Real Kaehler submanifolds in low codimension (Q1385089)

A real Kähler submanifold is an isometric immersion of a Kähler manifold \(M^{2n}\) into Euclidean space. The authors already analysed real Kähler hypersurfaces in [\textit{M. Dajczer} and \textit{D. Gromoll}, J. Differ. Geom. 22, 13-28 (1985; Zbl 0587.53051)] and the complete codimension two real Kähler submanifolds in [Invent. Math. 119, 235-242 (1995; Zbl 0827.53047)]. In this paper they show that generically any local example in codimension three arises as a complex hypersurface of a real Kähler submanifold \(N ^{2n+2}\) in codimension one. The only assumption they make is an upper on the index of relative nullity of real Kähler submanifold \(M^{2n}\) into Euclidean space \(\mathbb{R}^{2n+3}\). As a consequence, they prove that any real Kähler submanifold \(M^{2n}\) of \(\mathbb{R}^{2n+3}\) is locally isometrically rigid, unless the real Kähler hypersurface \(N ^{2n+2}\) is either flat or minimal.