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Some Optimal Inequalities for Anti-invariant Submanifolds of the Unit Sphere

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Abstract

In this paper, we study the rigidity phenomena on the \((n+1)\)-dimensional anti-invariant submanifolds of the unit sphere of dimension \((2n+1)\) from 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 form h and mean curvature vector field H. Secondly, the lower bound of the norm of H is further derived by means of a general inequality. Finally, in dealing with those minimal anti-invariant submanifolds with \(\eta \)-Einstein induced metrics, we obtain an inequality in terms of the Weyl curvature tensor, squared norm S of h, and scalar curvature. In particular, these inequalities above are optimal in the sense that all the submanifolds attaining the equalities are completely determined.

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Acknowledgements

The authors are much indebted to the referee for his/her valuable comments and suggestions toward the improvement of this work.

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Cheng Xing

  2. School of Mathematics and Statistics, Guangxi Normal University, Guilin, 541001, People’s Republic of China

    Jiabin Yin

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  1. Cheng Xing
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  2. Jiabin Yin
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Correspondence to Jiabin Yin.

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This project was supported Guangxi Science, Technology Project (Grant No. GuikeAD22035942), Guangxi Natural Science Foundation (Grant No. 2022GXNSFBA035465), Mathematics Tianyuan Fund Project (Grant No. 12226350) and National Natural Science Foundation of China (Grant Nos. 11971244, 12171437 and 12201138). C. Xing was also supported by China Postdoctoral Science Foundation (Grant No. 2023M731810) and the Fundamental Research Funds for the Central Universities (Grant No. 050-63233068).

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Xing, C., Yin, J. Some Optimal Inequalities for Anti-invariant Submanifolds of the Unit Sphere. J Geom Anal 34, 38 (2024). https://doi.org/10.1007/s12220-023-01481-w

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