Some Optimal Inequalities for Anti-invariant Submanifolds of the Unit Sphere
Some Optimal Inequalities for Anti-invariant Submanifolds of the Unit Sphere
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DOI：
10.1007/s12220-023-01481-w
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发表时间：
2023-12
期刊：
影响因子：
--
通讯作者：
Cheng Xing;Jiabin Yin
中科院分区：
文献类型：
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作者：
Cheng Xing;Jiabin Yin
本文分别从内禀和外禀两个方面研究了单位维球面中维反不变子流形上的刚性现象。首先，我们建立了这类子流形关于第二基本形式和平均曲率向量场H的协变微分范数的一个基本不等式。其次，利用一个一般不等式进一步导出了His范数的下界.最后，在处理具有-Einstein诱导度量的极小反不变子流形时，我们得到了关于Weyl曲率张量、平方范数Sofh和标量曲率的一个不等式。特别地，上述这些不等式是最优的，因为所有达到这些不等式的子流形都是完全确定的。
In this paper, we study the rigidity phenomena on the-dimensional anti-invariant submanifolds of the unit sphere of dimensionfrom the intrinsic and extrinsic aspects, respectively. First of all, we establish a basic inequality for such submanifolds relative to the norm of the covariant differentiation of both the second fundamental formhand mean curvature vector fieldH. Secondly, the lower bound of the norm ofHis further derived by means of a general inequality. Finally, in dealing with those minimal anti-invariant submanifolds with-Einstein induced metrics, we obtain an inequality in terms of the Weyl curvature tensor, squared normSofh, and scalar curvature. In particular, these inequalities above are optimal in the sense that all the submanifolds attaining the equalities are completely determined.