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

• Main Authors: Yanlin Li, Pişcoran Laurian-Ioan, Akram Ali, Ali H. Alkhaldi
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Symmetry
• Subjects: warped product submanifolds; euclidean spaces; homology group; homotopic; fundamental groups; stable currents
• Online Access: https://www.mdpi.com/2073-8994/13/9/1587

In this paper, we prove that, for compact warped product submanifolds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>M</mi><mi>n</mi></msup></semantics></math></inline-formula> in an Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">E</mi><mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msup></semantics></math></inline-formula>, there are no stable <i>p</i>-currents, homology groups are vanishing, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>M</mi><mn>3</mn></msup></semantics></math></inline-formula> is homotopic to the Euclidean sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">S</mi><mn>3</mn></msup></semantics></math></inline-formula> 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.