Праці міжнародного геометричного центру (Proceedings of the International Geometry Center)

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https://card-file.ontu.edu.ua/handle/123456789/6192

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2F-планарні відображення псевдоріманових просторів з f-структурою
(2018) Nadiia Konovenko, Irina Kurbatova, Katya Tsventoukh
Статтю присвячено проблемі дифеоморфізмів многовидів, на яких задано афінорну структуру певного типу. Поняття 2F-планарного відображення афіннозв’язних і ріманових просторів було запроваджено до розгляду Р.Дж.Кадемом. Воно є природним узагальненням F-планарного відображення і містить в собі такі відомі дифеоморфізми афіннозв’язних і ріманових просторів з афінорною структурою, як геодезичні, квазі-геодезичні, голоморфно-проективні відображення. Р.Дж.Кадем досліджував загальні питання теорії 2F-планарних відображень афіннозв’язних і ріманових просторів з афінорною структурою. Зокрема він довів, що таке відображення за необхідністю зберігає афінорну структуру. Курбатова І.М. вивчала 2F-планарні відображення псевдоріманових просторів з афінорною структурою F третього порядку, що задовольняє умовам Коновенко Н.Г. розглядала деякі питання 2F-планарних відображень псевдоріманових просторів з коваріантно сталою f- структурою F, яка визначається співвідношеннями В наявній статті продовжено дослідження 2F-планарних відображень псевдоріманових просторів з f- структурою. Доведено, що псевдорімановий простір з коваріантно сталою f- структурою становить добуток псевдоріманових просторів, один з яких є келеровим; клас псевдоріманових просторів з коваріантно сталою f- структурою замкнений відносно відображень, що розглядуються; за умови коваріантної сталості афінора f-структури 2F-планарні відображення можуть належати одному з трьох типів: повні і канонічні I,II типу; залежно від типу 2F-планарне відображення індукує на компонентах добутку відповідних просторів геодезичне, голоморфно-проективне або афінне відображення. В теорії дифеоморфізмів многовидів відомі потужні класи ріманових просторів, що дозволяють геодезичні відображення, і келерових просторів, що дозволяють голоморфно-проективні відображення зі збереженням комплексної структури. Тому висновки статті дають змогу будувати численні класи псевдоріманових просторів з коваріантно сталою f- структурою та їх 2F-планарні відображення.
A (CHR)3-flat trans-Sasakian manifold
(2019) Koji Matsumoto
In [4] M. Prvanovic considered several curvaturelike tensors defined for Hermitian manifolds. Developing her ideas in [3], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly (CHR)3-curvature tensor. Then, we mainly researched (CHR)3-curvature tensor in a Sasakian manifold. Also we proved, that a conformally (CHR)3-flat Sasakian manifold does not exist. In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the (CHR)3-curvature tensor in a trans-Sasakian manifold. Also, the (CHR)3-Ricci tensor ρ3 and the (CHR)3-scalar curvature τ3 in a trans-Sasakian manifold have been obtained. Moreover, we define the notion of the (CHR)3-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We proved that a (CHR)3-flat trans-Sasakian manifold is a generalized ɳ-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric g of a trans-Sasakian manifold, if the latter is (CHR)3-flat.
A calculation of periodic data of surface diffeomorphisms with one saddle orbit.
(2018) Олена В'ячеславівна Ноздрінова, Ольга Віталіївна Починка
In the paper it is proved that any orientable surface admits an orientation-preserving diffeomorphism with one saddle orbit. It distinguishes in principle the considered class of systems from source-sink diffeomorphisms existing only on the sphere. It is shown that diffeomorphisms with one saddle orbit of a positive type on any surface have exactly three node orbits. In addition, all possible types of periodic data for such diffeomorphisms are established. Namely, formulas are found expressing the periods of the sources through the periods of the sink and the saddle.
A dynamical approach to shape
(2022) Martin Shoptrajanov
In this paper we will discuss a dynamical approach to an open problem from shape theory. We will address the problem in compact metric spaces using the notion of Lebesgue number for a covering and the intrinsic approach to strong shape.
