Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster Reeb parallel shape operator

摘要

In a paper due to Jeong et al. (Kodai Math J 34(3):352-366, 2011) we have shown that there does not exist a hypersurface in G_2(?~(m+2)) with parallel shape operator in the generalized Tanaka-Webster connection (see Tanaka in Jpn J Math 20:131-190, 1976; Tanno in Trans Am Math Soc 314(1):349-379, 1989). In this paper, we introduce the notion of the Reeb parallel in the sense of generalized Tanaka-Webster connection for a hypersurface M in G_2(?~(m+2)) and prove that M is an open part of a tube around a totally geodesic G_2(?~(m+1)) in G_2(?~(m+2)).