Infinite-dimensional Manifold Research Articles

Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals

Abstract We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles.

Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge

AbstractWe consider an autonomous, indefinite Lagrangian L admitting an infinitesimal symmetry K whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point p to a flow line $$\gamma =\gamma (t)$$ γ = γ ( t ) of K that does not cross p. By utilizing the invariance of L under the flow of K, we simplify the problem into a two-point boundary problem. Consequently, we derive an equation that involves the differential of the “arrival time” t, seen as a functional on the infinite dimensional manifold of connecting paths satisfying the semi-holonomic constraint defined by the Noether charge. When L is positively homogeneous of degree 2 in the velocities, the resulting equation establishes a variational principle that extends the Fermat’s principle in a stationary spacetime. Furthermore, we also analyze the scenario where the Noether charge is affine.

Isometric Immersions and the Waving of Flags.

In this article we propose a novel geometric model to study the motion of a physical flag. In our approach, a flag is viewed as an isometric immersion from the square with values in satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. In the second part of the article we equip the space of isometric immersions with its natural kinetic energy and derive the corresponding equations of motion. This approach can be viewed in a spirit similar to Arnold's geometric picture for the motion of an incompressible fluid.

The space of closed G2-structures. I. Connections

Abstract In this article, we develop foundational theory for geometries of the space of closed G2-structures in a given cohomology class as an infinite-dimensional manifold. We construct Levi-Civita connections for Sobolev-type metrics, formulate geodesic equations and analyze the variational structures of torsion-free G2-structures under these Sobolev-type metrics.

On the differential geometry of some classes of infinite dimensional manifolds

On the differential geometry of some classes of infinite dimensional manifolds

Soft particles and infinite-dimensional geometry

In the sigma model, soft insertions of moduli scalars enact parallel transport of S-matrix elements about the finite-dimensional moduli space of vacua, and the antisymmetric double-soft theorem calculates the curvature of the vacuum manifold. We explore the analogs of these statements in gauge theory and gravity in asymptotically flat spacetimes, where the relevant moduli spaces are infinite-dimensional. These models have spaces of vacua parameterized by (trivial) flat connections on the celestial sphere, and soft insertions of photons, gluons, and gravitons parallel transport S-matrix elements about these infinite-dimensional manifolds. We argue that the antisymmetric double-soft gluon theorem in d + 2 bulk dimensions computes the curvature of a connection on the infinite-dimensional space Map, where G is the global part of the gauge group. The analogous metrics in abelian gauge theory and gravity are flat, as indicated by the vanishing of the antisymmetric double-soft theorems in those models. In other words, Feynman diagram calculations not only know about the vacuum manifold of Yang–Mills theory, they can also be used to compute its curvature. The results have interesting implications for flat space holography.

To the Theory of Decaying Turbulence

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension d>2. This solution family is equivalent to a fractal curve in complex space Cd with random steps parametrized by N Ising variables σi=±1, in addition to a rational number pq and an integer winding number r, related by ∑σi=qr. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in Rd space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size N→∞ or its chemical potential μ→0. The system with fixed N has different asymptotics at odd and even N→∞, but the limit μ→0 is well defined. The energy dissipation rate is analytically calculated as a function of μ using methods of number theory. It grows as ν/μ2 in the continuum limit μ→0, leading to anomalous dissipation at μ∝ν→0. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as t−λ, with a continuous spectrum of indexes λ in the local limit μ→0. The spectrum is determined by a resolvent, which is represented as an infinite product of 3⊗3 matrices depending of the element of the Euler ensemble.

Smooth Homotopy of Infinite-Dimensional 𝐶^{∞}-Manifolds

In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional C ∞ C^{\infty } -manifolds in convenient calculus. More precisely, we discuss the smoothing of maps, sections, principal bundles, and gauge transformations. We first introduce the notion of hereditary C ∞ C^\infty -paracompactness along with the semiclassicality condition on a C ∞ C^\infty -manifold, which enables us to use local convexity in local arguments. Then, we prove that for C ∞ C^\infty -manifolds M M and N N , the smooth singular complex of the diffeological space C ∞ ( M , N ) C^\infty (M,N) is weakly equivalent to the ordinary singular complex of the topological space C 0 ( M , N ) {\mathcal {C}^0}(M,N) under the hereditary C ∞ C^\infty -paracompactness and semiclassicality conditions on M M . We next generalize this result to sections of fiber bundles over a C ∞ C^\infty -manifold M M under the same conditions on M M . Further, we establish the Dwyer-Kan equivalence between the simplicial groupoid of smooth principal G G -bundles over M M and that of continuous principal G G -bundles over M M for a Lie group G G and a C ∞ C^\infty -manifold M M under the same conditions on M M , encoding the smoothing results for principal bundles and gauge transformations. For the proofs, we fully faithfully embed the category C ∞ C^{\infty } of C ∞ C^{\infty } -manifolds into the category D {\mathcal {D}} of diffeological spaces and develop the smooth homotopy theory of diffeological spaces via a homotopical algebraic study of the model category D {\mathcal {D}} and the model category C 0 {\mathcal {C}^0} of arc-generated spaces, also known as Δ \Delta -generated spaces. Then, the hereditary C ∞ C^\infty -paracompactness and semiclassicality conditions on M M imply that M M has the smooth homotopy type of a cofibrant object in D {\mathcal {D}} . This result can be regarded as a smooth refinement of the results of Milnor, Palais, and Heisey, which give sufficient conditions under which an infinite-dimensional topological manifold has the homotopy type of a C W CW -complex. We also show that most of the important C ∞ C^\infty -manifolds introduced and studied by Kriegl, Michor, and their coauthors are hereditarily C ∞ C^\infty -paracompact and semiclassical, and hence, results can be applied to them.

