The purpose of this paper is to find some conditions under which the tangent bundle TM has a dual... more The purpose of this paper is to find some conditions under which the tangent bundle TM has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle of a statistical manifold too. Moreover, we also study the mutual curvatures of a statistical manifold M and its tangent bundle TM and we investigate their relations. More precisely, we obtain the mutual curvatures of well-known connections on the tangent bundle TM (the complete, horizontal, and Sasaki connections) and we study the vanishing of them.
In this paper, we improve the Chen first inequality for special contact slant submanifolds and Le... more In this paper, we improve the Chen first inequality for special contact slant submanifolds and Legendrian submanifolds, respectively, in (α,β) trans-Sasakian generalized Sasakian space forms endowed with a semi-symmetric metric connection.
In this paper, we consider M θ , a pointwise slant submanifold and prove that every bi-warped pro... more In this paper, we consider M θ , a pointwise slant submanifold and prove that every bi-warped product M ⊥ × f1 M T × f2 M θ in a locally product Riemannian manifold satisfies a general inequality: σ 2 ≥ n 2 ∇ T (ln f 1) 2 + n 3 cos 2 θ ∇ θ (ln f 2) 2 , where n 2 = dim(M T), n 3 = dim(M θ) and σ is the second fundamental form and ∇ T (ln f 1) and ∇ θ (ln f 2) are the gradient components along M T and M θ , respectively. We also discuss the equality case of this inequality. Furthermore, we give some applications and non-trivial examples.
This volume is a selection of lectures presented at the International Conference “Riemannian Geom... more This volume is a selection of lectures presented at the International Conference “Riemannian Geometry and Applications” held in Brasov, June 21–25, 2007.
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2021
Introducing left invariant almost para-contact metric structures on Hom-Lie groups (Hom-Lie algeb... more Introducing left invariant almost para-contact metric structures on Hom-Lie groups (Hom-Lie algebras), we study the normal and para-Sasakian conditions on them. The left-invariant curvature tensor properties of left-invariant para-Sasakian Hom-Lie groups are investigated. Some examples are commented as support of the obtained results.
We consider a statistical connection ∇ on an almost complex manifold with (pseudo-) Riemannian me... more We consider a statistical connection ∇ on an almost complex manifold with (pseudo-) Riemannian metric, in particular the Norden metric. We investigate almost Norden (statistical) manifolds under the condition that the almost complex structure J is ∇-recurrent. We provide one example of a complex statistical connection.
Chen’s first inequality for statistical submanifolds in Hessian manifolds of constant Hessian cur... more Chen’s first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature was obtained by B.-Y. Chen et al. Other particular cases of Chen inequalities in a statistical setting were given by different authors. The objective of the present article is to establish the general Chen inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature.
In this paper, we obtain a geometric inequality for warped product pointwise semi-slant submanifo... more In this paper, we obtain a geometric inequality for warped product pointwise semi-slant submanifolds of complex space forms endowed with a semi-symmetric metric connection and discuss the equality case of this inequality. We provide some applications concerning the minimality and compactness of such submanifolds.
We consider a statistical connection ∇ on an almost complex manifold with (pseudo-) Riemannian me... more We consider a statistical connection ∇ on an almost complex manifold with (pseudo-) Riemannian metric, in particular the Norden metric. We investigate almost Norden (statistical) manifolds under the condition that the almost complex structure J is ∇-recurrent. We provide one example of a complex statistical connection.
The objective of the present article is to prove two geometric inequalities for submanifolds in S... more The objective of the present article is to prove two geometric inequalities for submanifolds in S-space forms. First, we establish inequalities for the generalized normalized δ-Casorati curvatures for bi-slant submanifolds in S-space forms and then we derive the generalized Wintgen inequality for Legendrian and bi-slant submanifolds in the same ambient space. We also discuss the equality cases of the inequalities. Further, we provide some immediate geometric applications of the results. Finally, we construct some examples of slant and Legendrian submanifolds, respectively.
In the present paper, weinvestigate some properties of the distributions involved in the definiti... more In the present paper, weinvestigate some properties of the distributions involved in the definition of a CR-statistical submanifold. The characterization of a CR-product in holomorphic statistical manifolds is given. By using an optimization technique, we establish a relationship between the Ricci curvature and the squared norm of the mean curvature of any submanifold in the same ambient space. The equality case is also discussed here. This paper finishes with some related examples.
