Real Kaehler submanifolds and uniqueness of the Gauss map

Abstract

The main purpose of this paper is to answer the question: To what extent is a euclidean submanifold determined by its Gauss map? More precisely, let /,/: M" -> R n+P be two isometric immersions of a connected riemannian manifold whose Gauss maps into the Grassmannian G np are congruent. When are / and / congruent? Classical examples of isometric noncongruent deformations with the same Gauss map are the associated families of minimal surfaces in R 3 . They are a special case of associated families which can be defined for certain real isometric immersions of Kaehler manifolds which we call circular. It will turn out that locally, all isometric immersions with the same Gauss map can be described in terms of circular submanifolds, whereas globally, additional phenomena arise.In §1, we discuss circular submanifolds in spaces of constant curvature. This is related to work of Calabi, Lawson, and others on minimal surfaces. §2 deals with circular hypersurfaces. More generally, we classify all Kaehler submanifolds of real codimension 1, which is of independent interest. In the remaining two sections we show that all isometric immersions M n -> R n+ ? with congruent Gauss maps form a compact abelian group, and we compute its structure.We would like to thank K. Nomizu for suggesting to look at the above question in the light of some of our previous work. The associated family of a circular immersionLet M n be a connected riemannian manifold of dimension n, and /: M n -» Q" +p an isometric immersion into a complete simply connected space of constant curvature c. We will always assume that / is substantial, i.e. f(M)

Moduli spaces of isometric pluriharmonic immersions of Kähler manifolds into indefinite Euclidean spaces

“…The following proposition is the indefinite version of a result due to Dajczer-Gromoll [4], which can be proved in the same way as in the positive definite case.…”

Complete real Kähler submanifolds in codimension two

“…However, we recall that any minimal real Kähler submanifold f : M 2n → R N is the real part of a holomorphic complex submanifold g: M 2n → C N = R N ⊕ R N , f = Re g; cf. Theorem 1.11 in [DG1] and Remark 8 in [FHZ].…”

Complete real Kähler Euclidean hypersurfaces are cylinders

“…The local classification for hypersurfaces without flat points was carried out in [3]. It was shown that they are abundant and can be parametrized by means of a pseudoholomorphic surface in S 2n (the Gauss image) and a smooth function over it.…”