Construction Of Manifolds Research Articles

Connected sum construction of constant Q-curvature manifolds in higher dimensions

For a compact Riemannian manifold (M,g2) with constant Q-curvature of dimension n≥6 satisfying nondegeneracy condition, we show that one can construct many examples of constant Q-curvature manifolds by gluing construction. We provide a general procedure of gluing together (M,g2) with any compact manifold (N,g1) satisfying a geometric assumption. In particular, we can prove that there exists a metric with constant Q-curvature on the connected sum N#M.

Quanta of geometry: noncommutative aspects.

In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres, which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin manifolds with large quantized volume are then obtained as solutions. The two algebras M_{2}(H) and M_{4}(C) are obtained, which are the exact constituents of the standard model. Using the two maps from M_{4} to S^{4} the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. We give several physical applications of this scheme such as quantization of the cosmological constant, mimetic dark matter, and area quantization of black holes.

Geometry and the quantum: basics

Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M_2(H) and M_4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.

Construction of Complex Contact Manifolds via Reduction

Kobayashi [13] introduced complex contact manifolds as a variant of real contact manifolds. Later, Ishihara and Konishi [11] defined normality of complex contact manifolds as for Sasakian manifolds in real contact geometry. In this paper, we construct normal complex contact manifolds via reduction from hyperk\{a}hler manifolds, and give a new example of normal complex contact manifolds. To check the normality for the new examples, we give a useful identity about sectional curvatures of normal complex contact manifolds. We also give an explicit example of a non-normal complex almost contact metric structure on $S^{4m+3} \times S^{4n+3}$.

The Calabi construction for compact flat orbifolds

In this paper we extend and generalize the Calabi construction for flat manifolds to flat orbifolds. Let Γ be an n-dimensional crystallographic group (n-space group) that is Z-reducible. Then the flat orbifold En/Γ has a nontrivial fibered orbifold structure. We prove that this structure can be described by a Calabi construction, that is, En/Γ is represented as the quotient of the Cartesian product of two flat orbifolds under the diagonal action of a structure group of isometries. We determine the structure group and prove that it is finite if and only if the fibered orbifold structure has an orthogonally dual fibered orbifold structure.A fibered orbifold structure on En/Γ corresponds to a space group extension 1→N→Γ→Γ/N→1. We give a condition for the splitting of a space group extension in terms of the structure group action that is strong enough to detect the splitting of all the space group extensions corresponding to the standard Seifert fibrations of a compact, connected, flat 3-orbifold.

Center Manifold Method in the Asymptotic Integration Problem for Functional Differential Equations with Oscillatory Decreasing Coefficients. I

In this paper, we study the asymptotic integration problem in the neighborhood of infinity for a certain class of linear functional differential systems. We construct the asymptotics for solutions of the considered systems in the critical case. Using the ideas of the center manifold method, we show the existence of the so-called critical manifold that is positively invariant for trajectories of the initial system. We establish that the asymptotics for solutions of the system on this manifold defines the asymptotics for all solutions of the initial system. In the first part of this work, we propose an algorithm for an approximate construction of the critical manifold. Moreover, we establish the unique solvability for auxiliary algebraic problems that occur within the algorithm implementation.

Example-Based Human Motion Extrapolation and Motion Repairing Using Contour Manifold

We propose a human motion extrapolation algorithm that synthesizes new motions of a human object in a still image from a given reference motion sequence. The algorithm is implemented in two major steps: contour manifold construction and object motion synthesis. Contour manifold construction searches for low-dimensional manifolds that represent the temporal-domain deformation of the reference motion sequence. Since the derived manifolds capture the motion information of the reference sequence, the representation is more robust to variations in shape and size. With this compact representation, we can easily modify and manipulate human motions through interpolation or extrapolation in the contour manifold space. In the object motion synthesis step, the proposed algorithm generates a sequence of new shapes of the input human object in the contour manifold space and then renders the textures of those shapes to synthesize a new motion sequence. We demonstrate the efficacy of the algorithm on different types of practical applications, namely, motion extrapolation and motion repair.

