this case, we say that is an almost contact metric structure and is almost contact metric manifold . Moreover, we have the propertyfor all on manifold with dimensional. The fundamental 2-form of the almost contact metric structure is the transformation such thatfor all , where

We can write the Riemann curvature tensor of an almost -manifold which has a constant sectional curvature by

If we take in (4), then we obtain as follows:

If we take in (4), then we have as follows:

Moreover, if we take in (9), then we obtain as follows:

Let us take an inner product of (7) by . Then, we obtain as follows:

Let for be a -type tensor field, a be a -type tensor field and is a Riemannian manifold. We define -tensor field byfor all , where

Lemma 1. Let be an almost manifold with dimensional. Then, we have as follows:for each , where is the Ricci operator and is the Ricci tensor of manifold .

If and are linearly dependent, that is, there is a constant providedthe almost manifold is called the Einstein manifold . In particular, if the Ricci tensor satisfies the relationfor each , M is called an Einstein manifold .

Let (M, ) be a Riemannian manifold. For two-dimensional subspace of the tangent, let space befor each . In this case, defined as followsis called the sectional curvature of the plane . If is a constant for each , is called a constant section curvature space or real space form . In this case, if the Riemann manifold is a real space form and constant section curvature, the Riemann curvature tensor of is as follows:for each .

#### 3. Pseudosymmetric and Ricci Pseudosymmetric Almost -Manifold

In this section, the cases of pseudosymmetry and Ricci pseudosymmetry of an almost manifold are investigated. According to Riemann, Ricci, projective, and concircular curvature tensors, the pseudosymmetrical and Ricci pseudosymmetrical cases of the almost manifold can be given as follows.

Definition 1. Let be an almost manifold with dimensional, be the Riemann curvature tensor of , and be the Ricci curvature tensor of .(i)If the pair and are linearly dependent, that is, if a function can be found on the set such thatthe manifold is called a Riemann pseudosymmetric manifold.(ii)If the pair and are linearly dependent, that is, if a function can be found on the set such thatthe manifold is called a Riemann Ricci pseudosymmetric manifold.;Particularly, if , then this manifold is said to be semisymmetric.Let us now investigate the cases of Riemann pseudosymmetry and Riemann Ricci pseudosymmetry.

Theorem 1. If a dimensional almost manifold is a Riemann pseudo-symmetric manifold, then .

Proof. Let us assume that the manifold is a Riemann pseudosymmetric manifold. Then, we can writefor each . In this case, we obtain as follows:If necessary arrangements are made in here, and we choose and using the expression (8) in (26), we obtain as follows:If we use the expression (8) again in (27) and make the necessary adjustments, then we obtain as follows:If we apply to both sides of (28) and make use of (11), then we obtain as follows:Since , it is clear thatThis completes our proof.

Corollary 1. If a dimensional almost manifold is a Riemann pseudosymmetric manifold, then is a Riemann semisymmetric manifold.

Corollary 2. If a dimensional almost manifold is a Riemann pseudosymmetric manifold, then is a real space form with a constant section curvature.

Theorem 2. If a dimensional almost manifold is a Riemann Ricci pseudosymmetric manifold, then is either co-Keahler manifold or .

Proof. Let us assume that the manifold is a Riemann Ricci pseudosymmetric manifold. Then, we can write as follows:for each . In this case, we obtain as follows:If necessary arrangements are made in here, and we choose and using (8), (15) in (32), we obtain as follows:If we use the expression (5) again in (33) and make the necessary adjustments, then we obtain as follows:If we apply to both sides of (34) and make use of (11), then we obtain as follows:Since , it is clear thatThus, either or is obtained. This completes our proof.

Corollary 3. If a dimensional almost manifold is a Riemann Ricci pseudosymmetric manifold, then is either co-Keahler manifold or a Riemann semisymmetric manifold.

Corollary 4. If a dimensional almost manifold is a Riemann Ricci pseudosymmetric manifold, then is a real space form with a constant section curvature.

Definition 2. Let be an almost manifold with dimensional, be the Riemann curvature tensor of and be the Ricci curvature tensor of . If the pair and are linearly dependent, that is, if a function can be found on the set such thatthe manifold is called a Ricci pseudosymmetric manifold.
Let us now investigate the case of Ricci pseudosymmetry.

Theorem 3. If a dimensional almost manifold is a Ricci pseudosymmetric manifold, then is either an Einstein manifold or .

