Differentiable Manifold Research Articles

Differential Geometry Approach to Continuous Model of Micro-Structural Defects in Finite Elasto-Plasticity

This paper concerns finite elasto-plasticity of crystalline materials with micro-structural defects. We revisit the basic concepts: plastic distortion and decomposition of the plastic connection. The body is endowed with a structure of differential manifold. The plastic distortion is an incompatible diffeomorphism. The metric induced by the plastic distortion on the intermediate configuration (considered to be a differential manifold) is a key point in the theory, in defining the defects related to point defects, or extra-matter. The so-called plastic connection is metric, with plastic metric tensor expressed in terms of the plastic distortion and its adjoint. We prove an appropriate decomposition of the plastic connection, without any supposition concerning the non-metricity of plastic connection. All types of the lattice defects, dislocations, disclinations, and point defects are described in terms of the densities related to the elements that characterize the decomposition theorem for plastic connection. As a novelty, the measure of the interplay of the possible lattice defects is introduced via the Cartan torsion tensor. To justify the given definitions, the proposed measures of defects are compared to their counterparts corresponding to a classical framework of continuum mechanics. Thus, their physical meanings can be emphasized at once.

A Unified Dissertation on Bearing Rigidity Theory

This work focuses on bearing rigidity theory, namely the branch of knowledge investigating the structural properties necessary for multi-element systems to preserve the inter-unit bearings under deformations. The contributions of this work are twofold. The first one consists in the development of a general framework for the statement of the principal definitions and properties of bearing rigidity. We show that this approach encompasses results existing in the literature, and also provides a systematic approach for studying bearing rigidity on any differential manifold in SE(3)^n, where n is the number of agents. The second contribution is the derivation of a general form of the rigidity matrix, a central construct in the study of rigidity theory. We provide a necessary and sufficient condition for the infinitesimal rigidity of a bearing framework as a property of the rank of the rigidity matrix. Finally, we present two examples of multi-agent systems not encountered in the literature and we study their rigidity properties using the developed methods.

Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits

The purpose of this study is to establish the topological properties of three-dimensional manifolds which admit Morse – Smale flows without fixed points (non-singular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be investigated additionally and refined in the case of a small number of orbits. For example, in the case of a flow with two non-twisted (having a tubular neighborhood homeomorphic to a solid torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an NMS-flow with two orbits is a lens space. Previously, it was believed that all prime manifolds admitting NMS-flows with at most three non-twisted orbits have the same topology. Methods. In this paper, we consider suspensions over Morse – Smale diffeomorphisms with three periodic orbits. These suspensions, in turn, are NMS-flows with three periodic trajectories. Universal coverings of the ambient manifolds of these flows and lens spaces are considered. Results. In this paper, we present a countable set of pairwise distinct simple 3-manifolds admitting NMS-flows with exactly three non-twisted orbits. Conclusion. From the results of this paper it follows that there is a countable set of pairwise distinct three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an NMS-flow with at most three orbits is lens space.

Manipulator Kinematics and Dynamics on Differentiable Manifolds: Part I Kinematics

Abstract Using basic tools of Euclidian space differential geometry, maximal singularity free components of the regular manipulator configuration space are defined, with conditions that establish the space as a differentiable manifold. This structure shows that the conventional categorization of manipulators as either serial or parallel is incomplete and that three distinct categories of manipulator must be accounted for; (1) serial manipulators in which inputs globally determine outputs, (2) explicit parallel manipulators in which outputs globally determine inputs, and (3) compound manipulators in which there is no global input or output mapping. Results of differential geometry are used to show that configuration space differentiable manifolds in each category are partitioned into maximal, disjoint, path-connected components in which the manipulator is singularity free and may be effectively controlled. This extends local analytical properties of manipulators that are used for analysis and control to global validity on maximal components of regular manipulator configuration space, providing explicit criteria for avoidance of singular behavior. Model manipulators in each of the three categories are analyzed to illustrate application of the differentiable manifold structure, using only multivariable calculus and linear algebra. Computational methods for forward and inverse kinematics and construction of ordinary differential equations of manipulator dynamics on differentiable manifolds are presented in Part II of this paper, in support of manipulator control.

