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Gerd Schmalz

Publications and source records attributed to Gerd Schmalz.

At least 19 recordsLinked to original sources
arXiv2026

Canonical Bochner-K\"ahler potentials via the Sasaki-K\"ahler correspondence

We introduce a new approach to the study of Bochner-K\"ahler manifolds, based on normal form techniques in CR geometry. As observed by Webster, Bochner-K\"ahler manifolds are intimately related to spherical Sasakian manifolds. Our approach utilises this relationship as well as normal forms for spherical Sasakian manifolds (known as rigid spheres). In this setting potentials of Bochner-K\"ahler manifolds are nothing but the defining equations of rigid spheres in rigid normal form. This leads to a canonical formula for Bochner-K\"ahler potentials, as well as streamlined proofs of known results on the structure and the symmetries of Bochner-K\"ahler manifolds. Our approach also provides a host of examples in terms of an implicit formula of the corresponding K\"ahler potentials.

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arXiv2024

On the classificarion of 3-dimensional spherical Sasakian manifolds

In this article we consider spherical hypersurfaces in $\mathbb C^2$ with a fixed Reeb vector field as 3-dimensional Sasakian manifolds. We establish the correspondence between three different sets of parameters, namely, those arising from representing the Reeb vector field as an automorphism of the Heisenberg sphere, the parameters used in Stanton's description of rigid spheres, and the parameters arising from the rigid normal forms. We also geometrically describe the moduli space for rigid spheres, and provide geometric distinction between Stanton's hypersurfaces and those found by Ezhov and Schmalz. Finally, we determine the Sasakian automorphism groups of the rigid spheres and detect the homogeneous Sasakian manifolds among them.

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arXiv2024

Quasi-Einstein shearfree spacetimes lifted from Sasakian manifolds

In this article we prove that a certain class of {\it smooth} Sasakian manifolds admits lifts to 4-dimensional quasi-Einstein shearfree spacetimes of Petrov type II or D. This is related to an analogous result by Hill, Lewandowski and Nurowski \cite{HLN} for general {\it real-analytic} CR manifolds. In particular, this holds for all tubular CR manifolds. Furthermore, we show that any Sasakian manifold with underlying Kähler-Einstein manifold with non-zero Einstein constant has a lift to a shearfree Einstein metric of Petrov type II or D.

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arXiv2024

Einstein manifolds with optical geometries of Kerr type

We classify the Ricci flat Lorentzian $n$-manifolds satisfying three particular conditions, encoding and combining some crucial features of the Kerr metrics and the Robinson-Trautman optical structures. We prove that: (a) If $n>4$, there is no Lorentzian manifold satisfying the considered Kerr type conditions, in unexpected contrast with what occurs for the metrics satisfying (very similar) Taub-NUT type conditions; (b) If $n=4$ there are two large classes of such Kerr type manifolds. Each class consists of manifolds fibering over open Riemann surfaces, equipped with a metric of constant Gaussian curvature $\kappa = 1$ or $\kappa = -1$. The first class includes a three parameter family of metrics admitting real analytic extensions to $(\mathbb R^3 \setminus\{0\}) \times \mathbb R = (S^2 \times \mathbb R_+) \times \mathbb R$ and a large class of other metrics not admitting this kind of extensions. The metrics of this first class admitting such extensions are all isometric to the well known Kerr metrics, with the three parameters corresponding to the three space-like components of the angular momentum of the gravitational field. The second class contains a subclass of metrics defined on $\big(\mathbb D\times \mathbb R_+\big)\times \mathbb R$, where $\mathbb D$ is the Lobachevsky Poincar\'e disc. This subclass is in bijection with the holomorphic functions on $\mathbb D$ satisfying an appropriate open condition. These and other results are consequences of a very simple way to construct totally explicit examples of Ricci flat Lorentzian manifolds.

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arXiv2023

The Levi-Civita connections of manifolds with prescribed optical geometries

We explicitly derive the Christoffel symbols in terms of adapted frame fields for the Levi-Civita connection of a Lorentzian $n$-manifold $(M, g)$, equipped with a prescribed optical geometry of Kähler-Sasaki type. The formulas found in this paper have several important applications, such as determining the geometric invariants of Lorentzian manifolds with prescribed optical geometries or solving curvature constraints.

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arXiv2021

Lorentzian manifolds with shearfree congruences and Kähler-Sasaki geometry

We study Lorentzian manifolds $(M, g)$ of dimension $n\geq 4$, equipped with a maximally twisting shearfree null vector field $p_o$, for which the leaf space $S = M/\{\exp t p_o\}$ is a smooth manifold. If $n = 2k$, the quotient $S = M/\{\exp t p_o\}$ is naturally equipped with a subconformal structure of contact type and, in the most interesting cases, it is a regular Sasaki manifold projecting onto a quantisable Kähler manifold of real dimension $2k -2$. Going backwards through this line of ideas, for any quantisable Kähler manifold with associated Sasaki manifold $S$, we give the local description of all Lorentzian metrics $g$ on the total spaces $M$ of $A$-bundles $π: M \to S$, $A = S^1, \mathbb R$, such that the generator of the group action is a maximally twisting shearfree $g$-null vector field $p_o$. We also prove that on any such Lorentzian manifold $(M, g)$ there exists a non-trivial generalized electromagnetic plane wave having $p_o$ as propagating direction field, a result that can be considered as a generalization of the classical $4$-dimensional Robinson Theorem. We finally construct a 2-parametric family of Einstein metrics on a trivial bundle $M = \mathbb R \times S$ for any prescribed value of the Einstein constant. If $\dim M = 4$, the Ricci flat metrics obtained in this way are the well-known Taub-NUT metrics.