Proof. Let us assume that the manifold is a Ricci pseudosymmetric manifold. Then, we can write as follows:for each . In this case, we can write as follows:In equation (39), first is chosen and then (8) and (15) are used, we obtain as follows:If we choose in (40) and make the necessary adjustments, we obtain as follows:and from here, we can write thatIt is clear from equation (42) that eitherThis completes the proof of the theorem.
The projective curvature tensor is defined as follows:for all each , by Yano and Sawaki . If , and are selected, respectively, in (45), then we obtain as follows:Let us take inner product of (45) by , we get

Definition 3. Let be a dimensional almost manifold, be the Riemann curvature tensor of , be the Ricci curvature tensor of , and be the projective curvature tensor of .(i)If the pair and are linearly dependent, that is, if a function can be found on the set such thatthe manifold is called a projective pseudosymmetric manifold.(ii)If the pair and are linearly dependent, that is, if a function can be found on the set such thatThe manifold is called a projective Ricci pseudosymmetric manifold.Let us now investigate the case of projective pseudosymmetry and projective Ricci pseudosymmetry.

Theorem 4. If a dimensional almost manifold is a projective pseudosymmetric manifold, then is either an Einstein manifold or .

Proof. Let us assume that the manifold is a projective pseudosymmetric manifold. Then, we can write as follows:for each . In this case, we obtain as follows:In the last equation, if is chosen and necessary corrections are made, we can writeIf (8), (46), and (47) are used in (54), we obtain follows:If is chosen in (55), equation (8) is used and necessary adjustments are made, we obtain as follows:If we apply to both sides of equation (54) and make use of equation (11), we obtain as follows:If is chosen in the last equation, we obtain as follows:that is,Thus, the proof of the theorem is complete.

Theorem 5. If a dimensional almost manifold is a projective Ricci pseudosymmetric manifold, then is either an Einstein manifold, co-Keahler or .

Proof. Let us assume that the manifold is a projective Ricci pseudosymmetric manifold. Then, we can writefor each . In this case, we obtain as follows:In the last equation, if is chosen and necessary corrections are made, we can writeIf (8), (46), and (47) are used in (62), we obtain as follows:If is chosen in (63), equation (8) is used and necessary adjustments are made, we obtain as follows:If we apply to both sides of equation (65) and make use of equation (11), we obtain as follows:If is chosen in the last equation, we yield as follows:that is,Thus, the proof of the theorem is complete.
Let be a Riemannian manifold with dimensional. Then, the concircular curvature tensor is defined asfor each . If we choose, respectively, , and in (68), then we obtain as follows:In addition, we choose in (69), we obtain as follows:and finally, we take inner product of (68) by , we have

Definition 4. Let be a dimensional almost manifold, be the Riemann curvature tensor of , be the Ricci curvature tensor of , and be the concircular curvature tensor of .(i)If the pair and are linearly dependent, that is, if a function can be found on the set such thatthe manifold is called a concircular pseudo-symmetric manifold.(ii)If the pair and are linearly dependent, that is, if a function can be found on the set such thatthe manifold is called a concircular Ricci pseudosymmetric manifold.Let us now investigate the cases of concircular pseudosymmetry and concircular Ricci pseudosymmetry.

Theorem 6. If a dimensional almost manifold is a concircular pseudosymmetric manifold, then we can see either or .

Proof. Let us assume that the manifold is a concircular pseudosymmetric manifold. Then, we can writefor each . In this case, we yield as follows:If necessary arrangements are made in here and we choose and using (8), (69) in (77), we obtain as follows:If we use the expression (69) again in (78) and make the necessary adjustments, then we obtain as follows:If we replace the expression (68) in (79) and make the necessary adjustments, we have the following:If we apply to both sides of (80) and make use of (11), then we obtain as follows:On the other hand, we know that there is a metric such thaton a dimensional almost contact manifold . Therefore, equation (81) can be written as follows:It is clear from the last equation thatThis completes our proof.

Corollary 5. If a dimensional almost manifold is a concircular pseudosymmetric manifold, then is either or a concircular semisymmetric manifold.

Corollary 6. If a dimensional almost manifold is a concircular pseudosymmetric manifold, then is a real space form with a constant section curvature.

Theorem 7. If a dimensional almost manifold is a concircular Ricci pseudosymmetric manifold, then is either a co-Keahler, or .

Proof. Let us assume that the manifold is a concircular Ricci pseudosymmetric manifold. Then, we can write as follows: