Annals of mathematics and computer science, Feb 17, 2024
The object of the paper is to investigate invariant submanifolds of an almost α-cosymplectic (k, ... more The object of the paper is to investigate invariant submanifolds of an almost α-cosymplectic (k, µ, ν)-space. Then we have proved that the necessary and sufficient conditions for an invariant submanifold of an almost α-cosymplectic (k, µ, ν)-space to be totally geodesic under the some conditions. Consequently, we have obtained some interesting results invariant submanifolds of an almost α-cosymplectic (k, µ, ν)-space.
This research article is about the geometry of the Kenmotsu manifold. Some important properties s... more This research article is about the geometry of the Kenmotsu manifold. Some important properties such as the W7 • W5 = 0, W7 • W6 = 0, W7 • W7 = 0 , W7 • W8 = 0, W7 • W9 = 0 and W7 • W ⋆ 0 = 0 curvature conditions of the Kenmotsu manifold have been investigated.
Mathematical sciences and applications e-notes :, Jan 12, 2024
In this paper we present invariant submanifolds of an almost α-cosymplectic (k, µ, ν)-space. Then... more In this paper we present invariant submanifolds of an almost α-cosymplectic (k, µ, ν)-space. Then, we gave some results for an invariant submanifold of an almost α-cosymplectic (k, µ, ν)-space to be totally geodesic. As a result, we have discovered some interesting conclusions about invariant submanifolds of an almost cosymplectic (k, µ, ν)-space.
In the present paper we have studied the curvature tensor of a normal paracontact metric manifold... more In the present paper we have studied the curvature tensor of a normal paracontact metric manifold satisfying the conditions $ R(\xi,X) \widetilde{C}=0 $, $ \widetilde{C}(\xi,X)S=0 $, $ \widetilde{C}(\xi,X)P=0 $, $ \widetilde{C}(\xi,X)\widetilde{Z}=0 $ and pseudo quasi conformal flat, where $ R $, $ P $, $ S $, $ \widetilde{Z} $ and $ \widetilde{C} $ are the Riemannian curvature, projective curvature, Ricci, concircular curvature and pseudo-quasi conformal curvature tensors, respectively.
In this article, pseudoparallel submanifolds for almost Kenmotsu (κ, µ, ν) −space are investigate... more In this article, pseudoparallel submanifolds for almost Kenmotsu (κ, µ, ν) −space are investigated. The almost Kenmotsu (κ, µ, ν) −space is considered on the concircular curvature tensor. Submanifolds of these manifolds with properties such as concircular pseudoparallel, concircular 2−pseudoparallel, concircular Ricci generalized pseudoparallel, and concircular 2−Ricci generalized pseudoparallel has been characterized. Necessary and sufficient conditions are given for the invariant submanifolds of almost Kenmotsu (κ, µ, ν) −space to be total geodesic according to the behavior of the κ, µ, ν functions.
In this article, semi-symmetric generalized Sasakian space forms are investigated on some special... more In this article, semi-symmetric generalized Sasakian space forms are investigated on some special curvature tensors. Characterizations of generalized Sasakian space forms are obtained on some specially selected σ −curvature tensors. By examining the flatness of these σ −curvature tensors, the properties of generalized sasakian space forms are given. More importantly, the cases of σ −semi-symmetric generalized Sasakian space forms are discussed and the behavior of the manifold is examined for each case. Again, necessary and sufficient conditions have been obtained for σ −symmetric generalized Sasakian space forms to be Einstein manifolds.
Hacettepe Journal of Mathematics and Statistics, Mar 1, 2011
Warped product semi-invariant submanifolds were defined and studied in locally Riemannian product... more Warped product semi-invariant submanifolds were defined and studied in locally Riemannian product manifolds in (K. Matsumoto, On submanifolds of locally product Riemannian manifolds, TRU Math. 18 (2), 145-157, 1982). In this paper, we go on to study warped product semi-invariant submanifolds in locally decomposable Riemannian manifolds. We prove several fundamental properties of warped product semi-invariant submanifolds, and establish a general inequality for a warped product semi-invariant submanifold in a locally decomposable Riemannian manifold. After that, we investigate warped product semiinvariant submanifolds in a locally decomposable Riemannian manifold which satisfy the equality case of the inequality and obtain some new results.
