Differentiable Manifold Research Articles

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].

Geometry of stochastic delay differential equations with jumps in manifolds

In this article we propose a model for stochastic delay differential equation with jumps (SDDEJ) in a differentiable manifold $M$ endowed with a connection $\nabla $. In our model, the continuous part is driven by vector fields with a fixed delay and the jumps are assumed to come from a distinct source of (càdlàg) noise, without delay. The jumps occur along adopted differentiable curves with some dynamical relevance (with fictitious time) which allow to take parallel transport along them. In the last section, using a geometrical approach, we show that the horizontal lift of the solution of an SDDEJ is again a solution of an SDDEJ in the linear frame bundle $BM$ with respect to a horizontal connection $\nabla ^H$ in $BM$.

Nonlinear Alignment and Its Local Linear Iterative Solution

In manifold learning, the aim of alignment is to derive the global coordinate of manifold from the local coordinates of manifold’s patches. At present, most of manifold learning algorithms assume that the relation between the global and local coordinates is locally linear and based on this linear relation align the local coordinates of manifold’s patches into the global coordinate of manifold. There are two contributions in this paper. First, the nonlinear relation between the manifold’s global and local coordinates is deduced by making use of the differentiation of local pullback functions defined on the differential manifold. Second, the method of local linear iterative alignment is used to align the manifold’s local coordinates into the manifold’s global coordinate. The experimental results presented in this paper show that the errors of noniterative alignment are considerably large and can be reduced to almost zero within the first two iterations. The large errors of noniterative/linear alignment verify the nonlinear nature of alignment and justify the necessity of iterative alignment.

Lie Group Methods for Eigenvalue Function

By considering a C∞ structure on the ordered non-increasing of elements of Rn, we show that it is a differentiable manifold. By using of Lie groups, we show that eigenvalue function is a submersion. This fact is used to prove some results. These results is applied to prove a few facts about spectral manifolds and spectral functions. Orthogonal matrices act on the real symmetric matrices as a Lie transformation group. This fact, also, is used to prove the results.

Gravitational Space-Time Curve Generation via Accelerated Charged Particles

A force with an acceleration that is equal to multiples greater than the speed of light per unit time is exerted on a cloud of charged particles. The particles are resultantly accelerated to within an infinitesimal fraction of the speed of light. As the force or acceleration increases, the particles’ velocity asymptotically approaches but never achieves the speed of light obeying relativity. The asymptotic increase in the particles’ velocity toward the speed of light as acceleration increasingly surpasses the speed of light per unit time does not compensate for the momentum value produced on the particles at sub-light velocities. Hence, the particles’ inertial mass value must increase as acceleration increases. This increase in the particles’ inertial mass as the particles are accelerated produce a gravitational field which is believed to occur in the oscillation of quarks achieving velocities close to the speed of light. The increased inertial mass of the density of accelerated charged particles becomes the source mass (or Big “M”) in Newton’s equation for gravitational force. This implies that a space-time curve is generated by the accelerated particles. Thus, it is shown that the acceleration number (or multiple of the speed of light greater than 1 per unit of time) and the number of charged particles in the cloud density are surjectively mapped to points on a differential manifold or space-time curved surface. Two aspects of Einstein’s field equations are used to describe the correspondence between the gravitational field produced by the accelerated particles and the resultant space-time curve. The two aspects are the Schwarzchild metric and the stress energy tensor. Lastly, the possibility of producing a sufficient acceleration or electromagnetic force on the charged particles to produce a gravitational field is shown through the Lorentz force equation. Moreover, it is shown that a sufficient voltage can be generated to produce an acceleration/force on the particles that is multiples greater than the speed of light per unit time thereby generating gravity.

Induced Riemannian structures on null hypersurfaces

Given a null hypersurface $L$ of a Lorentzian manifold, we construct a Riemannian metric $\widetilde{g}$ on it from a fixed transverse vector field $\zeta$. We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold $(L,\widetilde{g})$ and the vector field $\zeta$. As an application, we prove some new results on null hypersurfaces, as well as known ones, using Riemannian techniques.

$1$-complete semiholomorphic foliations

A semiholomorphic foliations of type (n, d) is a differentiable real manifold X of dimension 2n + d, foliated by complex leaves of complex dimension n. In the present work, we introduce an appropriate notion of pseudoconvexity (and consequently, q-completeness) for such spaces, given by the interplay of the usual pseudoconvexity, along the leaves, and the positivity of the transversal bundle. For 1-complete real analytic semiholomorphic foliations, we obtain a vanishing theorem for the CR cohomology, which we use to show an extension result for CR functions on Levi flat hypersurfaces and an embedding theorem in C^N . In the compact case, we introduce a notion of weak positivity for the transversal bundle, which allows us to construct a real analytic embedding in CP^N .

