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2024 | Book

Differential Geometric Structures and Applications

4th International Workshop on Differential Geometry, Haifa, Israel, May 10–13, 2023

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About this book

This proceedings contains a collection of selected, peer-reviewed contributions from the 4th International Workshop "Differential Geometric Structures and Applications" held in Haifa, Israel from May 10–13, 2023. The papers included in this volume showcase the latest advancements in modern geometry and interdisciplinary applications in fields ranging from mathematical physics to biology.

Since 2008, this workshop series has provided a platform for researchers in pure and applied mathematics, including students, to engage in discussions and explore the frontiers of modern geometry. Previous workshops in the series have focused on topics such as "Reconstruction of Geometrical Objects Using Symbolic Computations" (2008), "Geometry and Symbolic Computations" (2013), and "Geometric Structures and Interdisciplinary Applications" (2018).

Table of Contents

Frontmatter
Some Topics in Sasakian Geometry, a Survey
Abstract
In the seminal book Sasakian Geometry by C. Boyer and K. Galicki, the authors formulated a research program for studying topological properties and answering questions about the existence of Sasakian structures. We survey recent progress in this topic.
Aleksy Tralle
Einstein-Type Metrics and Ricci-Type Solitons on Weak f-K-Contact Manifolds
Abstract
A weak metric f-structure \((f,Q,\xi _i,\eta ^i,g)\ (i=1,\ldots ,s)\) on a smooth manifold generalizes the metric f-structure, i.e., the linear complex structure on the contact distribution is replaced with a nonsingular skew-symmetric tensor. We study geometry of a weak f-K-contact structure, which is a weak f-contact structure, whose characteristic vector fields are Killing. We show that the distribution \(\ker f\) of a weak f-contact manifold defines a \(\mathfrak {g}\)-foliation with an abelian Lie algebra, characterize weak f-K-contact manifolds among all weak metric f-manifolds by the property known for f-K-contact manifolds, and find when a Riemannian manifold endowed with a set of orthonormal Killing vector fields is a weak f-K-contact manifold. We prove that for \(s>1\), an Einstein weak f-K-contact manifold is Ricci flat and find sufficient conditions for a weak f-K-contact manifold with parallel Ricci tensor or with a generalized gradient Ricci soliton structure to be Ricci flat or a quasi Einstein manifold. We show positive definiteness of the Jacobi operators in the characteristic directions and use this to deform a weak f-K-contact structure to an f-K-contact structure. We define an \(\eta \)-Ricci soliton and \(\eta \)-Einstein structures on a weak metric f-manifold (which for \(s=1\), give the well-known structures on contact metric manifolds) and find sufficient conditions for a compact weak f-K-contact manifold with an \(\eta \)-Ricci soliton structure of constant scalar curvature to be \(\eta \)-Einstein.
Vladimir Rovenski
Weak -Kenmotsu Manifolds and -Ricci Solitons
Abstract
Weak contact metric structures on a smooth manifold have been recently introduced by V. Rovenski and R. Wolak in 2022. In this paper, we define a new structure of this kind called a weak \(\beta \)-Kenmotsu structure (that generalizes the notion by K. Kenmotsu with \(\beta =1\) and its extension for \(\beta \ne 0\) by Z. Olszak). We show that a weak \(\beta \)-Kenmotsu manifold is locally the warped product \((-\varepsilon ,\varepsilon )\times _\sigma \bar{M}\), where \((\partial _t\,\sigma )/\sigma =\beta \ne 0\), and \((\bar{M},\bar{g})\) is equipped with a parallel skew-symmetric (1,1)-tensor \(\bar{\phi }\) such that \(\bar{\phi }^{\,2}\) is negative definite. Then, we show that an \(\eta \)-Einstein weak \(\beta \)-Kenmotsu manifold with \(\beta =const\ne 0\) admitting an \(\eta \)-Ricci soliton structure is an Einstein manifold. Finally, we prove that a weak \(\beta \)-Kenmotsu manifold and admitting an \(\eta \)-Ricci soliton structure, whose non-zero potential vector field is weak contact or is collinear to \(\xi \), is an Einstein manifold.
Dhriti Sundar Patra, Vladimir Rovenski
Twistor Bundles of Foliated Riemannian Manifolds
Abstract
In this paper, we present an overview of the theory of twistors on foliated manifolds, which has been developed in several papers of ours. We construct the twistor space of the normal bundle of a foliation. We demonstrate that the classical constructions of the twistor theory lead to foliated objects and permit us to formulate and prove foliated versions of some well-known results on holomorphic mappings. Since any orbifold can be understood as the leaf space of a suitably defined Riemannian foliation we obtain orbifold versions of the classical results as a simple consequence of the results on foliated mappings. Finally, we recall suspension construction, which is used to construct examples of transversely harmonic maps between foliated manifolds and discuss the twistor methods in this case.
R. Mohseni, R. Wolak
Mixed 3-Sasakian Statistical Manifolds and Statistical Submersions