A Generalized Palais-Smale Condition in the Fr'{e}chet space setting
(2018) Kaveh Eftekharinasab
The Palais-Smale condition was introduced by Palais and Smale in the mid-sixties and applied to an extension of Morse theory to infinite dimensional Hilbert spaces. Later this condition was extended by Palais for the more general case of real functions over Banach-Finsler manifolds in order to obtain Lusternik-Schnirelman theory in this setting. Despite the importance of Fr'{e}chet spaces, critical point theories have not been developed yet in these spaces.Our aim in this paper is to extend the Palais-Smale condition to the cases of $C^1$-functionals on Fr'{e}chet spaces and Fr'{e}chet-Finsler manifolds of class $C^1$. The difficulty in the Fr'{e}chet setting is the lack of a general solvability theory for differential equations. This restricts us to adapt the deformation results (which are essential tools to locate critical points) as they appear as solutions of Cauchy problems. However, Ekeland proved the result, today is known as Ekleand’s variational principle, concerning the existence of almost-minimums for a wide class of real functions on complete metric spaces. This principle can be used to obtain minimizing Palais-Smale sequences. We use this principle along with the introduced conditions to obtain some customary results concerning the existence of minima in the Fr'{e}chet setting.Recently it has been developed the projective limit techniques to overcome problems (such as solvability theory for differential equations) with Fr'{e}chet spaces. The idea of this approach is to represent a Fr'{e}chet space as the projective limit of Banach spaces. This approach provides solutions for a wide class of differential equations and every Fr'{e}chet space and therefore can be used to obtain deformation results. This method would be the proper framework for further development of critical point theory in the Fr'{e}chet setting.
A metrizable Lawson semitopological semilattice with non-closed partial order
(2020) Taras Banakh, Serhii Bardyla, Alex Ravsky
We construct a metrizable Lawson semitopological semilattice $X$ whose partial order $le_X,={(x,y)in Xtimes X:xy=x}$ is not closed in $Xtimes X$. This resolves a problem posed earlier by the authors.
A new curvature-like tensor in an almost contact Riemannian manifold
(2017) Koji Matsumoto
In a M. Prvanović’s paper [5], we can find a new curvature-like tensor in an almost Hermitian manifold.In this paper, we define a new curvature-like tensor, named contact holomorphic Riemannian, briefly (CHR), curvature tensor in an almost contactRiemannian manifold. Then, using this tensor, we mainly research (CHR)-curvature tensor in a Kenmotsu and a Sasakian manifold. We introducethe flatness of a (CHR)-curvature tensor and show that a Kenmotsu anda Sasakian manifold with a flat (CHR)-curvature tensor is flat, see Theorems3.1 and 4.1. Next, we introduce the notion of an (CHR)-n-Einstein inan almost contact Riemannian manifold. In particular, in a Sasakian or aKenmotsu manifold, a (CHR)-n-Einstein manifold is n-Einstein, see Theorem5.3. Finally, from this tensor, we introduce a notion of a (CHR)-spaceform in an almost contact Riemannian manifold. In particular, if a Kenmotsuand a Sasakian manifold are (CHR)-space form, then the (CHR)-curvaturetensor satisfies a special equation, see Theorems 6.2 and 7.1.
A Physics-Based Estimation of Mean Curvature Normal Vector for Triangulated Surfaces
(2019) Sudip Kumar Das, Mirza Cenanovic, Junfeng Zhang
In this note, we derive an approximation for the mean curvature normal vector on vertices of triangulated surface meshes from the Young-Laplace equation and the force balance principle. We then demonstrate that the approximation expression from our physics-based derivation is equivalent to the discrete Laplace-Beltrami operator approach in the literature. This work, in addition to providing an alternative expression to calculate the mean curvature normal vector, can be further extended to other mesh structures, including non-triangular and heterogeneous meshes.
A survey of homotopy nilpotency and co-nilpotency
(2020) Marek Golasinski
We review known and state some new results on homotopy nilpotency and co-nilpotency of spaces. Next, we take up the systematic study of homotopy nilpotency of homogenous spaces G/K for a Lie group G and its closed subgroup K < G. The homotopy nilpotency of the loop spaces Ω(Gn,m(K)) and Ω(Vn,m(K)) of Grassmann Gn,m(K) and Stiefel Vn,m(K) manifolds for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaternions is shown.