Remarks on the behaviour of higher-order derivations on the gluing of differential spaces

This paper is about some geometric properties of the gluing of order k in the category of Sikorski differential spaces, where k is assumed to be an arbitrary natural number. Differential spaces are one of possible generalizations of the concept of an infinitely differentiable manifold. It is known that in many (very important) mathematical models, the manifold structure breaks down. Therefore it is important to introduce a more general concept. In this paper, in particular, the behaviour of kth order tangent spaces, their dimensions, and other geometric properties, are described in the context of the process of gluing differential spaces. At the end some examples are given. The paper is self-consistent, i.e., a short review of the differential spaces theory is presented at the beginning.

Material covariant constitutive laws for continua with internal structure

In this work, we study the transformation properties of the local form of the material (referential) balance of energy equation under the superposition of arbitrary material diffeomorphisms. For this purpose, the tensor analysis on manifolds is utilized. We show that the material balance of energy equation, in general, cannot be invariant; in fact an extra term appears in the transformed balance of energy equation, which is directly related to the work performed by the configurational stresses. By making the fundamental assumption that the body and the ambient space manifolds are always related in the course of deformation and by utilizing the metric concept, we determine this extra term. Building on this, we derive several constitutive equations for the material stress tensor. The compatibility of these constitutive equations with the second law of thermodynamics is evaluated. Finally, we postulate that the material balance of energy equation is covariant, and we study this case in detail, as well.

Geometric partial differentiability on manifolds: The tangential derivative and the chain rule

We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this notion is independent of local charts and thus well-defined for functions defined on a differentiable manifold. This notion is stronger than the classical directional derivative and equivalent to the latter for Lipschitz continuous functions. We characterize also the pairs of tangentially differentiable functions for which the chain rule holds.

Infinitesimal affine transformations of a Weil bundle of second order with complete lift connection

We obtain the canonical expansion of an arbitrary infinitesimal affine transformation of a Weil bundle of second order over a differentiable manifold with complete lift connection. We establish necessary and sufficient conditions under which a vector field is an infinitesimal affine transformation.

Extendability of parallel sections in vector bundles

I address the following question: Given a differentiable manifold M , what are the open subsets U of M such that, for all vector bundles E over M and all linear connections ∇ on E , any ∇ -parallel section in E defined on U extends to a ∇ -parallel section in E defined on M ? For simply connected manifolds M (among others) I describe the entirety of all such sets U which are, in addition, the complement of a C 1 submanifold, boundary allowed, of M . This delivers a partial positive answer to a problem posed by Antonio J. Di Scala and Gianni Manno (2014). Furthermore, in case M is an open submanifold of R n , n ≥ 2 , I prove that the complement of U in M , not required to be a submanifold now, can have arbitrarily large n -dimensional Lebesgue measure.

Geometric tools for exploring manifolds of light transport paths

Photorealistic images created using physical simulations of light have become a ubiquitous element of our everyday lives. The most successful techniques for producing such images replicate the key physical phenomena in a detailed software simulation, including the emission of light by sources, transport through space, and scattering in the atmosphere and at the surfaces of objects. Mathematically, this computation involves the approximation of many high-dimensional integrals, one for each pixel of the image, usually using Monte Carlo methods. Although a great deal of progress has been made on rendering algorithms, so that physically based rendering is now routinely used in many applications, commonly occurring situations can still cause these algorithms to become impractically slow, forcing users to make unrealistic scene modifications to obtain satisfactory results. Light transport is complex because light can flow along a great variety of different paths through a scene, though only a subset of these makes relevant contributes to the final image. The simulation becomes ineffective when it is difficult to find the important paths. Commonly occurring materials like smooth metal or glass surfaces can easily lead to such situations, where only very few lighting paths participate, leading to spiky integrands and poor convergence. How to efficiently handle such cases in general has been a long-standing problem.<!-- END_PAGE_1 --> In this paper, we provide a geometric solution to this problem by representing light paths as points in an abstract high-dimensional configuration space that is defined by a system of constraint equations. This configuration space is a differentiable manifold, which can be locally parameterized in the neighborhood of an existing path. Building on this framework, we propose Manifold Exploration, a rendering technique that efficiently explores the integration domain by taking geometrically informed steps on the manifold of light paths.

A new upper bound for the Dirac operators on hypersurfaces

We prove a new upper bound for the first eigenvalue of the Dirac operator of a compact hypersurface in any Riemannian spin manifold carrying a non-trivial twistor spinor without zeros on the hypersurface. The upper bound is expressed as the first eigenvalue of a drifting Schr\"odinger operator on the hypersurface. Moreover, using a recent approach developed by O. Hijazi and S. Montiel, we completely characterize the equality case when the ambient manifold is the standard hyperbolic space.

