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Robert Wolak

Publications and source records attributed to Robert Wolak.

15 recordsLinked to original sources
arXiv2026

Laplace and Dolbeault operators on Sasakian manifolds

Sasakian manifolds, the odd-dimensional analogues of K\"ahler manifolds, carry two natural pairs of first-order Dolbeault-type operators on the full complex of differential forms, extending the Kohn--Rossi differentials. On such Sasaki manifolds, we establish K\"ahler-type identities for these operators, and relate the resulting Dolbeault Laplacians to the Hodge Laplacian. Unlike the K\"ahler case, where $\Delta=2\Delta_{\overline\partial}$, the relation has extra terms coming from the Reeb flow. Using these formulas and a Lefschetz decomposition, we derive lower and upper eigenvalue estimates for $\Delta$ on forms depending on the eigenvalues of the Lie derivative of the Reeb vector field.

math.DG↗
arXiv2026

Basic Albanese maps of regular Riemannian foliations

In the paper we introduce the notion of basic Albanese map which we define for foliated Riemannian manifolds using basic 1-forms. We relate this mapping to the classical Albanese map for the ambient manifold. The study of general properties is supplemented with the description of several important examples.

math.DG↗
arXiv2024

Transverse geometric formality

A Riemannian metric on a closed manifold is said to be geometrically formal if the wedge product of any two harmonic forms is harmonic; equivalently, the interior product of any two harmonic forms is harmonic. Given a Riemannian foliation on a closed manifold, we say that a bundle-like metric is transversely geometrically formal if the interior product of any two basic harmonic forms is basic harmonic. In this paper, we examine the geometric and topological consequences of this condition.

math.DG↗
arXiv2023

Hard Lefschetz property for $\mathbb{S}^3$-actions

The Hard Lefschetz Property (HLP) has recently been formulated in the context of isometric flows without singularities on manifolds. In this category, two versions of the HLP (transverse and not) have been proven to be equivalent, thus generalizing what happens in the important cases of both K-contact and Sasakian manifolds. In this work we define both versions of the HLP for almost-free S3 -actions, and prove that they agree for actions satisfying a cohomological condition, which includes the important category of 3-Sasakian manifolds, where those two versions of the HLP are shown to be held. We also provide a family of examples of free actions of the 3-sphere which are not 3-Sasakian manifolds, but satisfy the HLP.

math.DG↗
arXiv2021

Hard Lefschetz Property for Isometric Flows

The Hard Lefschetz Property (HLP) is an important property which has been studied in several categories of the symplectic world. For Sasakian manifolds, this duality is satisfied by the basic cohomology (so, it is a transverse property), but a new version of the HLP has been recently given in terms of duality of the cohomology of the manifold itself in arXiv:1306.2896. Both properties were proved to be equivalent (see arXiv:1311.1431) in the case of K-contact flows. In this paper we extend both versions of the HLP (transverse and not) to the more general category of isometric flows, and show that they are equivalent. We also give some explicit examples which illustrate the categories where the HLP could be considered.

math.DG↗
arXiv2021

The Partial Ricci Flow on $\mathfrak{g}$-foliations

In the paper we introduce new metric structures on $\mathfrak{g}$-foliations that are less rigid than the well-known structures: almost contact and 3-quasi-Sasakian structures as well as $f$-structures with parallelizable kernel and almost para-$ϕ$-structures with complemented frames. We discuss the properties of the new structures in order to demonstrate similarities with the corresponding classical structures. Then using the flow of metrics on a $\mathfrak{g}$-foliation, we build deformation retraction of our structures with positive partial Ricci curvature onto the subspace of the aforementioned classical structures.

math.DG↗
arXiv2020

On Ricci curvature of metric structures on $\mathfrak{g}$-manifolds

We study the properties of Ricci curvature of ${\mathfrak{g}}$-manifolds with particular attention paid to higher dimensional abelian Lie algebra case. The relations between Ricci curvature of the manifold and the Ricci curvature of the transverse manifold of the characteristic foliation are investigated. In particular, sufficient conditions are found under which the ${\mathfrak{g}}$-manifold can be a Ricci soliton or a gradient Ricci soliton. Finally, we obtain a amazing (non-existence) higher dimensional generalization of the Boyer-Galicki theorem on Einstein K-manifolds for a special class of abelian ${\mathfrak{g}}$-manifolds.

