The main objective of this paper is to investigate topological properties from the view point of compact warped product submanifolds of a space form with the vanishing constant sectional curvature. That is, we prove the non-existence of stable integral $ p $-currents in a compact oriented warped product pointwise semi-slant submanifold $ M^n $ in the Euclidean space $ \mathbb{R}^{p+2q} $ which satisfies an operative condition involving the Laplacian of a warped function and a pointwise slant function, and show that their homology groups are zero on this operative condition. Moreover, under the assumption of extrinsic conditions, we derive new topological sphere theorems on a warped product submanifold $ M^n $, and prove that $ M^n $ is homeomorphic to $ \mathbb{S}^n $ if $ n = 4 $, and $ M^n $ is homotopic to $ \mathbb{S}^n $ if $ n = 3 $. Furthermore, the same results are generalized for CR-warped products and our results recovered [
Citation: Ali H. Alkhaldi, Akram Ali, Jae Won Lee. The Lawson-Simons' theorem on warped product submanifolds with geometric information[J]. AIMS Mathematics, 2021, 6(6): 5886-5895. doi: 10.3934/math.2021348
The main objective of this paper is to investigate topological properties from the view point of compact warped product submanifolds of a space form with the vanishing constant sectional curvature. That is, we prove the non-existence of stable integral $ p $-currents in a compact oriented warped product pointwise semi-slant submanifold $ M^n $ in the Euclidean space $ \mathbb{R}^{p+2q} $ which satisfies an operative condition involving the Laplacian of a warped function and a pointwise slant function, and show that their homology groups are zero on this operative condition. Moreover, under the assumption of extrinsic conditions, we derive new topological sphere theorems on a warped product submanifold $ M^n $, and prove that $ M^n $ is homeomorphic to $ \mathbb{S}^n $ if $ n = 4 $, and $ M^n $ is homotopic to $ \mathbb{S}^n $ if $ n = 3 $. Furthermore, the same results are generalized for CR-warped products and our results recovered [
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Ali H. Alkhaldi, Akram Ali, Jae Won Lee. The Lawson-Simons' theorem on warped product submanifolds with geometric information[J]. AIMS Mathematics, 2021, 6(6): 5886-5895. doi: 10.3934/math.2021348
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