Topology Of Manifolds Research Articles

Riemannian Properties of Engel Structures

Abstract This paper is about geometric and Riemannian properties of Engel structures. A choice of defining forms for an Engel structure $\mathcal{D}$ determines a distribution $\mathcal{R}$ transverse to $\mathcal{D}$ called the Reeb distribution. We study conditions that ensure integrability of $\mathcal{R}$. For example, if we have a metric $g$ that makes the splitting $TM=\mathcal{D}\oplus \mathcal{R}$ orthogonal and such that $\mathcal{D}$ is totally geodesic then there exists another Reeb distribution, which is integrable. We introduce the notion of K-Engel structures in analogy with K-contact structures, and we classify the topology of K-Engel manifolds. As natural consequences of these methods, we provide a construction that is the analogue of the Boothby–Wang construction in the contact setting, and we give a notion of contact filling for an Engel structure.

Exact ground state and elementary excitations of a topological spin chain

A novel Bethe Ansatz scheme is proposed to calculate physical properties of quantum integrable systems without $U(1)$ symmetry. As an example, the anti-periodic XXZ spin chain, a typical correlated many-body system embedded in a topological manifold, is examined. Conserved "momentum" and "charge" operators are constructed despite the absence of translational invariance and $U(1)$ symmetry. The ground state energy and elementary excitations are derived exactly. It is found that two intrinsic fractional (one half) zero modes accounting for the double degeneracy exist in the eigenstates. The elementary excitations show quite a different picture from that of a periodic chain. This method can be applied to other quantum integrable models either with or without $U(1)$ symmetry.

Persistence of normally hyperbolic invariant manifolds in the absence of rate conditions

We consider perturbations of normally hyperbolic invariant manifolds, under which they can lose their hyperbolic properties. We show that if the perturbed map which drives the dynamical system preserves the properties of topological expansion and contraction, then the manifold is perturbed to an invariant set. The main feature is that our results do not require the rate conditions to hold after the perturbation. In this case the manifold can be perturbed to an invariant set, which is not a topological manifold. We work in the setting of nonorientable Banach vector bundles, without needing to assume invertibility of the map.

Étale Steenrod operations and the Artin–Tate pairing

We prove a 1966 conjecture of Tate concerning the Artin–Tate pairing on the Brauer group of a surface over a finite field, which is the analog of the Cassels–Tate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of Poonen–Stoll on the Cassels–Tate pairing. Our method is based on studying a connection between the Artin–Tate pairing and (generalizations of) Steenrod operations in étale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate the relevant étale Steenrod operations in terms of characteristic classes.

Aeroelastic response of an airfoil to gusts: Prediction and control strategies from computed energy maps

A method to predict the aeroelastic pitch response of an airfoil to gusts is presented. The prediction is based on energy maps generated by high-fidelity fluid dynamic simulations of the airfoil with prescribed pitch oscillations. The energy maps quantify the exchange of energy between the pitching airfoil and the flow, and serve as manifolds over which the dynamical states of aeroelastic airfoil system grow, decay and attain stationary states. This method allows us to study the full nonlinear response of the system to large gusts, and predict the growth and saturation of aeroelastic pitch instabilities. We also show that the manifold topology in these maps can be used to make informed modifications to the system parameters in order to control the response to gusts.

Geometrical aspects and quantum brachistochrone problem for a collection of N spin-s system with long-range Ising-type interaction

We consider a physical system of N interacting qudits consisting of N spin-s particles coupled via the long-range interaction of Ising-type. We investigate the corresponding dynamics, define the associated quantum state manifold and we give the related Fubini-Study metric. We derive the Gaussian curvature and using the Gauss-Bonnet theorem, we show that the dynamics happen on a two-parametric manifold of spherical topology. We examine the geometrical phase acquired by the system under arbitrary and cyclic evolutions. Further, we study the quantum brachistochrone problem concerning the determination of the smallest possible time to realize a time-optimal evolution. By restricting our study to a two-qubit system under the Ising interaction, a detailed analysis is performed for the Fubini-Study metric, the Gaussian curvature, the geometrical phase and the optimal time in relation with the entanglement of the two qubits.

A fault diagnosis model based on singular value manifold features, optimized SVMs and multi-sensor information fusion

To achieve better fault diagnosis of rotating machinery, this paper presents a novel intelligent fault diagnosis model based on singular value manifold features (SVMF), optimized support vector machines (SVMs) and multi-sensor information fusion. Firstly, a new fault feature named SVMF is developed to better represent faults. SVMF is acquired by extracting manifold topology features of the singular spectrum. Compared with frequently-used fault features, the feature scale of SVMF is constant for variable rotating speed, and the extraction process of SVMF also has the effect of self-weighting. So SVMF has a better representation of faults. Then, to select optimal parameters for model training of SVMs, an improved fruit fly algorithm is proposed by introducing a guidance search mechanism and enhanced local search operation, and as a result both the convergence speed and accuracy are improved. Finally, the Dempster–Shafer evidence theory is introduced to fuse decision-level information from SVM models of multiple sensors. Information fusion eliminates the conflict of conclusions on fault diagnosis from multiple sensors, which leads to high robustness and accuracy of the fault diagnosis model. As a summary, the proposed method combines the advantages of SVMF in fault representation, SVMs in fault identification and the Dempster–Shafer evidence theory in information fusion, and as a result the proposed method will perform better at fault diagnosis. The proposed intelligent fault diagnosis model is subsequently applied to fault diagnosis of the gearbox. Experimental results show that the proposed diagnostic framework is versatile at detecting faults accurately.

