On the isolation phenomena of Einstein manifolds—submanifolds versions

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 $m\ge 3$.