In this article, generalized Saasakian space forms are discussed and invariant submanifolds of th... more In this article, generalized Saasakian space forms are discussed and invariant submanifolds of these space forms are examined. The curvature tensor chosen is of great importance when examining the characterization of a manifold. In this article, invariant submanifolds of generalized Sasakian space forms are characterized according to the W * 0 −curvature tensor and pseudoparallel submanifolds are investigated for these space forms.
Universal journal of mathematics and applications, Jul 1, 2023
In this paper, we consider Lorentz generalized Sasakian space forms admitting almost η−Ricci soli... more In this paper, we consider Lorentz generalized Sasakian space forms admitting almost η−Ricci solitons in some curvature tensors. Ricci pseudosymmetry concepts of Lorentz generalized Sasakian space forms admitting η−Ricci soliton have introduced according to the choice of some special curvature tensors such as Riemann, concircular, projective, M −projective, W 1 and W 2. Then, again according to the choice of the curvature tensor, necessary conditions are given for Lorentz generalized Sasakian space form admitting η−Ricci soliton to be Ricci semisymmetric. Then some characterizations are obtained and some classifications have made.
Fundamental journal of mathematics and applications, Mar 29, 2023
In this article, pseudoparallel submanifolds for almost Kenmotsu $\left( \kappa,\mu,\nu\right) -$... more In this article, pseudoparallel submanifolds for almost Kenmotsu $\left( \kappa,\mu,\nu\right) -$space are investigated. The almost Kenmotsu $\left( \kappa,\mu,\nu\right) -$space is considered on the concircular curvature tensor. Submanifolds of these manifolds with properties such as concircular pseudoparallel, concircular $2-$pseudoparallel, concircular Ricci generalized pseudoparallel, and concircular $2-$Ricci generalized pseudoparallel has been characterized. Necessary and sufficient conditions are given for the invariant submanifolds of almost Kenmotsu $\left( \kappa,\mu,\nu\right) -$space to be total geodesic according to the behavior of the $\kappa,\mu,\nu$ functions.
Communications in advanced mathematical sciences, Mar 31, 2023
In this paper, we consider pseudosymmetric Lorentz Sasakian space forms admitting almost η−Ricci ... more In this paper, we consider pseudosymmetric Lorentz Sasakian space forms admitting almost η−Ricci solitons in some curvature tensors. Ricci pseudosymmetry concepts of Lorentz Sasakian space forms admits η−Ricci soliton have introduced according to the choice of some special curvature tensors such as Riemann, concircular, projective, M −projective, W 1 and W 2. Then, again according to the choice of the curvature tensor, necessary conditions are given for Lorentz Sasakian space form admits η−Ricci soliton to be Ricci semisymmetric. Then some characterizations are obtained and some classifications have made under the some conditions.
In this paper we have investigated the existence of warped product semi-invariant submanifolds in... more In this paper we have investigated the existence of warped product semi-invariant submanifolds in almost para contact metric manifolds. Finally, we see that there exists no any warped product semiinvariant submanifold in almost para contact metric manifolds such that the contravariant vector field tangent to submanifold.
In this paper, the geometry of F-invariant submanifolds of a Kaehlerian product manifold is studi... more In this paper, the geometry of F-invariant submanifolds of a Kaehlerian product manifold is studied. The fundamental properties of these submanifolds are investigated such as pseudo umbilical, curvature invariant, totally geodesic, mixed geodesic submanifold and locally decomposable Riemannian product manifold.
