On the isolation phenomena of Einstein manifolds—submanifolds versions

Cheng, Xiuxiu; Hu, Zejun; Li, An-Min; Li, Haizhong

doi:10.1090/proc/13901

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

Proc. Amer. Math. Soc. 146 (2018), 1731-1740 Request permission

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 $m\ge 3$.

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