Manipulator Kinematics and Dynamics on Differentiable Manifolds: Part II Dynamics

Abstract The manipulator differentiable manifold kinematics formulation presented in Part I of this paper is employed for forward and inverse kinematic analysis and formulation of ordinary differential equations of motion on disjoint, maximal, singularity free, path-connected components of regular configuration space, for a broad spectrum of manipulators. Existence of forward and inverse configuration mappings throughout maximal singularity free manifold components yields analytical forward and inverse velocity mappings. Efficient computational algorithms for forward and inverse configuration and velocity analysis on a time grid are derived for each of three manipulator categories. Manifold parameterizations are used to transform variational equations of motion in manipulator coordinates to second-order ordinary differential equations of manipulator dynamics, with both input and output coordinates as state variables, eliminating the need for ad hoc derivation of equations of motion. Criteria that define manipulator differentiable manifolds are shown to guarantee that the equations of motion derived are well-posed on maximal singularity free components of manipulator configuration space. This process is illustrated by presenting terms required for the evaluation of equations of motion for three model manipulators. It is shown that computation involved in the evaluation of equations of manipulator kinematics and dynamics can be carried out in real-time on modern microprocessors, supporting the in-line implementation of modern methods of manipulator control.

Discrete gravity

We assume that the points in volumes smaller than an elementary volume (which may have a Planck size) are indistinguishable in any physical experiment. This naturally leads to a picture of a discrete space with a finite number of degrees of freedom per elementary volume. In such discrete spaces, each elementary cell is completely characterized by displacement operators connecting a cell to the neighboring cells and by the spin connection. We define the torsion and curvature of the discrete spaces and show that in the limiting case of vanishing elementary volume the standard results for the continuous curved differentiable manifolds are completely reproduced.

Numerical strategies for variational updates in large strain inelasticity with incompressibility constraint

AbstractIn finite deformation inelasticity, one often has to deal with the incompressibility constraint. In the past, this was dealt with using, for example, an exponential mapping approach, which yields exact volume preservation in plastic deformations. In this article however, the special‐linear update approach by Hurtado et al. The special‐linear update: An application of differential manifold theory to the update of isochoric plasticity flow rules, Int J Numer Methods Eng. 2014;97(4):298‐312, which utilizes a projection method to fulfill the incompressibility constraint is used. The model is applied to isotropic plasticity by a novel approximation of the logarithm and treats kinematic hardening without losing the symmetry of the internal variables. The model results are compared to models utilizing an exponential mapping approach in numerical experiments.

Leibniz cohomology and connections on differentiable manifolds

We show how an affine connection on a Riemannian manifold occurs naturally as a cochain in the complex for Leibniz cohomology of vector fields with coefficients in the adjoint representation. The Leibniz coboundary of the Levi-Civita connection can be expressed as a sum of two terms, one the Laplace-Beltrami operator and the other a Ricci curvature term. The vanishing of this coboundary has an interpretation in terms of eigenfunctions of the Laplacian. Separately, we compute the Leibniz cohomology with adjoint coefficients for a certain family of vector fields on Euclidean Rn corresponding to the affine orthogonal Lie algebra, n≥3.

Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold Ω p + q in the hyperbolic space ℍ m − 1 satisfy various extrinsic restrictions, then Ω p + q has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group π 1 Ω p + q is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.

Estimates for Local Approximations of Functions on Differential Manifold

Estimates for Local Approximations of Functions on Differential Manifold

Solutions of the KdV Equation through Analysis of Regular Symmetries

We investigate a case of the generalized Korteweg – De Vries Burgers equation. Our aim is to demonstrate the need for the application of further methods in addition to using Lie Symmetries. The solution is found through differential topological manifolds. We apply Lie’s theory to take the PDE to an ODE. However, this ODE is of third order and not easily solvable. It is through differentiable topological manifolds that we are able to arrive at a solution

Null Homology Groups and Stable Currents in Warped Product Submanifolds of Euclidean Spaces

In this paper, we prove that, for compact warped product submanifolds Mn in an Euclidean space En+k, there are no stable p-currents, homology groups are vanishing, and M3 is homotopic to the Euclidean sphere S3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature, and the Hessian tensor. The results in this paper can be considered an extension of Xin’s work in the framework of a compact warped product submanifold, when the base manifold is minimal in ambient manifolds.

Certain results on invariant submanifolds of an almost Kenmotsu (kappa ,mu ,nu )-space

In the present paper, we study invariant submanifolds of almost Kenmotsu structures whose Riemannian curvature tensor has (kappa ,mu ,nu )-nullity distribution. Since the geometry of an invariant submanifold inherits almost all properties of the ambient manifold, we research how the functions kappa ,mu and nu behave on the submanifold. In this connection, necessary and sufficient conditions are investigated for an invariant submanifold of an almost Kenmotsu (kappa ,mu ,nu )-space to be totally geodesic under some conditions.

The existence of homologically fibered links and solutions of some equations

There is one generalization of fibered links in 3-manifolds, called homologically fibered links. It is known that the existence of a homologically fibered link whose fiber surface has a given homeomorphic type is determined by the first homology group and its torsion linking form of the ambient 3-manifold. In this paper, we interpret the existence of homologically fibered links with that of a solution of some equation, in terms of the first homology group and its torsion linking form or a surgery diagram of the ambient manifold. As an application, we compute the invariant hc(⋅), defined through homologically fibered knots, for 3-manifolds whose torsion linking forms represent a generator of linkings.

