Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

Ali H. Alkhaldi; Pişcoran Laurian-Ioan; Izhar Ahmad; Akram Ali

doi:10.3390/math10203884

Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

Ali H. Alkhaldi, Pişcoran Laurian-Ioan^*, Izhar Ahmad, Akram Ali

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold (Formula presented.) of a complex space form (Formula presented.) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of (Formula presented.) are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold (Formula presented.), in which (Formula presented.) is homeomorphic to a sphere (Formula presented.) if (Formula presented.) and if (Formula presented.), then (Formula presented.) is homotopic to a sphere (Formula presented.) under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.

Original language	English
Article number	3884
Journal	Mathematics
Volume	10
Issue number	20
DOIs	https://doi.org/10.3390/math10203884
State	Published - Oct 2022

Bibliographical note

Publisher Copyright:
© 2022 by the authors.

Keywords

Dirichlet energy
Homology groups
complex space form
sphere theorem
stable currents
warped product submanifolds

ASJC Scopus subject areas

Computer Science (miscellaneous)
General Mathematics
Engineering (miscellaneous)

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10.3390/math10203884

Cite this

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title = "Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications",

abstract = "In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold (Formula presented.) of a complex space form (Formula presented.) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of (Formula presented.) are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold (Formula presented.), in which (Formula presented.) is homeomorphic to a sphere (Formula presented.) if (Formula presented.) and if (Formula presented.), then (Formula presented.) is homotopic to a sphere (Formula presented.) under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.",

keywords = "Dirichlet energy, Homology groups, complex space form, sphere theorem, stable currents, warped product submanifolds",

author = "Alkhaldi, {Ali H.} and Pi{\c s}coran Laurian-Ioan and Izhar Ahmad and Akram Ali",

note = "Publisher Copyright: {\textcopyright} 2022 by the authors.",

year = "2022",

month = oct,

doi = "10.3390/math10203884",

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journal = "Mathematics",

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TY - JOUR

T1 - Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

AU - Alkhaldi, Ali H.

AU - Laurian-Ioan, Pişcoran

AU - Ahmad, Izhar

AU - Ali, Akram

PY - 2022/10

Y1 - 2022/10

N2 - In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold (Formula presented.) of a complex space form (Formula presented.) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of (Formula presented.) are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold (Formula presented.), in which (Formula presented.) is homeomorphic to a sphere (Formula presented.) if (Formula presented.) and if (Formula presented.), then (Formula presented.) is homotopic to a sphere (Formula presented.) under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.

AB - In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold (Formula presented.) of a complex space form (Formula presented.) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of (Formula presented.) are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold (Formula presented.), in which (Formula presented.) is homeomorphic to a sphere (Formula presented.) if (Formula presented.) and if (Formula presented.), then (Formula presented.) is homotopic to a sphere (Formula presented.) under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.

KW - Dirichlet energy

KW - Homology groups

KW - complex space form

KW - sphere theorem

KW - stable currents

KW - warped product submanifolds

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U2 - 10.3390/math10203884

DO - 10.3390/math10203884

M3 - Article

AN - SCOPUS:85140627377

SN - 2227-7390

VL - 10

JO - Mathematics

JF - Mathematics

IS - 20

M1 - 3884

ER -

Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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