Embeddability in R 3 is NP-hard

We prove that the problem of deciding whether a two- or three-dimensional simplicial complex embeds into R 3 is NP -hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an S 3 filling is NP -hard. The former stands in contrast with the lower-dimensional cases, which can be solved in linear time, and the latter with a variety of computational problems in 3-manifold topology, for example, unknot or 3-sphere recognition, which are in NP ∩ co- NP. (Membership of the latter problem in co-NP assumes the Generalized Riemann Hypotheses.) Our reduction encodes a satisfiability instance into the embeddability problem of a 3-manifold with boundary tori, and relies extensively on techniques from low-dimensional topology, most importantly Dehn fillings of manifolds with boundary tori.

Dynamical trapping in the area-preserving Hénon map

Stickiness is a well known phenomenon in which chaotic orbits expend an expressive amount of time in specific regions of the chaotic sea. This phenomenon becomes important when dealing with area-preserving open systems because, in this case, it leads to a temporary trapping of orbits in certain regions of phase space. In this work, we propose that the different scenarios of dynamical trapping can be explained by analyzing the crossings between invariant manifolds. In order to corroborate this assertion, we use an adaptive refinement procedure to approximately obtain the sets of homoclinic and heteroclinic intersections for the area-preserving Hénon map, an archetype of open systems, for a generic parameter interval. We show that these sets have very different statistical properties when the system is highly influenced by dynamical trapping, whereas they present similar properties when stickiness is almost absent. We explain these different scenarios by taking into consideration various effects that occur simultaneously in the system, all of which are connected with the topology of the invariant manifolds.

Topological manifold bundles and the 𝐴-theory assembly map

We give a new proof of an index theorem for fiber bundles of compact topological manifolds due to Dwyer, Weiss, and Williams, which asserts that the parametrized A A -theory characteristic of such a fiber bundle factors canonically through the assembly map of A A -theory. Furthermore our main result shows a refinement of this statement by providing such a factorization for an extended A A -theory characteristic, defined on the parametrized topological cobordism category. The proof uses a convenient framework for bivariant theories and recent results of Gomez-Lopez and Kupers on the homotopy type of the topological cobordism category. We conjecture that this lift of the extended A A -theory characteristic becomes highly connected as the manifold dimension increases.

On the disposition of cubic and pair of conics in a real projective plane

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

Globally subanalytic CMC surfaces in ℝ3 with singularities

AbstractIn this paper we present a classification of a class of globally subanalytic CMC surfaces in ℝ3 that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic CMC surface in ℝ3 with isolated singularities and a suitable condition of local connectedness is a plane or a finite union of round spheres and right circular cylinders touching at the singularities. As a consequence, we obtain that a globally subanalytic CMC surface in ℝ3 that is a topological manifold does not have isolated singularities. It is also proved that a connected closed globally subanalytic CMC surface in ℝ3 with isolated singularities which is locally Lipschitz normally embedded needs to be a plane or a round sphere or a right circular cylinder. A result in the case of non-isolated singularities is also presented. It also presented some results on regularity of semialgebraic sets and, in particular, it proved a real version of Mumford's Theorem on regularity of normal complex analytic surfaces and a result about C1 regularity of minimal varieties.

A Remarkable Property of Concircular Vector Fields on a Riemannian Manifold

In this paper, we show that, given a non-trivial concircular vector field u on a Riemannian manifold ( M , g ) with potential function f, there exists a unique smooth function ρ on M that connects u to the gradient of potential function ∇ f . We call the connecting function of the concircular vector field u. This connecting function is shown to be a main ingredient in obtaining characterizations of n-sphere S n ( c ) and the Euclidean space E n . We also show that the connecting function influences on a topology of the Riemannian manifold.

