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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 2, pp. 521-536 (2002)
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On Homogeneous Hypersurfaces in Complex Grassmannians

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Alicia N. Garcia, Eduardo G. Hulett, Cristian U. Sanchez

Fa.M.A.F. - CIEM, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, Argentina, e-mail: agarcia@mate.uncor.edu, e-mail: hulett@mate.uncor.edu, e-mail: csanchez@mate.uncor.edu

**Abstract:** In the present article we consider a class of real hypersurfaces of the Grassmann manifold of $k$-planes in ${\mathbb C}^{n}$, $G_{k}({\mathbb C}^{n})$, for $k>2$. Namely the family of tubes around $G_{k}({\mathbb C}^{m})$ with $m<n$ and around the quaternionic Grassmann manifold of $k/2$-quaternionic planes in ${\mathbb H}^{n/2}$, $G_{k/2}({\mathbb H}^{n/2})$, when $k$ and $n$ are even. We determine which of those tubes are homogeneous and for them we find the spectral decomposition of the shape operator. As a consequence we show that they are Hopf hypersurfaces.

**Keywords:** complex Grassmannians, real hypersurfaces, tubes, shape operator, Kaehler structure

**Classification (MSC2000):** 53C30, 53C35, 53C42

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