Differential Geometry of Real Hypersurfaces in Hermitian Symmetric Spaces with Rank 2 Jürgen Berndt and Young

In this talk, first we introduce the classification of homogeneous hypersurfaces in some Hermitian symmetric spaces of rank 1 or rank 2. In particular, we give a full expression of the geometric structures for hypersurfaces in complex two-plane Grassmannians G2(C) or in complex hyperbolic twoplane Grassmannians G2(C). Next by using the isometric Reeb flow we give a complete classification for hypersurfaces M in complex two-plane Grassmannians G2(C), complex hyperbolic two-plane Grassmannians G2(C), complex quadric Qn and its noncompact dual Qn∗. Moreover, we give a classification of contact hypersurfaces with constant mean curvature in the complex quadric Qn = SOn+2/SOnSO2 and its noncompact dual Qn∗ = SOo n,2/SOnSO2 for n ≥ 3 1. Compact Hermitian Symmetric Space with rank 2 The study of real hypersurfaces in non-flat complex space forms or quaternionic space forms which belong to HSSP with rank 1 of compact type in section 1 is a classical topic in differential geometry. For instance, there have been many investigations for homogeneous hypersurfaces of type A1, A2, B, C, D and E in complex projective space CP. They are completely classified by Cecil and Ryan [9], Kimura [13] and Takagi [30]. Here, explicitly, we mention that A1 : Geodesic hyperspheres, A2 : a tube around a totally geodesic complex projective spaces CP , B : a tube around a complex quadric Qm−1 and can be viewed as a tube around a real projective space RP, C : a tube around the Segre embedding of CP 1 ×CP k into CP 2k+1 for some k≥2, D : a tube around the Plücker embedding into CP 9 of the complex Grassmannian manifold G2(C) of complex 2-planes in C and E : a tube around the half spin embedding into CP 15 of the Hermitian symmetric space SO(10)/U(5). Now let us study hypersurfaces in complex two-plane Grassmanians G2(C) which is a kind of HSSP with rank two of compact type. The ambient space G2(C) is known to be the unique compact irreducible Riemannian symmetric space equipped with both a Kähler structure J and a quaternionic Kähler structure J not containing J . 2000 Mathematics Subject Classification. Primary 53C40; Secondary 53C15.