Design of exhaust manifold for pulse converters considering fatigue strength due to vibration

The design of the exhaust manifold for the pulse converters of a 4 strokes high power medium-speed diesel engine is presented in terms of fatigue analysis. The said system undergoes thermal expansion due to high temperature of exhaust gas and is exposed to intrinsic vibration of the internal combustion engine. Moreover, the exhaust pulse generates pressure pulsating along the runner inside manifold. Under such circumstances, the design and construction of exhaust manifold must be carried out in a way to prevent early failure due to fracture. To validate the design concept, a test rig was developed to simulate the combination of thermal and vibrational movements, simultaneously. Experimental results showed that a certain sense of reliability can be achieved by considering a field factor obtained from the results of engine bench tests.

On stable manifolds for planar fractional differential equations

In this paper, we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations. The construction of this stable manifold is based on the associated Lyapunov–Perron operator. An example is provided to illustrate the result.

The Generalized Frankel Conjecture in Sasaki Geometry

We prove some structure results for \emph{transverse reducible} Sasaki manifolds. In particular, we show Sasaki manifolds with positive Ricci curvature is transversely irreducible, and so there is no join (product) construction for irregular Sasaki-Einstein manifolds, as opposed to the quasi-regular case done by Wang-Ziller and Boyer-Galicki. As an application, we classify compact Sasaki manifolds with non-negative transverse bisectional curvature, which can be viewed as the generalized Frankel conjecture (N. Mok's theorem) in Sasaki geometry.

Homology equivalences of manifolds and zero-in-the-spectrum examples

Working with group homomorphisms, a construction of manifolds is introduced to preserve homology groups. The construction gives as special cases Qullien's plus construction with handles obtained by Hausmann, the existence of one-sided $h$-cobordism of Guilbault and Tinsley, the existence of homology spheres and higher-dimensional knots proved by Karvaire. We also use it to get counter-examples to the zero-in-the-spectrum conjecture found by Farber-Weinberger, and by Higson-Roe-Schick.

Numerical construction of balanced state manifold for single-support legged mechanism in sagittal plane

Legged mechanisms can walk and run but sometimes encounter a risk of falling. In this article, a general numerical framework of balance criteria for a single-support legged mechanism is proposed and demonstrated. Explicit forms of necessary and sufficient conditions for balancing are identified and the balanced state manifold is constructed accordingly in the extended phase space of joint angle, joint velocity, and actuation limit. Within the iteration loops for partitioned joint angle and actuation limit, a nonlinear constrained optimization problem is formulated where the dynamic models of the legged mechanism are incorporated. The necessary conditions for balancing, such as the Zero-Moment Point, positive normal reaction, friction, and the ability to end up at a final static equilibrium, are implemented along with the system parameters for generality. The sequential quadratic programming method numerically solves for the velocity extrema within the complete feasible domain to construct the balanced state manifold as a viability kernel, which is a reachable superset of all possible controller-based domains. The balanced state manifold, along with its demonstration using the proposed optimal balancing motion for minimum energy and biped walking motions in sagittal plane, shows valid features that are physically consistent. The framework can be extended to systems in three-dimension with higher complexity, both in single and double support phases, for the development and stability analysis of walking robots and humans.

Conditions for Factorization of Linear Differential-Difference Equations

Abstract The paper deals with a linear system of differential equations of the form with constant coefficients, a small parameter and complex deviating argument. Sufficient conditions for factorizing of this system are presented. These conditions are obtained by construction of an integral manifold of solutions to the considered system.

