On some submanifolds of a locally product manifold

Pitiş, Gheorghe

doi:10.2996/kmj/1138037261

Cited by 14 publications

(12 citation statements)

References 5 publications

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“…Using an analogy of a locally product manifold ( [29]), we can define locally metallic (or locally Golden) Riemannian manifold, as follows ( [25]):…”

Section: Slant and Semi-slant Submanifolds In Metallic Riemannian Man...mentioning

confidence: 99%

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Slant and Semi-Slant Submanifolds in Metallic Riemannian Manifolds

Hreƫcanu

Blaga

2018

Journal of Function Spaces

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In this paper, we study the existence of proper warped product submanifolds in metallic (or Golden) Riemannian manifolds and we discuss about semi-invariant, semi-slant and, respectively, hemi-slant warped product submanifolds in metallic and Golden Riemannian manifolds. Also, we provide some examples of warped product submanifolds in Euclidean spaces.2010 Mathematics Subject Classification. 53B20, 53B25, 53C42, 53C15. Key words and phrases. Metallic Riemannian structure, Golden Riemannian structure, warped product submanifold, bi-slant submanifold, semi-invariant submanifold, semi-slant submanifold, hemi-slant submanifold. 1 2 CRISTINA E. HRETCANU AND ADARA M. BLAGA of a Kähler manifold and he found some properties of warped product submanifolds of the form M ⊥ × f M θ ([27]).Semi-invariant submanifolds in locally product Riemannian manifolds were studied in ([25], [2]). Semi-slant submanifolds in locally Riemannian product manifolds were studied by M. Atçeken which found that, in a locally Riemannian product manifold does not exist any warped product semi-slant submanifold of the forma proper slant submanifold and M ⊥ is an anti-invariant submanifold, but he found some examples of warped product semi-slant submanifolds of the form M θ × f M T and of the form M θ × f M ⊥ ([1]). Warped product submanifolds of the form M θ × f M T and M θ × f M ⊥ in Riemannian product manifolds were also studied by F. R. Al-Solamy and M. A. Khan ([28]). Warped product pseudo-slant (named also hemi-slant) submanifolds of the form M θ × f M ⊥ , where M θ and M ⊥ are proper slant and, respectively, anti-invariant submanifolds, in a locally product Riemannian manifold were studied by S. Uddin et al. in ([30]). Recently, warped product bi-slant submanifolds in Kähler manifolds were studied by S. Uddin et al. and some examples of this type of submanifolds in complex Euclidean spaces were constructed ([31]). Moreover, L. S. Alqahtani et al. have shown that there is no proper warped product bi-slant submanifold other than pseudo-slant warped product in cosymplectic manifolds ([4]).The authors of the present paper studied some properties of invariant, antiinvariant and slant submanifolds ([7]), semi-slant submanifolds ([19]) and, respectively, hemi-slant submanifolds ([18]) in metallic and Golden Riemannian manifolds and they obtained integrability conditions for the distributions involved in these types of submanifolds. Moreover, properties of metallic and Golden warped product Riemannian manifolds were presented in the two previews works of the authors ([6], [9]).In the present paper, we study the existence of proper warped product bi-slant submanifolds in locally metallic Riemannian manifolds. In Sections 2 and 3, we remind the main properties of metallic and Golden Riemannian manifolds and of their submanifolds. In Section 4, we discuss about slant and bi-slant submanifolds (with their particular cases: semi-slant and hemi-slant submanifolds) in locally metallic (or Golden) Riemannian manifolds. In Section 5, we find some properties of...

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“…Using an analogy of a locally product manifold ( [29]), we can define locally metallic (or locally Golden) Riemannian manifold, as follows ( [25]):…”

Section: Slant and Semi-slant Submanifolds In Metallic Riemannian Man...mentioning

confidence: 99%

“…Using an analogy of a locally product manifold ( [9]), we can define locally metallic Riemannian manifold, as follows: Definition 2. If (M , g, J) is a metallic Riemannian manifold and J is parallel with respect to the Levi-Civita connection ∇ on M (i.e.…”

Section: σ-Metallic Riemannian Structuresmentioning

confidence: 99%

Slant and Semi-Slant Submanifolds in Metallic Riemannian Manifolds

Hreƫcanu

Blaga

2018

Journal of Function Spaces

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“…where J is a (1, 1) tensor field on M, I is the identity operator on the Lie algebra Γ(T M ) of vector fields on M and p, q are real numbers. This structure can be also viewed as a generalization of following well known structures : · If p = 0, q = 1, then J is called almost product or almost para complex structure and denoted by F [15], [12], · If p = 0, q = −1, then J is called almost complex structure [17], · If p = 1, q = 1, then J is called golden structure [6], [7], · If p is positive integer and q = −1, then J is called poly-Norden structure [16], · If p = 1, q = −3 2 , then J is called almost complex golden structure [3], · If p and q are positive integers, then J is called metallic structure [11]. If a differentiable manifold endowed with a metallic structure J then the pair (M, J) is called metallic manifold.…”

Section: Introductionmentioning

confidence: 99%

“…• If p = 0 , q = 1 , then J is called an almost product or almost para complex structure and denoted by F [12,16],;…”

Section: Introductionmentioning

confidence: 99%

A Neutral relation between metallic structure and almost quadratic φ -structure

Gönül¹,

Erken²,

Yazla³

et al. 2019

Turk J Math

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“…Furthermore, the submanifolds of a locally product Riemannian manifold have been studied by many geometers. For example, Adati [1] defined and studied invariant and anti-invariant submanifolds, while Bejancu [5] and Pitis [15] studied semi-invariant submanifolds. Slant and semi-slant submanifolds of a locally product Riemannian manifold are examined by Şahin [17] and Li and Liu [12].…”

Section: Introductionmentioning

confidence: 99%

The geometry of hemi-slant submanifolds of a locally product Riemannian manifold

Taştan

Özdemir²

2015

Turk J Math

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In the present paper, we study hemi-slant submanifolds of a locally product Riemannian manifold. We prove that the anti-invariant distribution involved in the definition of hemi-slant submanifold is integrable and give some applications of this result. We get a necessary and sufficient condition for a proper hemi-slant submanifold to be a hemi-slant product. We also study these types of submanifolds with parallel canonical structures. Moreover, we give two characterization theorems for the totally umbilical proper hemi-slant submanifolds. Finally, we obtain a basic inequality involving Ricci curvature and the squared mean curvature of a hemi-slant submanifold of a certain type of locally product Riemannian manifolds.

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On some submanifolds of a locally product manifold

Cited by 14 publications

References 5 publications

Slant and Semi-Slant Submanifolds in Metallic Riemannian Manifolds

Slant and Semi-Slant Submanifolds in Metallic Riemannian Manifolds

A Neutral relation between metallic structure and almost quadratic φ -structure

The geometry of hemi-slant submanifolds of a locally product Riemannian manifold

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