Balkan Journal of Geometry and its Applications (BJGA), 2003
In this paper, the geometry of submanifolds of a Riemannian product manifold is studied. Fundamen... more In this paper, the geometry of submanifolds of a Riemannian product manifold is studied. Fundamental properties of these submanifolds are investigated such as integrability of distributions, totally umbilical semi-invariant submanifold. Finally, necessary and sufficient conditions are given on a semi-invariant submanifold of a Riemannian product manifold to be a locally Riemannian manifold.
The aim of the present paper is to study some geometric conditions for an invariant submanifold o... more The aim of the present paper is to study some geometric conditions for an invariant submanifold of an almost ?-cosymplectic (?,?,?)-space to be totally geodesic by means of the curvature tensors.
In this paper, we generalize the geometry of the invariant submanifolds of Riemannian product man... more In this paper, we generalize the geometry of the invariant submanifolds of Riemannian product manifold to the geometry of the invariant submanifolds of Riemannian warped product manifold. We investigate some properties of an invariant submanifolds of a Riemannian warped product manifold. We show that every invariant submanifold of the Riemannian warped product manifold is a Riemannian warped product manifold. Also, we give a theorem on the pseudo-umbilical invariant submanifold. Further, we obtain that integral manifolds on an invariant submanifold are curvature-invariant submanifolds. Finally, we give a necessary condititon on a totally umbilical invariant submanifold to be totally geodesic.
International electronic journal of geometry, Oct 30, 2012
In [3], a new structure is defined in an even dimensional on a differentiable Riemannian manifold... more In [3], a new structure is defined in an even dimensional on a differentiable Riemannian manifold which is called almost pseudo contact structure. In this paper, we introduce the fundamental properties of submanifolds of an almost pseudo contact metric manifold and research the induced structures on submanifold. They have been categorized. 2000 Mathematics Subject Classification. 53C15, 53C25. Key words and phrases. Almost Hermite structure, Almost contact metric structure, almost pseudo contact metric structure and invariant submanifold.
Annals of mathematics and computer science, Feb 17, 2024
The object of the paper is to investigate invariant submanifolds of an almost α-cosymplectic (k, ... more The object of the paper is to investigate invariant submanifolds of an almost α-cosymplectic (k, µ, ν)-space. Then we have proved that the necessary and sufficient conditions for an invariant submanifold of an almost α-cosymplectic (k, µ, ν)-space to be totally geodesic under the some conditions. Consequently, we have obtained some interesting results invariant submanifolds of an almost α-cosymplectic (k, µ, ν)-space.
This research article is about the geometry of the Kenmotsu manifold. Some important properties s... more This research article is about the geometry of the Kenmotsu manifold. Some important properties such as the W7 • W5 = 0, W7 • W6 = 0, W7 • W7 = 0 , W7 • W8 = 0, W7 • W9 = 0 and W7 • W ⋆ 0 = 0 curvature conditions of the Kenmotsu manifold have been investigated.
Mathematical sciences and applications e-notes :, Jan 12, 2024
In this paper we present invariant submanifolds of an almost α-cosymplectic (k, µ, ν)-space. Then... more In this paper we present invariant submanifolds of an almost α-cosymplectic (k, µ, ν)-space. Then, we gave some results for an invariant submanifold of an almost α-cosymplectic (k, µ, ν)-space to be totally geodesic. As a result, we have discovered some interesting conclusions about invariant submanifolds of an almost cosymplectic (k, µ, ν)-space.
In the present paper we have studied the curvature tensor of a normal paracontact metric manifold... more In the present paper we have studied the curvature tensor of a normal paracontact metric manifold satisfying the conditions $ R(\xi,X) \widetilde{C}=0 $, $ \widetilde{C}(\xi,X)S=0 $, $ \widetilde{C}(\xi,X)P=0 $, $ \widetilde{C}(\xi,X)\widetilde{Z}=0 $ and pseudo quasi conformal flat, where $ R $, $ P $, $ S $, $ \widetilde{Z} $ and $ \widetilde{C} $ are the Riemannian curvature, projective curvature, Ricci, concircular curvature and pseudo-quasi conformal curvature tensors, respectively.
