Structure Of Differentiable Manifold Research Articles

A DIFFERENTIABLE STRUCTURE ON A FINITE DIMENSIONAL REAL VECTOR SPACE AS A MANIFOLD

There are three conditions for a topological space to be said a topological manifold of dimension : Hausdorff space, second-countable, and the existence of homeomorphism of a neighborhood of each point to an open subset of or -dimensional locally Euclidean. The differentiable structure is given if the intersection of two charts is an empty chart or its transition map is differentiable. In this article, we study a differentiable manifold on finite dimensional real vector spaces. The aim is to prove that any finite-dimensional vector space is a differentiable manifold. First of all, it is proved that a finite dimensional vector space is a topological manifold by constructing a norm as its topology. Given a metric which is induced by a norm. Two norms on a finite dimensional vector space are always equivalent and they are determine the same topology. Secondly, it is proved that the transition map in the finite dimensional vector space is differentiable. As conclusion, we have that any finite dimensional vector space with independent norm topology choice is a differentiable manifold. As a matter of discussion, it can be studied that the vector space of all linear operators of a finite dimensional vector space has a differentiable manifold structure as well.

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.

Multibody Dynamics on Differentiable Manifolds

Abstract Topological and vector space attributes of Euclidean space are consolidated from the mathematical literature and employed to create a differentiable manifold structure for holonomic multibody kinematics and dynamics. Using vector space properties of Euclidean space and multivariable calculus, a local kinematic parameterization is presented that establishes the regular configuration space of a multibody system as a differentiable manifold. Topological properties of Euclidean space show that this manifold is naturally partitioned into disjoint, maximal, path connected, singularity free domains of kinematic and dynamic functionality. Using the manifold parameterization, the d'Alembert variational equations of multibody dynamics yield well-posed ordinary differential equations of motion on these domains, without introducing Lagrange multipliers. Solutions of the differential equations satisfy configuration, velocity, and acceleration constraint equations and the variational equations of dynamics, i.e., multibody kinematics and dynamics are embedded in these ordinary differential equations. Two examples, one planar and one spatial, are treated using the formulation presented. Solutions obtained are shown to satisfy all three forms of kinematic constraint to within specified error tolerances, using fourth-order Runge–Kutta numerical integration methods.

On the geometry of realizable Markov parameters by SIMO and MISO systems

Let p, m, n, d be positive integers and let Ln(d) denote the set of sequences L=(L1,…,Ln) of p×m real or complex matrices which are realizable by systems of minimal order d. It was shown in [5,14] that Ln(d) can be endowed with a structure of differentiable manifold when p=m=1; that is, when the sequences are realizable by Single Input/Single Output (SISO) systems. In this paper a similar result is obtained for more general sequences. Specifically, we will consider the set Ln(r_,s_) of sequences L which are realizable by systems of minimal order d and having r_ and s_ as Brunovsky indices of controllability and observability, respectively. It is shown in this paper that when one of the two collections of indices r_ or s_ is constant, then Ln(r_,s_) can be provided with a structure of differentiable manifold and a formula of its dimension is given. The special cases r_=(1,…,1) or s_=(1,…,1) correspond to sequences realizable, respectively, by Single Input/Multi Output (SIMO) or Multi Input/Single Output (MISO) systems.

Defining the space in a general spacetime

A global vector field [Formula: see text] on a “spacetime” differentiable manifold [Formula: see text], of dimension [Formula: see text], defines a congruence of world lines: the maximal integral curves of [Formula: see text], or orbits. The associated global space [Formula: see text] is the set of these orbits. A “[Formula: see text]-adapted” chart on [Formula: see text] is one for which the [Formula: see text] vector [Formula: see text] of the “spatial” coordinates remains constant on any orbit [Formula: see text]. We consider non-vanishing vector fields [Formula: see text] that have non-periodic orbits, each of which is a closed set. We prove transversality theorems relevant to such vector fields. Due to these results, it can be considered plausible that, for such a vector field, there exists in the neighborhood of any point [Formula: see text] a chart [Formula: see text] that is [Formula: see text]-adapted and “nice”, i.e. such that the mapping [Formula: see text] is injective — unless [Formula: see text] has some “pathological” character. This leads us to define a notion of “normal” vector field. For any such vector field, the mappings [Formula: see text] build an atlas of charts, thus providing [Formula: see text] with a canonical structure of differentiable manifold (when the topology defined on [Formula: see text] is Hausdorff, for which we give a sufficient condition met in important physical situations). Previously, a local space manifold [Formula: see text] had been associated with any “reference frame” [Formula: see text], defined as an equivalence class of charts. We show that, if [Formula: see text] is made of nice [Formula: see text]-adapted charts, [Formula: see text] is naturally identified with an open subset of the global space manifold [Formula: see text].