Abstract
Mixed 3-structures are known to be the counterpart in odd dimension of the quaternionic structures of second kind, also termed in the literature as paraquaternionic structures. The aim of this paper is to investigate the statistical structures in paraquaternionic and mixed 3-Sasakian geometry. We derive the main properties of statistical manifolds equipped with such kind of structures, focusing on the case of mixed 3-Sasakian statistical manifolds and paraquaternionic Kähler-like statistical manifolds. Moreover, we investigate the geometry of statistical submersions with total space a mixed 3-Sasakian manifold or a paraquaternionic Kähler-like statistical manifold. Moreover, we provide some illustrative examples.
Crina Daniela Neacşu
Minimal Unit Vector Fields on Oscillator Groups
Abstract
In this paper we treat minimal left-invariant unit vector fields on oscillator group and their relations with the ones that defines a harmonic map. Particularly, if all structure constants of the oscillator group are equal to each other, then all unit left-invariant vector fields that define a harmonic map into the unit tangent bundle with Sasaki metric are minimal.
Alexander Yampolsky
Growth and Structure of Equicontinuous Foliated Spaces
Abstract
Molino’s description of Riemannian foliations on compact manifolds extends to compact equicontinuous foliated spaces as developed by Àlvarez Lòpez and Manuel Moreira. This extension is particularly considered when leaves are densely packed, and the pseudogroup exhibits strong quasi-analytic behavior. Notably, this extension leads to the establishment of an association with a structural local group within such a foliated space. Application of this framework results in a partial generalization of Carrière and Breuillard-Gelander’s results, creating a connection between the structural local group and the growth of leaves. Additionally, we present an illustrative examples, as well as a scenario with zero topological codimension. In this context, instances of weak solenoids embodying this characteristic have been previously established by Dyer, Hurder, and Lukina.
Manuel F. Moreira Galicia
Lichnerowicz-Type Laplacians in the Bochner Technique
Abstract
In the well-known monograph of A. Besse the following is written: the Bochner technique is a method of proving vanishing theorems for null space of a Laplace operator admitting a Weitzenböck decomposition and further of estimating its lowest nonzero eigenvalue. In this article, we consider a generalized form of the well-known Lichnerowicz Laplacian, show how the Bochner technique works for this operator and provide some its important examples.
Vladimir Rovenski, Sergey Stepanov, Irina Tsyganok
On General Solutions of Sinyukov Equations on Two-Dimensional Equidistant (pseudo-)Riemannian Spaces
Abstract
The paper is devoted to study of Sinyukov equations on two-dimensional equidistant (pseudo-) Riemannian spaces. The general solution of Sinyukov equations is found beyond these spaces under minimal requirements for the differentiability of the studied objects.
Patrik Peška, Lenka Vítková, Josef Mikeš, Irina Kuzmina
Fundamental Equations on Conformal Fedosov Spaces
Abstract
The present work is devoted to the study of the basic equations of conformally Fedosov structures. These equations are obtained in the form of closed linear equations in covariant derivatives of the Cauchy type. It is established that the general solution depends on no more than \(1/2n(n + 1)\) numerical parameters. The maximum is achieved in projectively Euclidean spaces. Estimates are found for these equations’ dependence on spaces that are not projectively Euclidean.
Chakibek Almazbekov, Nadezda Guseva, Josef Mikeš
Rotary Mappings of Equidistant Spaces
Abstract
The paper is devoted to study of rotary mappings equations of two-dimensional equidistant (pseudo-) Riemannian spaces. The general solution of these equations is found beyond these spaces under minimal requirements for the differentiability of the studied objects.
Lenka Vítková
Smith-Gysin Sequence
Abstract
Given a smooth semifree action of \(S^3\) on a manifold M, we have the Smith-Gysin sequence:
$$\begin{aligned} \cdots \rightarrow {H}^{^{*}}{\big ( M \big )} {\rightarrow } {H}^{^{*-3}}{\big ( M/S^3, M^{S^3} \big )} \oplus {H}^{^{*}}{\big ( M^{S^3} \big )} {\rightarrow } {H}^{^{*+1}}{\big ( M/S^3, M^{S^3} \big )} {\rightarrow } {H}^{^{*+1}}{\big ( M \big )}\rightarrow \cdots \end{aligned}$$
In this paper, we construct a Smith-Gysin sequence that does not require the semifree condition. This sequence includes a new term, referred to as the exotic term, which depends on the subset \(M^{S^1}\):
$$\begin{aligned} \cdots \rightarrow {H}^{^{*}}{\big ( M \big )} \rightarrow {H}^{^{*-3}}{\big ( M/S^3, \Sigma /S^3 \big )} \oplus {H}^{^{*}}{\big ( M^{S^3} \big )} \oplus \big ({H}^{^{*-2}}{\big ( M^{S^1} \big )}\big )^{-{\mathbb {Z}}_{_2}} \\ \rightarrow {H}^{^{*+1}}{\big ( M/S^3,M^{S^3} \big )} \rightarrow {H}^{^{*+1}}{\big ( M \big )} \rightarrow \cdots \end{aligned}$$
Here, \(\Sigma \subset M\) is the subset of points in M whose isotropy group is infinite. The group \(\mathbb {Z}_2\) acts on \(M^{S^1}\) by \(j \in S^3\).