In this article, pseudoparallel submanifolds for almost Kenmotsu (κ, µ, ν) −space are investigate... more In this article, pseudoparallel submanifolds for almost Kenmotsu (κ, µ, ν) −space are investigated. The almost Kenmotsu (κ, µ, ν) −space is considered on the concircular curvature tensor. Submanifolds of these manifolds with properties such as concircular pseudoparallel, concircular 2−pseudoparallel, concircular Ricci generalized pseudoparallel, and concircular 2−Ricci generalized pseudoparallel has been characterized. Necessary and sufficient conditions are given for the invariant submanifolds of almost Kenmotsu (κ, µ, ν) −space to be total geodesic according to the behavior of the κ, µ, ν functions.
In this article, semi-symmetric generalized Sasakian space forms are investigated on some special... more In this article, semi-symmetric generalized Sasakian space forms are investigated on some special curvature tensors. Characterizations of generalized Sasakian space forms are obtained on some specially selected σ −curvature tensors. By examining the flatness of these σ −curvature tensors, the properties of generalized sasakian space forms are given. More importantly, the cases of σ −semi-symmetric generalized Sasakian space forms are discussed and the behavior of the manifold is examined for each case. Again, necessary and sufficient conditions have been obtained for σ −symmetric generalized Sasakian space forms to be Einstein manifolds.
Hacettepe Journal of Mathematics and Statistics, Mar 1, 2011
Warped product semi-invariant submanifolds were defined and studied in locally Riemannian product... more Warped product semi-invariant submanifolds were defined and studied in locally Riemannian product manifolds in (K. Matsumoto, On submanifolds of locally product Riemannian manifolds, TRU Math. 18 (2), 145-157, 1982). In this paper, we go on to study warped product semi-invariant submanifolds in locally decomposable Riemannian manifolds. We prove several fundamental properties of warped product semi-invariant submanifolds, and establish a general inequality for a warped product semi-invariant submanifold in a locally decomposable Riemannian manifold. After that, we investigate warped product semiinvariant submanifolds in a locally decomposable Riemannian manifold which satisfy the equality case of the inequality and obtain some new results.
In this article, generalized Saasakian space forms are discussed and invariant submanifolds of th... more In this article, generalized Saasakian space forms are discussed and invariant submanifolds of these space forms are examined. The curvature tensor chosen is of great importance when examining the characterization of a manifold. In this article, invariant submanifolds of generalized Sasakian space forms are characterized according to the W * 0 −curvature tensor and pseudoparallel submanifolds are investigated for these space forms.
Universal journal of mathematics and applications, Jul 1, 2023
In this paper, we consider Lorentz generalized Sasakian space forms admitting almost η−Ricci soli... more In this paper, we consider Lorentz generalized Sasakian space forms admitting almost η−Ricci solitons in some curvature tensors. Ricci pseudosymmetry concepts of Lorentz generalized Sasakian space forms admitting η−Ricci soliton have introduced according to the choice of some special curvature tensors such as Riemann, concircular, projective, M −projective, W 1 and W 2. Then, again according to the choice of the curvature tensor, necessary conditions are given for Lorentz generalized Sasakian space form admitting η−Ricci soliton to be Ricci semisymmetric. Then some characterizations are obtained and some classifications have made.