Transportation-cost inequalities on path spaces over manifolds carrying geometric flows

Let Lt:=Δt+Zt for a C1,1-vector field Z on a differential manifold M possibly with a boundary ∂M, where Δt is the Laplacian operator induced by a time dependent metric gt differentiable in t∈[0,Tc). In this article, by constructing suitable coupling, transportation-cost inequalities on the path space of the (reflecting if ∂M≠∅) diffusion process generated by Lt are proved to be equivalent to a new curvature lower bound condition and the convexity of the geometric flow (i.e., the boundary keeps convex). Some of them are further extended to non-convex flows by using conformal changes of the flows. As an application, these results are applied to the Ricci flow with the umbilic boundary.

Pseudo f-Manifolds with Complemented Frames

In this paper we study pseudo \({\varphi}\)-structure on a differential manifold in which is a tensor field of type (1, 1) satisfying \({\varphi^{3}\,-\,\varphi=0}\). We obtain basic curvature tensor fields for pseudo f-manifolds with complemented frame.

Coxeter transformation groups and reflection arrangements in smooth manifolds

Artin groups are a natural generalization of braid groups and are well-understood in certain cases. Artin groups are closely related to Coxeter groups. There is a faithful representation of a Coxeter group $W$ as a linear reflection group on a real vector space $V$. The group acts properly and fixes a union of hyperplanes. The $W$-action extends as the covering space action to the complexified complement of these hyperplanes. The fundamental groups of the complement and the orbit space are the pure Artin group and the Artin group respectively. For the Artin groups of finite type Deligne proved that the associated complement is aspherical. Using the Coxeter group data Salvetti gave a construction of a cell complex which is a $W$-equivariant deformation retract of the complement. This construction was independently generalized by Charney and Davis to the Artin groups of infinite type. A lot of algebraic properties of these groups were discovered using combinatorial and topological properties of this cell complex. In this paper we represent a Coxeter group as a subgroup of diffeomorphisms of a smooth manifold. These so-called Coxeter transformation groups fix a union of codimension-$1$ (reflecting) submanifolds and permute the connected components of the complement. Their action naturally extends to the tangent bundle of the ambient manifold and fixes the union of tangent bundles of these reflecting submanifolds. Fundamental group of the tangent bundle complement and that of its quotient serve as the analogue of pure Artin group and Artin group respectively. The main aim of this paper is to prove Salvetti's theorems in this context. We show that the combinatorial data of the Coxeter transformation group can be used to construct a cell complex homotopy equivalent to the tangent bundle complement and that this homotopy equivalence is equivariant.

Sunflower model: time-dependent coefficients and topology of the periodic solutions set

We investigate the structure of the set of periodic solutions of a time-dependent generalized version of the sunflower equation (in fact of the delayed Lienard equation), where the coefficients can vary periodically, thus allowing for environmental oscillations. Our result stems from a more general analysis, based on fixed point index and degree-theoretic methods, of the set of T-periodic solutions of T-periodically perturbed coupled delay differential equations on differentiable manifolds.

Geodesic Hypothesis Testing for Comparing Location Parameters in Elliptical Populations

In this paper we study the geometry of the differentiable manifold associated with two samples of symmetric distributions in the real line equipped with the Fisher information as Riemannian metric. Expressions for the entries of the information matrix are obtained under different assumptions. The geodesic or Rao distance induced by this geometry is used to construct asymptotic parametrization-invariant testing procedures for comparing location parameters. As special cases, we obtain new asymptotic tests for the two sample Behrens-Fisher and Fieller-Creasy problems. Testing equality of several location parameters is also considered. It is shown that when scale parameters are equal, the geodesic test statistic is a strictly monotone increasing function of the Wald statistic. Empirical results for the Student-t distribution provide evidence that the geodesic test statistic has good sampling properties in terms of level and power.

Jacobi stability analysis of the Lorenz system

We perform the study of the stability of the Lorenz system by using the Jacobi stability analysis, or the Kosambi–Cartan–Chern (KCC) theory. The Lorenz model plays an important role for understanding hydrodynamic instabilities and the nature of the turbulence, also representing a nontrivial testing object for studying nonlinear effects. The KCC theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach, we describe the evolution of the Lorenz system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a nonlinear connection and a Berwald type connection, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In order to apply the KCC theory, we reformulate the Lorenz system as a set of two second-order nonlinear differential equations. The geometric invariants associated to this system (nonlinear and Berwald connections), and the deviation curvature tensor, as well as its eigenvalues, are explicitly obtained. The Jacobi stability of the equilibrium points of the Lorenz system is studied, and the condition of the stability of the equilibrium points is obtained. Finally, we consider the time evolution of the components of the deviation vector near the equilibrium points.