# zbMATH — the first resource for mathematics

Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians. (English) Zbl 1015.53034
Let $$(\overline M,g,J)$$ be an almost Hermitian manifold and let $$M$$ be an orientable hypersurface and $$V$$ a unit normal vector field on it. Then, $$\xi=JV$$ is the Reeb (or characteristic) vector field of an almost contact metric structure on $$M$$. The classification of those $$M$$ such that $$\xi$$ is a Killing vector field is an interesting problem which has been achieved for some specific $$\overline M$$. For example, the classification is completely known for $$\overline M\in \{\mathbb{C},\mathbb{C} P^n, \mathbb{C} H^n\}$$ and it turns out that all these hypersurfaces $$M$$ are homogeneous when they are supposed to be complete.
In this paper, the authors consider the case when $$\overline M$$ is the complex Grassmann manifold $$G_2 (\mathbb{C}^{m+2})$$, $$m\geq 3$$, of all two-dimensional linear subspaces in $$\mathbb{C}^{m+2}$$. These manifolds are Riemannian symmetric spaces of rank two equipped with a Kähler and a quaternionic Kähler structure. For $$m=1$$, $$G_2 (\mathbb{C}^3)$$ is isometric to a $$\mathbb{C} P^2$$ and for $$m=2$$, $$G_2(\mathbb{C}^4)$$ is isometric to the real Grassmann manifold $$G^+_2(\mathbb{R}^6)$$ of oriented two-dimensional linear subspaces of $$\mathbb{R}^6$$. It is proved here that the connected orientable hypersurfaces $$M$$ in $$G_2(\mathbb{C}^{m+2})$$ with isometric Reeb vector field are just the open parts of tubes around totally $$G_2(\mathbb{C}^{m+1})$$ in $$G_2 (\mathbb{C}^{m+2})$$ and when $$M$$ is complete they are again homogeneous.

##### MSC:
 53C40 Global submanifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C35 Differential geometry of symmetric spaces
Full Text: