Twisted products are generalizations of warped products, namely the warping function may depend on the points of both factors. The two canonical foliations of a twisted product are mutually perpendicular and their leaves are totally geodesic, resp. totally umbilic. The main result is a decomposition theorem of de Rham type: If on a simply connected, geodesically complete pseudo-Riemannian manifoldM two foliations with the above properties are given, thenM is a twisted product.

Twisted products in pseudo-Riemannian geometry

Geometry of submanifolds and its applications

We introduce anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and check the harmonicity of such submersions. We also find necessary and sufficient conditions for a Langrangian Riemannian submersion, a special anti-invariant Riemannian submersion, to be totally geodesic. Moreover, we obtain decomposition theorems for the total manifold of such submersions.

Anti-invariant Riemannian submersions from almost Hermitian manifolds

The geometry and, especially, the geodesics of a class of spacetimes generalizing Robertson-Walker ones (without any assumption on the fiber) is studied, under a global point of view. Our study covers geodesic connectedness, geodesic completeness and stability of completeness.

On the Geometry of Generalized Robertson-Walker Spacetimes: Geodesics

In this paper we introduce the notion of generalized quasi-Einstein manifold that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi-Einstein manifolds. We prove that a complete generalized quasi-Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature is locally a warped product with (n − 1)-dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.

Generalized quasi-Einstein manifolds with harmonic Weyl tensor

Ruled real hypersurfaces of complex space forms are investigated by using the fact that such hypersurfaces can be constructed by moving a 1-codimensional complex totally geodesic submanifold of the ambient space along a curve Among other results, a classification of minimal ruled real hypersurfaces and an example of a homogeneous ruled real hypersurface are given

On Ruled Real Hypersurfaces in Complex Space Forms

By definition an orthogonal net on a pseudoriemannian manifold is a family of complementary foliations which intersect perpendicularly. There are derived generalizations of de Rham’s decomposition theorem by characterizing those pseudoriemannian manifolds equipped with an orthogonal net, which locally resp. globally allow a representation as a twisted resp. warped product. The results are applied for studying hypersurfaces with harmonic curvature.

Decomposition of twisted and warped product nets

By definition a net on a manifold is a family of complementary foliations. Nets and net morphisms between netted manifolds allow to develop basic tools for the decomposability of netted manifolds and of differentiable maps between them. In this way new generalizations of de Rham’s decomposition theorem are obtained.

De Rham decomposition of netted manifolds

Let M be a Riemannian manifold, ω a differentiable form on M with values in a bundle Hom (TM, ζ), and let G be an open subset of M such that Kerω forms a vector bundle ξ of constant fibre dimension k>0 over G. We prove: If ω satisfies some analytical conditions, then ξ is completely integrable, the integral manifolds of ξ are spherically bent in M, and in some interesting cases they are complete.

Helmut Reckziegel

Papers

Twisted products in pseudo-Riemannian geometry

On Ruled Real Hypersurfaces in Complex Space Forms

Decomposition of twisted and warped product nets

De Rham decomposition of netted manifolds

Über sphärische Blätterungen und die Vollständigkeit ihrer Blätter.