Бифуркации интегрируемых механических систем с магнитным полем на поверхностях вращения

На поверхности, гомеоморфной 2-мерной сфере, изучается натуральная механическая система с магнитным полем, инвариантная относительно $S^1$-действия. Для особых точек ранга 0 отображения момента получен критерий невырожденности, определен тип невырожденных особых точек (центр-центр и фокус-фокус), описаны бифуркации типичных вырожденных особых точек (интегрируемая гамильтонова бифуркация Хопфа двух типов). Для семейств особых окружностей ранга 1 отображения момента (состоящих из относительных положений равновесия системы) получено их параметрическое задание, доказан критерий невырожденности, определен тип невырожденных (эллиптические и гиперболические) и типичных вырожденных (параболические) особых окружностей. Получено параметрическое задание бифуркационной диаграммы отображения момента. Описаны геометрические свойства бифуркационной диаграммы и бифуркационного комплекса в случае, когда задающие систему функции находятся в общем положении. Определена топология неособых изоэнергетических 3-мерных многообразий, описана топология слоения Лиувилля на них с точностью до грубой лиувиллевой эквивалентности (в терминах атомов и молекул Фоменко). Описаны “расщепляющиеся” гиперболические особенности ранга 1, являющиеся топологически неустойчивыми бифуркациями слоения Лиувилля.

On Sectional Curvature of Metric Connection with Vectorial Torsion

Papers of many mathematicians are devoted to the study of semisymmetric connections or metric connections with vector torsion on Riemannian manifolds. This type of connectivity is one of the three main types discovered by E. Cartan and finds its application in modern physics, geometry, and topology of manifolds. Geodesic lines and the curvature tensor of a given connection were studied by I. Agricola, K. Yano, and other mathematicians. In particular, K. Yano proved an important theorem on the connection of conformal deformations and metric connections with vector torsion. Namely: a Riemannian manifold admits a metric connection with vector torsion and the curvature tensor being equal to zero if and only if it is conformally flat. Although the curvature tensor of a hemisymmetric connection has a smaller number of symmetries compared to the Levi-Civita connection, it is still possible to define the concept of sectional curvature in this case. The question naturally arises about the difference between the sectional curvature of a semisymmetric connection and the sectional curvature of a Levi-Civita connection.This paper is devoted to the study of this issue, and the authors find the necessary and sufficient conditions for the sectional curvature of the semisymmetric connection to coincide with the sectional curvature of the Levi-Civita connection. Non-trivial examples of hemisymmetric connections are constructed when possible.

The topology of moment-angle manifolds—on a conjecture of S. Gitler and S. López de Medrano

In this paper, we study the topology of moment-angle manifolds and prove a conjecture of S. Gitler and S. Lopez de Medrano concerned with the behavior of the moment-angle manifold under the surgery ‘cutting off a vertex’ on a simple polytope. Let P be a simple polytope of dimension n with m facets and Pv be a polytope obtained from P by cutting off one vertex v. Let Z = Z(P) and Zv = Z(Pv) be the corresponding moment-angle manifolds. S. Gitler and S. Lopez de Medrano conjectured that: Zv is diffeomorphic to $$\partial \left[ {\left( {Z - {D^{n + m}}} \right) \times {D^2}} \right]\# \,\# _{j = 1}^{m - n}\left( {_j^{m - n}} \right)\left( {{S^{j + 2}} \times {S^{m + n - j - 1}}} \right)$$ , and they proved the conjecture in the case m < 3n. In this paper we prove the conjecture in the general case.

Topological manifold‐based monitoring method for train‐centric virtual coupling control systems

Train-centric virtual coupling is one of the promising techniques available to increase the capacity of the railway, and is much more complex than the existing train control systems. The safety issue for the system needs to be addressed since virtually coupled trains are allowed to run closer than under the traditional fixed/moving block principle. In this study, the authors proposed a topological manifold-based monitoring method to guarantee the safety of the train-centric virtual coupling systems. The basic signalling equipment and dynamic train behaviour are described as topological elements and manifolds. Based on that, several safety monitoring theorems are proved to ensure the safety of the virtual coupling control logic. The theorem for the verification of trains running information is proved, along with theorems for the correctness of the virtual coupling control strategy. The trains' dynamic behaviour under specific control commands can also be supervised by a safety theorem. Finally, a case study is performed to illustrate the proposed method. The results show that the method is feasible for monitoring the train-centric virtual coupling control systems.

On H\xf6lder regularity of the singular set of energy minimizing harmonic maps into closed manifolds

Energy minimizing harmonic maps between manifolds are known to be smooth outside a rectifiable set of codimension 3, called the singular set. The possibility that this set is not a manifold, but has arbitrarily many small gaps in it, is not excluded in general. Here we prove that some part of the singular set—characterized by topological and analytic properties of tangent maps—is a topological manifold. In the special case of maps into the sphere {mathbb S}^2, we conclude that the whole top-dimensional part of the singular set is a manifold—this generalizes a similar result in four-dimensional domains, due to Hardt and Lin.

A cryptographic application of the Thurston norm

We discuss some applications of 3-manifold topology to cryptography. In particular, we propose a public-key and a symmetric-key cryptographic scheme based on the Thurston norm on the first cohomology of hyperbolic manifolds.

Toward and after virtual specialization in $3$-manifold topology

Toward and after virtual specialization in $3$-manifold topology