Fundamental journal of mathematics and applications, Mar 29, 2023
In this article, pseudoparallel submanifolds for almost Kenmotsu $\left( \kappa,\mu,\nu\right) -$... more In this article, pseudoparallel submanifolds for almost Kenmotsu $\left( \kappa,\mu,\nu\right) -$space are investigated. The almost Kenmotsu $\left( \kappa,\mu,\nu\right) -$space is considered on the concircular curvature tensor. Submanifolds of these manifolds with properties such as concircular pseudoparallel, concircular $2-$pseudoparallel, concircular Ricci generalized pseudoparallel, and concircular $2-$Ricci generalized pseudoparallel has been characterized. Necessary and sufficient conditions are given for the invariant submanifolds of almost Kenmotsu $\left( \kappa,\mu,\nu\right) -$space to be total geodesic according to the behavior of the $\kappa,\mu,\nu$ functions.
Communications in advanced mathematical sciences, Mar 31, 2023
In this paper, we consider pseudosymmetric Lorentz Sasakian space forms admitting almost η−Ricci ... more In this paper, we consider pseudosymmetric Lorentz Sasakian space forms admitting almost η−Ricci solitons in some curvature tensors. Ricci pseudosymmetry concepts of Lorentz Sasakian space forms admits η−Ricci soliton have introduced according to the choice of some special curvature tensors such as Riemann, concircular, projective, M −projective, W 1 and W 2. Then, again according to the choice of the curvature tensor, necessary conditions are given for Lorentz Sasakian space form admits η−Ricci soliton to be Ricci semisymmetric. Then some characterizations are obtained and some classifications have made under the some conditions.
In this paper we have investigated the existence of warped product semi-invariant submanifolds in... more In this paper we have investigated the existence of warped product semi-invariant submanifolds in almost para contact metric manifolds. Finally, we see that there exists no any warped product semiinvariant submanifold in almost para contact metric manifolds such that the contravariant vector field tangent to submanifold.
In this paper, the geometry of F-invariant submanifolds of a Kaehlerian product manifold is studi... more In this paper, the geometry of F-invariant submanifolds of a Kaehlerian product manifold is studied. The fundamental properties of these submanifolds are investigated such as pseudo umbilical, curvature invariant, totally geodesic, mixed geodesic submanifold and locally decomposable Riemannian product manifold.
Balkan Journal of Geometry and its Applications (BJGA), 2003
In this paper, the geometry of submanifolds of a Riemannian product manifold is studied. Fundamen... more In this paper, the geometry of submanifolds of a Riemannian product manifold is studied. Fundamental properties of these submanifolds are investigated such as integrability of distributions, totally umbilical semi-invariant submanifold. Finally, necessary and sufficient conditions are given on a semi-invariant submanifold of a Riemannian product manifold to be a locally Riemannian manifold.
The aim of the present paper is to study some geometric conditions for an invariant submanifold o... more The aim of the present paper is to study some geometric conditions for an invariant submanifold of an almost ?-cosymplectic (?,?,?)-space to be totally geodesic by means of the curvature tensors.
In this paper, we generalize the geometry of the invariant submanifolds of Riemannian product man... more In this paper, we generalize the geometry of the invariant submanifolds of Riemannian product manifold to the geometry of the invariant submanifolds of Riemannian warped product manifold. We investigate some properties of an invariant submanifolds of a Riemannian warped product manifold. We show that every invariant submanifold of the Riemannian warped product manifold is a Riemannian warped product manifold. Also, we give a theorem on the pseudo-umbilical invariant submanifold. Further, we obtain that integral manifolds on an invariant submanifold are curvature-invariant submanifolds. Finally, we give a necessary condititon on a totally umbilical invariant submanifold to be totally geodesic.
International electronic journal of geometry, Oct 30, 2012
In [3], a new structure is defined in an even dimensional on a differentiable Riemannian manifold... more In [3], a new structure is defined in an even dimensional on a differentiable Riemannian manifold which is called almost pseudo contact structure. In this paper, we introduce the fundamental properties of submanifolds of an almost pseudo contact metric manifold and research the induced structures on submanifold. They have been categorized. 2000 Mathematics Subject Classification. 53C15, 53C25. Key words and phrases. Almost Hermite structure, Almost contact metric structure, almost pseudo contact metric structure and invariant submanifold.
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