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.