A characterization of certain Einstein Kähler hypersurfaces in a complex Grassmann manifold of 2-planes
Homogeneous hypersurfaces in a wide class of Kähler spaces were described in [\textit{K. Konno}, J. Math. Soc. Japan 40, No.~4, 687--703 (1988; Zbl 0683.53059)]. Most of them are either symmetric or parallel (i.e., the second fundamental form is parallel). For Grassman manifolds Gr\(_k({\mathbb C}^n)\) and simply connected hypersurfaces in them the exception is the series \(M_l={}\)Sp\((l)/\)U\((2)\cdot\)Sp\((l-2)\subset{\,}\)Gr\(_2({\mathbb C}^{2l})\), \(l>2\). Such a manifold \(M\) satisfies the condition \(\Im T^\bot M\subset TM\), where \(T^\bot M\) is the normal bundle of \(M\) and \(\Im\) is the quaternionic Kähler structure (an SU\((n)\)-invariant subbundle of End\((T(\)Gr\({}_2({\mathbb C}^n)))\) of the real rank 3; \(\Im T^\bot M\) has the real rank 6). In this paper, it is shown that this property characterizes \(M_l\) locally. More precisely, the main result of the paper is the following theorem: {If an Einstein Kähler hypersurface \(M\) of Gr\(_2({\mathbb C}^n)\) satisfies the condition \(\Im T^\bot M\subset TM\) then \(n\) is even and \(M\) is locally congruent to \(M_{n/2}\)}. Note that homogeneity is not assumed. Also, all homogeneous Kähler hypersurfaces in Grassman manifolds are Einstein. The result is deduced from the equivalence of the first focal sets of these hypersurfaces. It is proved that the first focal set is a totally real complete quaternionic Kähler submanifold of Gr\(_2({\mathbb C}^n)\). Then it is totally geodesic according to [\textit{D. V. Alekseevskij}, Funkts. Anal. Prilozh. 2, No. 2, 11--20 (1968; Zbl 0175.19001)] and is congruent to a projective quaternionic space \(\mathbb{H}\mathbb{P}^{{n\over2}-1}\) by a result of [\textit{H. Tasaki}, Tôhoku Math. J. (2 38, 513--538 (1986; Zbl 0593.53037)].
scientific article; zbMATH DE number 5015939
scientific article