A.K. Bakhtin. Scientific legacy
(2023) Iryna Denega, Yaroslav Zabolotnyi
In the paper we give a brief overview of the O. Bakhtin' scientific results
Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere
(2019) Anna Kravchenko, Sergiy Maksymenko
Let $M$ be a compact two-dimensional manifold and, $f in C^{infty}(M, R)$ be a Morse function, and $Gamma$ be its Kronrod-Reeb graph.Denote by $O(f)={f o h | h in D(M)}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $D(M)$ onC^{infty}$, and by $S(f)={hin D(M) | f o h = f }$ the coresponding stabilizer of this function.It is easy to show that each $hin S(f)$ induces an automorphism of the graph $Gamma$.Let $D_{id}(M)$ be the identity path component of $D(M)$, $S'(f) = S(f) cap D_{id}(M)$ be the subgroup of $D_{id}(M)$ consisting of diffeomorphisms preserving $f$ and isotopic to identity map, and $G$ be the group of automorphisms of the Kronrod-Reeb graph induced by diffeomorphisms belonging to $S'(f)$. This group is one of key ingredients for calculating the homotopy type of the orbit $O(f)$. In the previous article the authors described the structure of groups $G$ for Morse functions on all orientable surfacesdistinct from $2$-torus and $2$-sphere. The present paper is devoted to the case $M = S^2$. In this situation $Gamma$ is always a tree, and therefore all elements of the group $G$ have a common fixed subtree $Fix(G)$, which may even consist of a unique vertex. Our main result calculates the groups $G$ for all Morse functions $f: S^2 to R$ whose fixed subtree $Fix(G)$ consists of more than one point.
Bypassing dynamical systems: a simple way to get the box-counting dimension of the graph of the Weierstrass function
(2018) Claire David
In the following, bypassing dynamical systems tools, we propose a simple means of computing the box dimension of the graph of the classical Weierstrass function defined, for any real number~$x$, by[{mathcal W}(x)= sum_{n=0}^{+infty} lambda^n,cos left ( 2, pi,N_b^n,x right),]where $lambda$ and $N_b$ are two real numbers such that $0 1$, using a sequence a graphs that approximate the studied one.
Canonical quasi-geodesic mappings of special pseudo-Riemannian spaces
(2022) Irina Kurbatova, M. Pistruil
The present paper continues the study of quasi-geodesic mappings f:(Vn, gij, Fih) → (V'n,g'ij, Fih) of pseudo-Riemannian spaces Vn, V'n with a generalized-recurrent structure Fih of parabolic type. By a generalized recurrent structure of parabolic type on Vn we mean an almost Hermitian affinor structure of parabolic type for which the covariant derivative of the structural affinor Fih satisfies the condition F(i,j)h=q(i Fj)h. In the previous paper by the authors [Proc. Intern. Geom. Center, 13:3 (2020) 18-32] it was proved that the class of pseudo-Riemannian spaces with generalized-recurrent structure of parabolic type is closed with respect to the considered mappings and the generalized recurrence vectors in (Vn, gij,Fih) and (V'_n, g'ij, Fih) may be distinct. In this article, it is assumed that the mapping f preserves the generalized recurrence vector qi. We construct geometric objects that are invariant under the quasi-geodesic mapping of generalized-recurrent spaces of parabolic type and recurrent-parabolic spaces. A number of conditions are given on these objects, which lead to the fact that a generalized-recurrent space of parabolic type admits a parabolic K-structure, and a recurrent-parabolic space admits a Kählerian structure of parabolic type. We study special types of these mappings that preserve some tensors of an intrinsic nature.
