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
Full text of the article: