On the isolation phenomena of Einstein manifolds—submanifolds versions
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- by Xiuxiu Cheng, Zejun Hu, An-Min Li and Haizhong Li PDF
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Abstract:
In this paper, we study the isolation phenomena of Einstein manifolds from the viewpoint of submanifolds theory. First, for locally strongly convex Einstein affine hyperspheres we prove a rigidity theorem and as its direct consequence we establish a unified affine differential geometric characterization of the noncompact symmetric spaces $\mathrm {E}_{6(-26)}/\mathrm {F}_4$ and $\mathrm {SL}(m,\mathbb {R})/\mathrm {SO}(m)$, $\mathrm {SL}(m,\mathbb {C})/\mathrm {SU}(m)$, $\mathrm {SU}^*(2m)/\mathrm {Sp}(m)$ for each $m\ge 3$. Second and analogously, for Einstein Lagrangian minimal submanifolds of the complex projective space $\mathbb {C}P^n(4)$ with constant holomorphic sectional curvature $4$, we prove a similar rigidity theorem and as its direct consequence we establish a unified differential geometric characterization of the compact symmetric spaces $\mathrm {E}_{6}/\mathrm {F}_4$ and $\mathrm {SU}(m)/\mathrm {SO}(m)$, $\mathrm {SU}(m)$, $\mathrm {SU}(2m)/\mathrm {Sp}(m)$ for each $m\ge 3$.References
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Additional Information
- Xiuxiu Cheng
- Affiliation: School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China
- MR Author ID: 892484
- Email: chengxiuxiu1988@163.com
- Zejun Hu
- Affiliation: School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China
- MR Author ID: 346519
- ORCID: 0000-0003-2744-5803
- Email: huzj@zzu.edu.cn
- An-Min Li
- Affiliation: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of China
- MR Author ID: 190196
- Email: anminliscu@126.com
- Haizhong Li
- Affiliation: Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, People’s Republic of China
- MR Author ID: 255846
- Email: hli@math.tsinghua.edu.cn
- Received by editor(s): January 5, 2017
- Published electronically: January 12, 2018
- Additional Notes: The first and second authors were supported by grants of NSFC-11371330 and 11771404, the third author was supported by grants of NSFC-11521061, and the fourth author was supported by grants of NSFC-11671224.
- Communicated by: Lei Ni
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1731-1740
- MSC (2010): Primary 53C24; Secondary 53A15, 53C25, 53D12
- DOI: https://doi.org/10.1090/proc/13901
- MathSciNet review: 3754356