Geometric Algebra of Singular Ruled Surfaces

Li, Yanlin; Wang, Zhigang; Zhao, Tie-Hong

doi:10.1007/s00006-020-01097-1

Cited by 33 publications

(18 citation statements)

References 22 publications

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“…e geometric structure and topological properties of submanifolds in different spaces have been studied on a large scale during the past few years [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Many results showed that there is a closed relationship between stable currents which are nonexistent and the vanished homology groups of submanifolds in a different class of the ambient manifold obtained by imposing conditions on the second fundamental form (1).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

Liu

Ali

Mofarreh

et al. 2021

Journal of Mathematics

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In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold Ω p + q in the hyperbolic space ℍ m − 1 satisfy various extrinsic restrictions, then Ω p + q has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group π 1 Ω p + q is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

Liu

Ali

Mofarreh

et al. 2021

Journal of Mathematics

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“…More recently, geometric, topological, and differentiable rigidity theorems of the Riemannian submanifold connecting to parallel mean curvature in space forms such that c + H 2 > 0 have been obtained in terms of Ricci curvature in [5]. In some articles such as [3,[6][7][8][9][10][11][12][13][14][15][16][17], several results have been derived on topological and differentiable structures of singular submanifolds and submanifolds with specific effective conditions for the second fundamental form, sectional curvatures, and Ricci curvatures.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Topology of Warped Product Pointwise Semi-Slant Submanifolds with Positive Curvature

et al. 2021

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In this paper, we obtain some topological characterizations for the warping function of a warped product pointwise semi-slant submanifold of the form Ωn=NTl×fNϕk in a complex projective space CP2m(4). Additionally, we will find certain restrictions on the warping function f, Dirichlet energy function E(f), and first non-zero eigenvalue λ1 to prove that stable l-currents do not exist and also that the homology groups have vanished in Ωn. As an application of the non-existence of the stable currents in Ωn, we show that the fundamental group π1(Ωn) is trivial and Ωn is simply connected under the same extrinsic conditions. Further, some similar conclusions are provided for CR-warped product submanifolds.

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“…We proved a compact warped product submanifold M n in a Euclidean space E n+k , that there are no stable p-currents, homology groups vanish, and that M 3 is homotopic to the Euclidean sphere S 3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature and the Hessian tensor. In our next work, we will combine the singularity theory presented in [31][32][33][34] to study compact warped product submanifold M n in a Euclidean space E n+k .…”

Section: Proof Of Corollarymentioning

confidence: 99%

Null Homology Groups and Stable Currents in Warped Product Submanifolds of Euclidean Spaces

et al. 2021

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In this paper, we prove that, for compact warped product submanifolds Mn in an Euclidean space En+k, there are no stable p-currents, homology groups are vanishing, and M3 is homotopic to the Euclidean sphere S3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature, and the Hessian tensor. The results in this paper can be considered an extension of Xin’s work in the framework of a compact warped product submanifold, when the base manifold is minimal in ambient manifolds.

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Geometric Algebra of Singular Ruled Surfaces

Cited by 33 publications

References 22 publications

Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

On the Topology of Warped Product Pointwise Semi-Slant Submanifolds with Positive Curvature

Null Homology Groups and Stable Currents in Warped Product Submanifolds of Euclidean Spaces

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