In this work, several pinched conditions on the Laplacian and gradient of the warping function are found in consideration of warped product submanifolds structure that force to homology groups vanish with no stable currents. Also, it is proved that a warped product pointwise semi-slant submanifold [Formula: see text] that is compact and oriented in an odd-dimensional spheres [Formula: see text] and [Formula: see text], has no stable integral [Formula: see text]-currents and [Formula: see text]-currents, respectively, and their homology groups are null, provided squared norm of the gradient for warping function satisfies some extrinsic restrictions including the Laplacian of the warping function, pointwise slant functions in addition to dimension of fiber of warped product immersions. Moreover, under assumption of extrinsic condition on the warping function, it is show [Formula: see text] being homeomorphic to a standard sphere [Formula: see text] with [Formula: see text] and homotopic to a standard sphere [Formula: see text] with [Formula: see text]. Further, the same results are generalized for contact CR-warped product submanifolds of same ambient spaces.

中文翻译：

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单位球内翘曲积子流形的同源性及其应用

在这项工作中，考虑到扭曲的产品子流形结构，在没有稳定电流的情况下迫使同调群消失，发现了拉普拉斯算子和扭曲函数梯度的几个收缩条件。此外，证明了在奇维球体[公式：见文本]和[公式：见文本]中紧致且定向的翘曲乘积点状半斜子流形[公式：见文本]，没有稳定积分[公式：见正文]-currents和[公式：见正文]-currents，并且它们的同调群为空，前提是翘曲函数的梯度平方范数满足一些外在限制，包括翘曲函数的拉普拉斯算子，逐点倾斜函数除了翘曲产品浸入的纤维尺寸外。而且，在翘曲函数的外在条件假设下，显示[公式：见文本]同胚于标准球体[公式：见文本]与[公式：见文本]并同伦于标准球体[公式：见文本]用 [公式：见正文]。此外，相同的结果被推广到相同环境空间的接触 CR 扭曲产品子流形。