The purpose of this paper is to find some conditions under which the tangent bundle TM has a dual... more The purpose of this paper is to find some conditions under which the tangent bundle TM has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle of a statistical manifold too. Moreover, we also study the mutual curvatures of a statistical manifold M and its tangent bundle TM and we investigate their relations. More precisely, we obtain the mutual curvatures of well-known connections on the tangent bundle TM (the complete, horizontal, and Sasaki connections) and we study the vanishing of them.
In this paper, we improve the Chen first inequality for special contact slant submanifolds and Le... more In this paper, we improve the Chen first inequality for special contact slant submanifolds and Legendrian submanifolds, respectively, in (α,β) trans-Sasakian generalized Sasakian space forms endowed with a semi-symmetric metric connection.
In this paper, we consider M θ , a pointwise slant submanifold and prove that every bi-warped pro... more In this paper, we consider M θ , a pointwise slant submanifold and prove that every bi-warped product M ⊥ × f1 M T × f2 M θ in a locally product Riemannian manifold satisfies a general inequality: σ 2 ≥ n 2 ∇ T (ln f 1) 2 + n 3 cos 2 θ ∇ θ (ln f 2) 2 , where n 2 = dim(M T), n 3 = dim(M θ) and σ is the second fundamental form and ∇ T (ln f 1) and ∇ θ (ln f 2) are the gradient components along M T and M θ , respectively. We also discuss the equality case of this inequality. Furthermore, we give some applications and non-trivial examples.
This volume is a selection of lectures presented at the International Conference “Riemannian Geom... more This volume is a selection of lectures presented at the International Conference “Riemannian Geometry and Applications” held in Brasov, June 21–25, 2007.
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2021
Introducing left invariant almost para-contact metric structures on Hom-Lie groups (Hom-Lie algeb... more Introducing left invariant almost para-contact metric structures on Hom-Lie groups (Hom-Lie algebras), we study the normal and para-Sasakian conditions on them. The left-invariant curvature tensor properties of left-invariant para-Sasakian Hom-Lie groups are investigated. Some examples are commented as support of the obtained results.
We consider a statistical connection ∇ on an almost complex manifold with (pseudo-) Riemannian me... more We consider a statistical connection ∇ on an almost complex manifold with (pseudo-) Riemannian metric, in particular the Norden metric. We investigate almost Norden (statistical) manifolds under the condition that the almost complex structure J is ∇-recurrent. We provide one example of a complex statistical connection.
Chen’s first inequality for statistical submanifolds in Hessian manifolds of constant Hessian cur... more Chen’s first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature was obtained by B.-Y. Chen et al. Other particular cases of Chen inequalities in a statistical setting were given by different authors. The objective of the present article is to establish the general Chen inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature.
In this paper, we obtain a geometric inequality for warped product pointwise semi-slant submanifo... more In this paper, we obtain a geometric inequality for warped product pointwise semi-slant submanifolds of complex space forms endowed with a semi-symmetric metric connection and discuss the equality case of this inequality. We provide some applications concerning the minimality and compactness of such submanifolds.
We consider a statistical connection ∇ on an almost complex manifold with (pseudo-) Riemannian me... more We consider a statistical connection ∇ on an almost complex manifold with (pseudo-) Riemannian metric, in particular the Norden metric. We investigate almost Norden (statistical) manifolds under the condition that the almost complex structure J is ∇-recurrent. We provide one example of a complex statistical connection.
The objective of the present article is to prove two geometric inequalities for submanifolds in S... more The objective of the present article is to prove two geometric inequalities for submanifolds in S-space forms. First, we establish inequalities for the generalized normalized δ-Casorati curvatures for bi-slant submanifolds in S-space forms and then we derive the generalized Wintgen inequality for Legendrian and bi-slant submanifolds in the same ambient space. We also discuss the equality cases of the inequalities. Further, we provide some immediate geometric applications of the results. Finally, we construct some examples of slant and Legendrian submanifolds, respectively.
In the present paper, weinvestigate some properties of the distributions involved in the definiti... more In the present paper, weinvestigate some properties of the distributions involved in the definition of a CR-statistical submanifold. The characterization of a CR-product in holomorphic statistical manifolds is given. By using an optimization technique, we establish a relationship between the Ricci curvature and the squared norm of the mean curvature of any submanifold in the same ambient space. The equality case is also discussed here. This paper finishes with some related examples.
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Papers by Ion Mihai