Minimal Immersions of Kahler manifolds into Euclidean Spaces

It is proved here that a minimal isometric immersion of a Kähler-Einstein or homogeneous Kähler-manifold into an Euclidean space must be totally geodesic

On an assertion in Riemann's Habilitationvortrag

We study an assertion in Riemann's Habilitation Lecture of 1854. Namely, the determination of the metric given $n\frac{n-1}{2}$ sectional curvatures. We give several counterexamples to a Riemann's clai

Reducibility of complex submanifolds of the complex euclidean spaces

Let M be a simply connected complex submanifold of CN. We prove that M is irreducible, up a totally geodesic factor,if and only if the normal holonomy group acts irreducibly. This is an extrinsic analogue of the well-known De Rham decomposition theorem for a complex manifold. Our result is not valid in the real context, as it is shown by many counterexamples

Kahler immersions of the disc into polydiscs

We give a concrete example of non totally geodesic Kahler immersion of a disc into a polydis

Autoparallel distributions and splitting theorems

We study some links between autoparallel distributions and the factorization of a riemannian manifold. Finally, we prove a splitting theorem for Lie groups with biinvariant metric

On submanifolds whose shape operator is unipotent

The object of this article is to characterize submanifolds of the Euclidean space whose shape operator is unipoten

Minimal homogeneous submanifolds in euclidean spaces

We prove that minimal (extrinsically) homogeneous submanifolds of the Euclidean space are totally geodesic. As an application, we obtain that a complex (intrisecally) homogeneous submanifold of a complex Euclidean space must be totally geodesic

Kahler manifolds and their relatives

Let M1 and M2 be two K¨ahler manifolds. We call M1 and M2 relatives if they share a non-trivial K¨ahler submanifold S, namely, if there exist two holomorphic and isometric immersions (K¨ahler immersions) h1 : S → M1 and h2 : S → M2. Moreover, two K¨ahler manifolds M1 and M2 are said to be weakly relatives if there exist two locally isometric (not necessarily holomorphic) K¨ahler manifolds S1 and S2 which admit two K¨ahler immersions into M1 and M2 respectively. The notions introduced are not equivalent (cf. Example 2.3). Our main results in this paper are Theorem 1.2 and Theorem 1.4. In the first theorem we show that a complex bounded domain D ⊂ Cn with its Bergman metric and a projective K¨ahler manifold (i.e. a projective manifold endowed with the restriction of the Fubini-Study metric) are not relatives. In the second theorem we prove that a Hermitian symmetric space of noncompact type and a projective K¨ahler manifold are not weakly relatives. Notice that the proof of the second result does not follows trivially from the first one. We also remark that the above results are of local nature, i.e. no assumptions are used about the compactness or completeness of the manifolds involve

Submanifolds with curvature normals of constant length and the Gauss map

We show that a submanifold with curvature normal of constant length has constant principal curvatures under suitable global hypothesis. We construct local counterexamples to show that the global hypothesis can not be dropped