J. I. Royo Prieto, M. Saralegi-Aranguren, R. Wolak
An Ay-Lê-Jost-Schwachhöfer Type Characterization of Quantitatively Weakly Sufficient Statistics
Abstract
We explain the notion of a \(\delta \)-almost sufficient statistic, which is a quantitative weak version of sufficient statistics introduced in [11], based on the example of coin tosses. We review the basic notions in information geometry, e.g., parametrized measure models, statistics and Fisher quadratic forms. Then we state characterizations of sufficient statistics due to Ay-Jost-Lê-Schwachhöfer and an analogous characterizations of sufficient statistics due to [11].
Kaori Yamaguchi, Hiraku Nozawa
Lower Bounds for High Derivatives of Smooth Functions With Given Zeros
Abstract
Let \(f: B^n \rightarrow {\mathbb R}\) be a \(d+1\) times continuously differentiable function on the unit ball \(B^n\), with \(\max _{z\in B^n} |f(z)|=1\). A well-known fact is that if f vanishes on a set \(Z\subset B^n\) with a non-empty interior, then for each \(k=1,\ldots ,d+1\) the norm of the k-th derivative \(\Vert f^{(k)}\Vert \) is at least \(M=M(n,k)>0\). A natural question to ask is: what happens for other sets Z? In particular, for finite, but sufficiently dense sets? This question was partially answered in [16] and [2022]. This study is naturally related to a certain special settings of the Whitney’s smooth extension problem. Our goal in this paper is threefold: first, to provide an overview of the relevant questions and existing results in the general Whitney’s problem. Second, we provide an overview of our specific setting and some available results. Third, we provide some new results in our direction, which extend the recent result of [21], where an answer to the above question is given via the topological information on Z.
Gil Goldman, Yosef Yomdin
Interactions Between Differential Geometry and Production Theory
Abstract
The use of differential geometry in the theory of production is an interesting topic that has attracted the attention of many researchers in the last two decades, both from the fields of mathematics and economics. The purpose of this work is to provide a comprehensive survey on the properties of the homogeneous and quasi-homogeneous production models, most of these properties being of geometric nature and obtained using a differential geometric treatment. Some open problems are also discussed at the end of the article.
Alina-Daniela Vîlcu, Gabriel-Eduard Vîlcu
A Lagrangian Program Detecting the Weighted Fermat-Steiner-Fréchet Multitree for a Fréchet N-multisimplex in Euclidean N-space
Abstract
We introduce the Fermat-Steiner-Fréchet (FSFR) problem for a given \(\frac{1}{2} N(N+1)\)-tuple of positive real numbers determining the edge lengths of an N-simplex in \(\mathbb {R}^{N}\) in order to study its solution called the “FSFR multitree”, which consist of a union of Fermat-Steiner (FS) trees for all derived pairwise incongruent N-simplexes in the sense of Blumenthal, Herzog for \(N=3\) and Dekster-Wilker for \(N\ge 3\). We obtain a method to determine the FSFR multitree in \(\mathbb {R}^{N}\) based on the theory of Lagrange multipliers, whose equality constraints depend on \(N-1\) independent solutions of the inverse weighted Fermat problem for an N-simplex in \(\mathbb {R}^{N}\). A fundamental application of the Lagrangian program for the FSFR problem in \(\mathbb {R}^{N}\) is the detection of the FS tree with global minimum length having \(N-1\) equally weighted FS points among \(\frac{[\frac{1}{2} N(N+1)]!}{(N+1)!}\) incongruent N-simplexes determined by an \(\frac{1}{2} N(N+1)\)-tuple of consecutive natural numbers controlled by Dekster-Wilker, Blumenthal-Herzog conditions and enriched with the fundamental evolutionary processes of Nature (Minimum communication networks, minimum mass transfer, maximum volume of incongruent simplexes). Furthermore, we obtain the unique solution of the inverse weighted Fermat problem, referring to the unique set of \((N+1)\) weights, which correspond to the vertices of an N-boundary simplex in \(\mathbb {R}^{N}\). Additionally, we give a negative answer for an intermediate weighted Fermat-Steiner-Fréchet multitree having one node (weighted Fermat point) for m boundary closed polytopes (\(m\ge N+2\)), which is determined by m prescribed rays meeting at a fixed weighted Fermat point, by deriving a linear dependence for the m variable weights (Plasticity of an Intermediate FSFR multitree for m boundary closed polytopes). By entering in the plasticity of an intermediate FSFR multitree for m boundary closed polytopes a two-way mass transport from k vertices to the unique weighted Fermat point and from this point to the \(m-k\) vertices, and reversely, we derive the equations of “mutation” of intermediate FSFR multitrees in \(\mathbb {R}^{N}\).
Anastasios N. Zachos
Metadata
Title
Differential Geometric Structures and Applications
Editors
Vladimir Rovenski
Paweł Walczak
Robert Wolak
Copyright Year
2024
Electronic ISBN
978-3-031-50586-7
Print ISBN
978-3-031-50585-0
DOI
https://doi.org/10.1007/978-3-031-50586-7

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