Centralizers of elements in Lie algebras of vector fields with polynomial coefficients
(2022) Анатолій Петрович Петравчук
abstract{ukrainian}{Нехай $mathbb K$ -- алгебраїчно замкнене поле харатеристики нуль,$A = mathbb K[x_1,dots,x_n]$ -- кільце многочленів і$R = mathbb K(x_1,dots,x_n)$ -- поле раціональних функцій від $n$ змінних. Позначимо через $W_n = W_n(mathbb K)$ алгебру Лі всіх$mathbb K$-диференціювань на $A$(у випадку $mathbb C$ це алгебра Лі всіх векторних полів на $ mathbb C^n$ з поліноміальними коефіцієнтами). Для заданого $D in W_n(mathbb K)$ будова централізатора$C_{W_n (mathbb K)}(D)$ залежить від поля констант$Ker D = {phi in R | D(phi)=0}$(тут ми природнім чином розширюємо кожне диференціювання $D$ на $A$ на поле $R$).Досліджено випадок, коли $tr.deg_{mathbb K} Ker D le 1$, охарактеризована будова підалгебри $C_{W_n(mathbb K)}(D)$, зокрема доведено, що якщо $Ker D$ не містить несталих многочленів, то$C_{W_n(mathbb K)}(D)$ скінченновимірний над $mathbb K$. Отримано деякі результати про централізатори лінійних диференціювань в $W_n(mathbb K).$}
Classification of curves on de Sitter plane
(2020) Irina Streltsova
In 1917, de Sitter used the modified Einstein equation and proposed a model of the Universe without physical matter, but with a cosmological constant. De Sitter geometry, as well as Minkowski geometry, is maximally symmetrical. However, de Sitter geometry is better suited to describe gravitational fields. It is believed that the real Universe was described by the de Sitter model in the very early stages of expansion (inflationary model of the Universe). This article is devoted to the problem of classification of regular curves on the de Sitter space. As a model of the de Sitter plane, the upper half-plane on which the metric is given is chosen. For this purpose, an algebra of differential invariants of curves with respect to the motions of the de Sitter plane is constructed. As it turned out, this algebra is generated by one second-order differential invariant (we call it by de Sitter curvature) and two invariant differentiations. Thus, when passing to the next jets, the dimension of the algebra of differential invariants increases by one. The concept of regular curves is introduced. Namely, a curve is called regular if the restriction of de Sitter curvature to it can be considered as parameterization of the curve. A theorem on the equivalence of regular curves with respect to the motions of the de Sitter plane is proved. The singular orbits of the group of proper motions are described.
Complex hyperbolic triangle groups with 2-fold symmetry
(2017) Джон Р. Паркер, Лі-Джі Сан
In this paper we will consider the 2-fold symmetric complex hyperbolic triangle groups generated by three complex reflections through angle 2Π/p with p ≥ 2. We will mainly concentrate on the groups where some elements are elliptic of finite order. Then we will classify all such groups which are candidates for being discrete. There are only 4 types.
Deformations of circle-valued Morse functions on 2-torus
(2021) Bohdan Feshchenko
In this paper we give an algebraic description of fundamental groups of orbits of circle-valued Morse functions on T2 with respect to the action of the group of diffeomorphisms of T2
Dynamics and exact solutions of the generalized Harry Dym equation
(2020) Ruslan Matviichuk
The Harry Dym equation is the third-order evolutionary partial differential equation. It describes a system in which dispersion and nonlinearity are coupled together. It is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It has an infinite number of conservation laws and does not have the Painleve property. The Harry Dym equation has strong links to the Korteweg – de Vries equation and it also has many properties of soliton solutions. A connection was established between this equation and the hierarchies of the Kadomtsev – Petviashvili equation. The Harry Dym equation has applications in acoustics: with its help, finite-gap densities of the acoustic operator are constructed. The paper considers a generalization of the Harry Dym equation, for the study of which the methods of the theory of finite-dimensional dynamics are applied. The theory of finite-dimensional dynamics is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of evolutionary differential equations. In our case, this approach allows us to obtain some classes of exact solutions of the generalized equation, and also indicates a method for numerically constructing solutions.
Explicit formulae for Chern-Simons invariants of the hyperbolic J(2n,-2m) knot orbifolds
(2023) Ji-Young Ham, Joongul Lee
We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of double twist knots.
f-атоми складності функцій Морса на замкнених оріорієнтованих двовимірних многовидах
(2017) О. О. Пришляк, Д. М. Скочко
В роботі було досліджено та знайдено всі можливі f-атоми складност 4 функцій Морса на замкнених орієнтованих двовимірних многовидах.

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