math.DS↗
arXiv2018

Cohomological Tautness of Singular Riemannian Foliations

For a Riemannian foliation F on a compact manifold M , J. A. Álvarez López proved that the geometrical tautness of F , that is, the existence of a Riemannian metric making all the leaves minimal submanifolds of M, can be characterized by the vanishing of a basic cohomology class (the Álvarez class). In this work we generalize this result to the case of a singular Riemannian foliation K on a compact manifold X. In the singular case, no bundle-like metric on X can make all the leaves of K minimal. In this work, we prove that the Álvarez classes of the strata can be glued in a unique global Álvarez class. As a corollary, if X is simply connected, then the restriction of K to each stratum is geometrically taut, thus generalizing a celebrated result of E. Ghys for the regular case.

math.DG↗
arXiv2017

Growth of some transversely homogeneous foliations

For transversely homogeneous foliations on compact manifolds whose global holonomy group has connected closure, it is shown that either all holonomy covers of the leaves have polynomial growth with degree bounded by a common constant, or all holonomy covers of the leaves have exponential growth. This is an extension of a recent answer given by Breuillard and Gelander to a question of Carrière. Examples of transversely projective foliations satisfying the above condition were constructed by Chihi and ben Ramdane.

math.GT↗
arXiv2016

On normality of f.pk-structures on g-manifolds

We consider higher dimensional generalisations of normal almost contact structures, the so called f.pk-structures where parallelism spans a Lie algebra g (f.pk-g-structures). Two types of these structures are discussed. In the first case, we construct an almost complex structure on a product manifold mirroring K-structures. We show that the natural normality condition can be satisfied only when g is abelian. The second case we consider is when the Lie algebra in question is 3-dimensional, but the almost complex structure on a product is constructed in a different manner. In both cases the normality conditions are expressed in terms of the structure tensors.

math.DG↗
arXiv2016

Sasakian structures. A foliated approach

Recent renewed interest in Sasakian manifolds is due mainly to the fact that they can provide examples of generalized Einstein manifolds, manifolds which are of great interest in mathematical models of various aspects of physical phenomena. Sasakian manifolds are odd dimensional counterparts of Kählerian manifolds to which they are closely related. The book of Ch. Boyer and K. Galicki, Sasakian Geometry is both the best introduction to the subject and at the same time it gathers state of the art information and results on these manifolds. However, although the authors are well aware that a Sasakian structure is a very special one-dimensional Riemannian foliation with Kählerian transverse structure, they use this fact only in a few very special cases. The paper presents an approach to Sasakian manifolds on which the author gave several lectures, most recently at the Workshop on almost hermitian and contact geometry at the Banach Center in Bȩdlewo in October 2015 and at University of the Basque Country in February 2016. The first lectures on the topic the author gave at Universidad de Sevilla in October 1988 and then presented the consequence for the geometry of Sasakian manifolds, in particular the relationns between various curvatures and those of the transverse Kähler manifold. The results were published in several sections of \cite{WO_S} as well as in \cite{Wo_deb}. The most general theory of geometrical structures "adapted" to a foliation was presented in \cite{WO_T}, see also \cite{WO_S}. The paper concentrates on cohomological properties of Sasakian manifolds and of transversely holomorphic and Kählerian foliations. These properties permit to formulate obstructions to the existence of Sasakian structures on compact manifolds. The presented results are due to the author as well as his former and present Ph.D. students.

math.DG↗
arXiv2016

Orbifolds, geometric structures and foliations. Applications to harmonic maps

In recent years a lot of attention has been paid to topological spaces which are a bit more general than smooth manifolds - orbifolds. Orbifolds are intuitively speaking manifolds with some singularities. The formal definition is also modelled on that of manifolds, an orbifold is a topological space which locally is homeomorphic to the orbit space of a finite group acting on $R^n$. Orbifolds were defined by Satake, as V-manifolds, then studied by W. Thurston, who introduced the term "orbifold". Due to their importance in physics, and in particular in the string theory, orbifolds have been drawing more and more attention. In this paper we propose to show that the classical theory of geometrical structures, easily translates itself to the context of orbifolds and is closely related to the theory of foliated geometrical structures, cf. \cite{Wo0}. Finally, we propose a foliated approach to the study of harmonic maps between Riemannian orbifolds based on our previous research into transversely harmonic maps.

math.DG↗
arXiv2012

Deforming metrics of foliations

Our results concern geometry of a manifold endowed with a pair of complementary orthogonal distributions (plane fields) and a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies conformally along one of the distributions. Then we introduce the Extrinsic Geometric Flow depending on the mean curvature vector field of the distribution, and show existence/uniquenes and convergence of a solution as $t\to\infty$, when the complementary distribution is integrable with compact leaves. We apply the method to the problem of prescribing mean curvature vector field of a foliation, and give examples for harmonic and umbilical foliations and for the double-twisted product metrics, including the codimension-one case.

math.DG↗
arXiv2008

A First Approximation for Quantization of Singular Spaces

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group - that commutes with reduction - can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals.

math.DG↗