Superrigidity of Hyperbolic Buildings

In this paper, we study the behavior of harmonic maps into complexes with branching differentiable manifold structure. The main examples of such target spaces are Euclidean and hyperbolic buildings. We show that a harmonic map from an irreducible symmetric space of noncompact type other than real or complex hyperbolic into these complexes are non-branching. As an application, we prove rank-one and higher-rank superrigidity for the isometry groups of a class of complexes which includes hyperbolic buildings as a special case.

The differentiable manifold model of quantum-chemical reaction networks

The methods of general and differential topology (topology ≈ “rubber geometry”) are particularly suitable for the quantum-mechanical description of nonrigid molecular systems. The concept of nuclear configuration space 3NR is replaced by a metric space M and subsequently by a topological space (M, TC) where points of nuclear geometries, as fundamental entities, are replaced by open sets. These open sets provide a quantum-mechanical description of molecular structures, reaction mechanisms, and reaction networks. Exploiting the special properties of the metric of M, the topological space (M, TC) can be provided with a differentiable manifold structure. This differentiable manifold is proposed as a common basis for both local and global analysis of reacting molecular systems.

Absolute space-time measurements and gravitation

A method of direct («absolute») space-time measurements by means of free trajectories is proposed, and its application to general relativity is discussed. Only the existence and order of the intersections of trajectories are used. Top speed is defined independently at all space-time points and for all pairs of bodies. The ratios of (infinite) oscillation numbers of photons (top-speed particles) provide all the means for geometry: topology, differentiable manifold structure etc. Metric is replaced with these ratios in that a correspondence of affine parameters on different null geodesics can be obtained. So, one is able to define an absolute integration and to construct the Green’s functional demanding the field-determining path integral to be finite and expressed solely via oscillation numbers. In particular, the simple geometric structure and immediate physical meaning of the so-called «tail term» become evident in connection with Huygens’ principle.

Nonlinear coordinate representations of smooth optimization problems

Nonlinear coordinate representations of smooth optimization problems are investigated from the point of view of variable metric algorithms. In other words, nonlinear coordinate systems, in the sense of differential geometry, are studied by taking into consideration the structure of smooth optimization problems and variable metric methods. Both the unconstrained and constrained cases are discussed. The present approach is based on the fact that the nonlinear coordinate transformation of an optimization problem can be replaced by a suitable Riemannian metric belonging to the Euclidean metric class. In the case of equality and inequality constraints, these questions are related closely to the right inverses of full-rank matrices; therefore, their characterization is a starting point of the present analysis. The main results concern a new subclass of nonlinear transformations in connection with the common supply of coordinates to two Riemannian manifolds, one immersed in the other one. This situation corresponds to the differentiable manifold structure of nonlinear optimization problems and improves the insight into the theoretical background of variable metric algorithms. For a wide class of variable metric methods, a convergence theorem in invariant form (not depending on coordinate representations) is proved. Finally, a problem of convexification by nonlinear coordinate transformations and image representations is studied.

Geodesics and near-geodesics in the manifolds of projector frames

The set of projector frames (i.e. finite sequences of commuting projectors which sum to the identity) can be given a differentiable manifold structure with an affine connection. We compare the resulting geodesic arcs with other naturally arising paths and find them cubically close. We also discuss Riemannian geodesics in an appropriate Hilbert-space context.

Differentiable structures on function spaces

A C s {C^s} differentiable manifold structure is constructed for spaces of maps from a compact C r {C^r} manifold M M to a C r + s {C^{r + s}} manifold N N . The method (1) is inspired by Douady; (2) does not require any additional structure on N N (such as sprays); (3) includes the case when N N is an analytic manifold and concludes that the mapping space is also an analytic manifold; (4) can be used to treat all the classical mapping spaces ( C r {C^r} functions, C r {C^r} functions with Hölder conditions, and Sobolev functions). Several interesting aspects of these manifolds are investigated such as their tangent spaces, their behavior with respect to functions, and realizations of Lie group structures on them. Differentiable structures are also exhibited for spaces of compact maps with noncompact domain.