Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces
Yanlin Li, Akram Ali, Fatemah Mofarreh, Nadia Alluhaibi
Abstract
In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:msup> <a:mi mathvariant="normal">Ω</a:mi> <a:mrow> <a:mrow> <a:mi>p</a:mi> </a:mrow> <a:mo>+</a:mo> <a:mrow> <a:mi>q</a:mi> </a:mrow> </a:mrow> </a:msup> </a:math> in the hyperbolic space <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" id="M2"> <d:msup> <d:mrow> <d:mi mathvariant="normal">ℍ</d:mi> </d:mrow> <d:mrow> <d:mrow> <d:mi>m</d:mi> </d:mrow> </d:mrow> </d:msup> <d:mfenced open="(" close=")" separators="|"> <d:mrow> <d:mo>−</d:mo> <d:mn>1</d:mn> </d:mrow> </d:mfenced> </d:math> satisfy various extrinsic restrictions, then <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M3"> <j:msup> <j:mi mathvariant="normal">Ω</j:mi> <j:mrow> <j:mrow> <j:mi>p</j:mi> </j:mrow> <j:mo>+</j:mo> <j:mrow> <j:mi>q</j:mi> </j:mrow> </j:mrow> </j:msup> </j:math> has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M4"> <m:msub> <m:mrow> <m:mi>π</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mfenced open="(" close=")" separators="|"> <m:mrow> <m:msup> <m:mi mathvariant="normal">Ω</m:mi> <m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:mrow> </m:msup> </m:mrow> </m:mfenced> </m:math> 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.