Warped Product Submanifolds of Riemannian Product Manifolds

Al-Solamy, Falleh R.; Khan, Meraj Ali

doi:10.1155/2012/724898

Cited by 8 publications

(7 citation statements)

References 12 publications

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“…Warped product manifolds have already proven themselves to be a rich source of examples in a wide range of distinct geometrical objects, of which solitons are an example (cf. [18,28,22,20,21,4]). A warped product manifold like the one in Definition 1.1 is suggestively written in the form B n × f F d and its components B, F and f are called base, fiber and warping function, in this order.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On warped product gradient Yamabe solitons

Tokura

Adriano

Pina

et al. 2019

Journal of Mathematical Analysis and Applications

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In this paper, we look for properties of gradient Yamabe solitons on top of warped product manifolds. Utilizing the maximum principle, we find lower bound estimates for both the potential function of the soliton and the scalar curvature of the warped product. By slightly modifying Li-Yau's technique so that we can handle drifting Laplacians, we were able to find three different gradient estimates for the warping function, one for each sign of the scalar curvature of the fiber manifold. As an application, we exhibit a nonexistence theorem for gradient Yamabe solitons possessing certain metric properties on the base of the warped product.2010 Mathematics Subject Classification. 53C21, 53C50, 53C25.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…From the Bochner formula for the m-Bakry-Émery Ricci tensor and the lower bound hypothesis (18) we obtain…”

Section: Proofsmentioning

confidence: 99%

On warped product gradient Yamabe solitons

Tokura

Adriano

Pina

et al. 2019

Journal of Mathematical Analysis and Applications

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“…By using (27)(ii), we have g(A N W Z, X) = g(h(Z, X), N W ) = 0 and we get g(h(Z, W ), N X) = −g(T W, ∇ Z X) + g(W, ∇ Z T X) and using Lemma 1 (2) we obtain (28). Thus, for the non-null vector field Z = W ∈ Γ(T M ⊥ ), we obtain (29).…”

Section: Semi-invariant Semi-slant and Hemi-slant Warped Product Submentioning

confidence: 96%

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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“…Also, this notion is slightly different from the definition of the skew semi-invariant submanifold [14]. For more details, we refer to [2,4,6,24].…”

Section: Generalized Semi-invariant Submanifolds In Locally Product Riemannian Manifoldsmentioning

confidence: 99%

Multiply Warped Product Generalized Semi-Invariant Submanifolds

Traore¹,

Taştan²,

Aydin³

2021

International Electronic Journal of Geometry

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We define generalized semi-invariant submanifolds in locally product Riemannian manifolds. Then we study multiply warped product generalized semi-invariant submanifolds in the same structure. We give an existence theorem for such submanifolds. We also give necessary and sufficient conditions for such a submanifold to be a multiply direct product submanifold. Moreover, we establish a general inequality for such submanifolds.

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Warped Product Submanifolds of Riemannian Product Manifolds

Abstract: We study warped product of the typeNθ×fNTandNθ×fN⊥, whereNθ,NT, andN⊥are proper slant, invariant, and anti-invariant submanifolds, respectively, and we prove some basic results and finally obtain some inequalities for squared norm of second fundamental form.

Cited by 8 publications

References 12 publications

On warped product gradient Yamabe solitons

On warped product gradient Yamabe solitons

Slant and Semi-Slant Submanifolds in Metallic Riemannian Manifolds

Multiply Warped Product Generalized Semi-Invariant Submanifolds

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