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arXiv2018

On a criterion for local embeddability of 3-dimensional CR-structures

We introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a 3-dimensional CR-structure, which we call quasi-Fefferman metrics. These metrics generalise the Fefferman metric but allow for more control of the Ricci curvature. Our main result is a criterion for embaddability of 3-dimensional CR-structures in terms of the Ricci curvature of the quasi-Fefferman metrics in the spirit of the results by Hill et al.

math.CV↗
arXiv2016

Normal forms of para-CR hypersurfaces

We consider hypersurfaces of finite type in a direct product space ${\mathbb R}^2 \times {\mathbb R}^2$, which are analogues to real hypersurfaces of finite type in ${\mathbb C}^2$. We shall consider separately the cases where such hypersurfaces are regular and singular, in a sense that corresponds to Levi degeneracy in hypersurfaces in ${\mathbb C}^2$. For the regular case, we study formal normal forms and prove convergence by following Chern and Moser. The normal form of such an hypersurface, considered as the solution manifold of a 2nd order ODE, gives rise to a normal form of the corresponding 2nd order ODE. For the degenerate case, we study normal forms for weighted $\ell$-jets. Furthermore, we study the automorphisms of finite type hypersurfaces.

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arXiv2014

Singular multicontact structures

We describe the automorphisms of a singular multicontact structure, that is a generalisation of the Martinet distribution. Such a structure is interpreted as a para-CR structure on a hypersurface M of a direct product space R^2 x R^2. We introduce the notion of a finite type singularity analogous to CR geometry and, along the way, we prove extension results for para-CR functions and mappings on embedded para-CR manifolds into the ambient space.

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arXiv2014

On the generalization of Forelli's theorem

The purpose of this paper is to present a solution to perhaps the final remaining case in the line of study concerning the generalization of Forelli's theorem on the complex analyticity of the functions that are: (1) $\mathcal{C}^\infty$ smooth at a point, and (2) holomorphic along the complex integral curves generated by a contracting holomorphic vector field with an isolated zero at the same point.

math.CV↗
arXiv2013

Explicit description of spherical rigid hypersurfaces in C^2

We provide an explicit description of all rigid hypersurfaces that are equivalent to a Heisenberg sphere. These hypersurfaces are determined by 4 real parameters. The defining equations of the rigid spheres can also be viewed as the complete solution of a non-linear PDE that expresses the vanishing Cartan curvature condition for rigid hypersurfaces.

math.CV↗
arXiv2012

Free CR distributions

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions $n$ and codimensions $n^2$ are among the very few possibilities of the so called parabolic geometries. Indeed, the homogeneous model turns out to be $\PSU(n+1,n)/P$ with a suitable parabolic subgroup $P$. We study the geometric properties of such real $(2n+n^2)$-dimensional submanifolds in $\mathbb C^{n+n^2}$ for all $n>1$. In particular we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry.

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arXiv2011

A generalization of Forelli's theorem

The purpose of this paper is to present a generalization of Forelli's theorem. In particular, we prove an all dimensional version of the two-dimensional theorem of Chirka of 2005.

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arXiv2008

Functions holomorphic along holomorphic vector fields

The main result of the paper is the following generalization of Forelli's theorem: Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with the eigenvalues whose ratios are positive reals. Then any function $ϕ$ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary.

math.CV↗
arXiv2005

Elliptic CR-manifolds and shear invariant ODE with additional symmetries

We classify the ODEs that correspond to elliptic CR-manifolds with maximal isotropy. It follows that the dimension of the isotropy group of an elliptic CR-manifold can be only 10 (for the quadric), 4 (for the listed examples) or less. This is in contrast with the situation of hyperbolic CR-manifolds, where the dimension can be 10 (for the quadric), 6 or 5 (for semi-quadrics) or less than 4. We also prove that, for all elliptic CR-manifolds with non-linearizable istropy group, except for two special manifolds, the points with non-linearizable isotropy form exactly some complex curve on the manifold.

math.CV↗
arXiv2005

Canonical Cartan connection and holomorphic invariants on Engel CR manifolds

We describe a complete system of invariants for 4-dimensional CR manifolds of CR dimension 1 and codimension 2 with Engel CR distribution by constructing an explicit canonical Cartan connection. We also investigate the relation between the Cartan connection and the normal form of the defining equation of an embedded Engel CR manifold.

math.CV↗
arXiv2004

Explicit construction of a Chern-Moser connection for CR manifolds of codimension two

In the present paper we suggest an explicit construction of a Cartan connection for an elliptic or hyperbolic CR manifold M of dimension six and codimension two, i.e. a pair (P, w), consisting of a principal bundle P over M and of a Cartan connection form w on P, satisfying the following property: the (local) CR transformations of M are in one to one correspondence with the (local) automorphisms of P which preserve w. For any point x in M, this construction determines an explicit immersion of the stability subalgebra Lie(aut(M)_x) into the Lie algebra Lie(H) of the structure group H of P.

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