ABSOLUTE CONTINUITY THEOREM FOR RANDOM DYNAMICAL SYSTEMS ON Rd

In this paper we provide a proof of the so-called absolute continuity theorem for random dynamical systems on Rd which have an invariant probability measure. First we present the construction of local stable manifolds in this case. Then the absolute continuity theorem basically states that for any two transversal manifolds to the family of local stable manifolds, the induced Lebesgue measures on these transversal manifolds are absolutely continuous under the map that transports every point on the first manifold along the local stable manifold to the second manifold, the so-called Poincaré map or holonomy map. In contrast to known results, we have to deal with the non-compactness of the state space and the randomness of the random dynamical system.

Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings

We study the topology of Hamiltonian-minimal Lagrangian submanifolds N in C^m constructed from intersections of real quadrics in a work of the first author. This construction is linked via an embedding criterion to the well-known Delzant construction of Hamiltonian toric manifolds. We establish the following topological properties of N: every N embeds as a submanifold in the corresponding moment-angle manifold Z, and every N is the total space of two different fibrations, one over the torus T^{m-n} with fibre a real moment-angle manifold R, and another over a quotient of R by a finite group with fibre a torus. These properties are used to produce new examples of Hamiltonian-minimal Lagrangian submanifolds with quite complicated topology.

Singularity Theory and Integrable Systems

The workshop brought together three very different areas of mathematics, namely singularity theory, integrable systems and quantum cohomology. They are linked by their applications in topological quantum field theory and by constructions of (often isomorphic) Frobenius manifolds. The first and second are related by a version of mirror symmetry, the link of the second and third has attracted much attention. The connection of the first and third is the least developed and was at the focus of the workshop.

Invariant manifolds for parabolic equations under perturbation of the domain

We study the effect of domain perturbation on invariant manifolds for semilinear parabolic equations subject to the Dirichlet boundary condition. Under the Mosco convergence assumption on the domains, we prove the upper and lower semicontinuity of both the local unstable invariant manifold and the local stable invariant manifold near a hyperbolic equilibrium. The continuity results are obtained by keeping track of the construction of invariant manifolds in [P.W. Bates, C.K.R.T. Jones, Invariant manifolds for semilinear partial differential equations, in: Dynamics Reported, Vol. 2, in: Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 1–38].

A quotient method for designing nonlinear controllers

An algorithmic method is proposed to design stabilizing control laws for a class of nonlinear systems that comprises single-input feedback-linearizable systems and a particular set of single-input non-feedback-linearizable systems. The method proceeds iteratively and consists of two stages: (i) the forward stage converts the system into a sequence of cascade forms while reducing the dimension at every step by creating quotient manifolds and (ii) the backward stage constructs the feedback law iteratively. The paper shows the construction of these quotient manifolds and provides algorithms for both stages. A strong feature of the approach is the possibility to use residual degrees of freedom to overcome singularities. A Lyapunov function is introduced to prove global asymptotic stability. Furthermore, the relation of the quotient method to backstepping is discussed. The approach is illustrated via the simulation of a field-controlled DC motor.

SLOW INTEGRAL MANIFOLDS WITH A CHANGE OF STABILITY

The paper is devoted to the investigation of slow integral manifolds with a change of stability in the case of vector fast variable. The existence conditions of the gluing function are given. The problem of construction of the integral manifold with a change of stability for systems with vector fast and slow variables is solved.

Mini-Workshop: Boundary Value Problems and Spectral Geometry

Boundary value problems and spectral geometry is an attractive and rapidly developing area in modern mathematical analysis. The interaction of PDE methods with concepts from operator theory and differential geometry is particularly challenging and leads directly to new insights and applications in various branches of pure and applied mathematics, e.g., analysis on manifolds, global analysis and mathematical physics. Some recent contributions in the field of boundary value problems and spectral geometry concern, e.g., construction of isospectral manifolds with boundary, eigenvalue and resonance distribution for large energies, multidimensional inverse spectral problems, singular perturbations, new regularity techniques, Dirichletto-Neumann maps and Titchmarsh-Weyl functions.