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$\mathfrak{D}^\perp$-invariant real hypersurfaces in complex
Grassmannians of rank two
Volume 61, no. 2
(2020),
pp. 197–207
https://doi.org/10.33044/revuma.v61n2a01
Abstract
Let $M$ be a real hypersurface in a complex Grassmannian of rank
two. Denote by $\mathfrak{J}$ the quaternionic Kähler structure
of the ambient space, $TM^\perp$ the normal bundle over $M$, and
$\mathfrak{D}^\perp=\mathfrak{J}TM^\perp$. The real hypersurface
$M$ is said to be $\mathfrak{D}^\perp$-invariant if
$\mathfrak{D}^\perp$ is invariant under the shape operator of $M$.
We show that if $M$ is $\mathfrak{D}^\perp$-invariant, then $M$ is
Hopf. This improves the results of Berndt and Suh [Int. J.
Math. 23 (2012) 1250103] and [Monatsh. Math.
127 (1999), 1–14]. We also classify $\mathfrak{D}^\perp$
real hypersurfaces in complex Grassmannians of rank two of
noncompact type with constant principal curvatures.
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Published by the Unión Matemática Argentina |