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| $mathcal{A}$-manifolds and $mathcal{B}$-manifolds, introduced by A.Gray, are two significant classes of Einstein-like Riemannian manifolds. A Riemannian manifold is Ricci parallel if and only if it is simultaneously an $mathcal{A}$-manifold and a $mathcal{B}$-manifold. The present paper proves that both focal submanifolds of each isoparametric hypersurface in unit spheres with $g=4$ distinct principal curvatures are $mathcal{A}$-manifolds. As for the focal submanifolds with $g=6$, $m=1$ or $2$, only one is an $mathcal{A}$-manifold, and neither is a $mathcal{B}$-manifold. |
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| Keywords:differential geometry, isoparametric hypersurface, focal submanifold, $mathcal{A}$-manifold, $mathcal{B}$-manifold |
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