Convergence of Spectral Triples on Fuzzy Tori to Spectral Triples on Quantum Tori

Fuzzy tori are finite dimensional C*-algebras endowed with an appropriate notion of noncommutative geometry inherited from an ergodic action of a finite closed subgroup of the torus, which are meant as finite dimensional approximations of tori and more generally, quantum tori. A mean to specify the geometry of a noncommutative space is by constructing over it a spectral triple. We prove in this paper that we can construct spectral triples on fuzzy tori which, as the dimension grow to infinity and under other natural conditions, converge to a natural spectral triple on quantum tori, in the sense of the spectral propinquity. This provides a formal assertion that indeed, fuzzy tori approximate quantum tori, not only as quantum metric spaces, but as noncommutative differentiable manifolds -- including convergence of the state spaces as metric spaces and of the quantum dynamics generated by the Dirac operators of the spectral triples, in an appropriate sense.

On the entropy of parabolic Allen–Cahn equation

We define a local (mean curvature flow) entropy for Radon measures in \mathbb{R}^n or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen–Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula for the parabolic Allen–Cahn equation. As an application, we show that when the entropy of the initial data is small enough (less than twice of the energy of the one-dimensional standing wave), the limit measure of the parabolic Allen–Cahn equation has unit density for all future time.

Formula omitted]-structures on manifolds

In this paper, we define and study two new structures on a differentiable manifold called by us an f(a,b)(3,2,1)-structure and a framed f(a,b)(3,2,1)-structure as a generalization of some geometric structures determined by polynomial structures, where a,b∈R and b≠0. At beginning, we present some examples regarding f(a,b)(3,2,1)-structures and establish their some fundamental properties. We also give a necessary condition for an f(a,b)(3,2,1)-structure to be an almost quadratic ϕ-structure. Later, it is shown that the existence of two semi-Riemannian metrics on differentiable manifolds admitting a framed f(a,b)(3,2,1)-structure, i.e., framed f(a,b)(3,2,1)-manifolds. In particular, a framed f(a,b)(3,2,1)-manifold endowed with the first semi-Riemannian metric mentioned above is called a framed metric f(a,b)(3,2,1)-manifold. Finally, we construct some examples to illustrate the existence of framed metric f(a,b)(3,2,1)-manifolds.

Minimum Time Problem Controlled by Affine Connection

Geometrically, the affine connection is the main ingredient that underlies the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on a finite-dimensional differentiable manifold. In this paper, we come up with a new idea of controllability and observability of states by using auto-parallel curves, and the minimum time problem controlled by the affine connection. The main contributions refer to the following: (i) auto-parallel curves controlled by a connection, (ii) reachability and controllability on the tangent bundle of a manifold, (iii) examples of equiaffine connections, (iv) minimum time problem controlled by a connection, (v) connectivity by stochastic perturbations of auto-parallel curves, and (vi) computing the optimal time and the optimal striking time. The connections with bounded pull-backs result in bang–bang optimal controls. Some significative examples on bi-dimensional manifolds clarify the intention of our paper and suggest possible applications. At the end, an example of minimum striking time with simulation results is presented.

Existence, uniqueness and regularity of the projection onto differentiable manifolds

We investigate the maximal open domain {mathscr {E}}(M) on which the orthogonal projection map p onto a subset Msubseteq {{mathbb {R}}}^d can be defined and study essential properties of p. We prove that if M is a C^1 submanifold of {{mathbb {R}}}^d satisfying a Lipschitz condition on the tangent spaces, then {mathscr {E}}(M) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is C^2 or if the topological skeleton of M^c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a C^k-submanifold M with kge 2, the projection map is C^{k-1} on {mathscr {E}}(M), and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion Msubseteq {mathscr {E}}(M) is that M is a C^1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with Msubseteq {mathscr {E}}(M), then M must be C^1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between {mathscr {E}}(M) and the topological skeleton of M^c.

Physical measures for partially hyperbolic flows with mostly contracting centre

In this paper, we study the physical measures for a class of partially hyperbolic flows with mostly contracting centre. Let X be a vector field on a compact Riemannian manifold M with partially hyperbolic splitting . We prove that if the centre direction exhibits the asmptotically sectionally contracting behaviour with respect to Gibbs u-states, then X admits finitely many physical measures, and their basins cover Lebesgue almost all points of the ambient manifold. Moreover, when the unstable manifolds are dense, we prove that X admits only one physical measure whose basin covers a full Lebesgue measure subset. By tracing the physical measures via typical hyperbolic periodic orbits, we study the statistical stability of this kind of partially hyperbolic flows.