Divergence of Multivector Fields on Infinite-Dimensional Manifolds

Divergence of Multivector Fields on Infinite-Dimensional Manifolds

FIXED POINT APPROXIMATION USING UNIFIED ITERATION SCHEME FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN CAT(0) SPACES

In the present paper, we focus on reexamination of unified three step iteration scheme in more general infinite-dimensional manifolds i.e. in geodesic CAT(0) spaces for asymptotically non-expansive mappings. The findings hold true for both asymptotically non-expansive type mappings and asymptotically quasi nonexpansive mappings. Since, numerous iteration schemes have been introducing for so long and also claimed new and different from other which shows huge lacking of existing iteration based literature. It is to be noted that there are several iteration schemes which are claimed to be different and unique but is special case of some existing scheme. Our results improve the existing iteration scheme based literature.

Divergence of multivector fields on infinite-dimensional manifolds

UDC 514.763.2+515.164.17 We study the divergence of multivector fields on Banach manifolds with a Radon measure. We propose an infinite-dimensional version of divergence consistent with the classical divergence from finite-dimensional differential geometry. We then transfer certain natural properties of the divergence operator to the infinite-dimensional setting. Finally, we study the relation between the divergence operator on a manifold M and the divergence operator on a submanifold S ⊂ M .

Energy-adaptive Riemannian optimization on the Stiefel manifold

This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross–Pitaevskii and Kohn–Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energy-adaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes.

Chaotic behaviour of countable products of homeomorphism groups

ABSTRACT Relationships between a chaotic behaviour and closely related properties of topological transitivity, sensitivity to initial conditions, density of closed orbits of homeomorphism groups and their countable products are investigated. We provide a large number of new examples of chaotic groups of homeomorphisms of countable products of various metrizable topological spaces, including infinite-dimensional topological manifolds, whose factors can be as noncompact surfaces, so triangulable closed manifolds of an arbitrary dimension.

An infinite-dimensional index theorem and the Higson–Kasparov–Trout algebra

We have been studying the index theory for some special infinite-dimensional manifolds with a "proper cocompact" actions of the loop group LT of the circle T, from the viewpoint of the noncommutative geometry. In this paper, we will introduce the LT-equivariant KK-theory and we will construct three KK-elements: the index element, the Clifford symbol element and the Dirac element. These elements satisfy a certain relation, which should be called the (KK-theoretical) index theorem, or the KK-theoretical Poincar\'e duality for infinite-dimensional manifolds. We will also discuss the assembly maps.

Integrated Construction of Multimodal Atlases with Structural Connectomes in the Space of Riemannian Metrics

The structural network of the brain, or structural connectome, can be represented by fiber bundles generated by a variety of tractography methods. While such methods give qualitative insights into brain structure, there is controversy over whether they can provide quantitative information, especially at the population level. In order to enable population-level statistical analysis of the structural connectome, we propose representing a connectome as a Riemannian metric, which is a point on an infinite-dimensional manifold. We equip this manifold with the Ebin metric, a natural metric structure for this space, to get a Riemannian manifold along with its associated geometric properties. We then use this Riemannian framework to apply object-oriented statistical analysis to define an atlas as the Fréchet mean of a population of Riemannian metrics. This formulation ties into the existing framework for diffeomorphic construction of image atlases, allowing us to construct a multimodal atlas by simultaneously integrating complementary white matter structure details from DWMRI and cortical details from T1-weighted MRI. We illustrate our framework with 2D data examples of connectome registration and atlas formation. Finally, we build an example 3D multimodal atlas using T1 images and connectomes derived from diffusion tensors estimated from a subset of subjects from the Human Connectome Project.

Projection Method for Droplet Dynamics on Groove-Textured Surface with Merging and Splitting

The geometric motion of small droplets placed on an impermeable textured substrate is mainly driven by the capillary effect, the competition among surface tensions of three phases at the moving contact lines, and the impermeable substrate obstacle. After introducing an infinite dimensional manifold with an admissible tangent space on the boundary of the manifold, by Onsager's principle for an obstacle problem, we derive the associated parabolic variational inequalities. These variational inequalities can be used to simulate the contact line dynamics with unavoidable merging and splitting of droplets due to the impermeable obstacle. To efficiently solve the parabolic variational inequality, we propose an unconditional stable explicit boundary updating scheme coupled with a projection method. The explicit boundary updating efficiently decouples the computation of the motion by mean curvature of the capillary surface and the moving contact lines. Meanwhile, the projection step efficiently splits the difficulties brought by the obstacle and the motion by mean curvature of the capillary surface. Furthermore, we prove the unconditional stability of the scheme and present an accuracy check. The convergence of the proposed scheme is also proved using a nonlinear Trotter-Kato's product formula under the pinning contact line assumption. After incorporating the phase transition information at splitting points, several challenging examples including splitting and merging of droplets are demonstrated.

A dual pair for the contact group

AbstractGeneralizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with equivariant moment maps, respectively, giving rise to a dual pair, called the EPContact dual pair. Via symplectic reduction, this dual pair provides a conceptual identification of non-linear Grassmannians of weighted submanifolds with certain coadjoint orbits of the contact group. Moreover, the EPContact dual pair gives rise to singular solutions for the geodesic equation on the group of contact diffeomorphisms. For the projectivized cotangent bundle, the EPContact dual pair is closely related to the EPDiff dual pair due to Holm and Marsden.

Accelerated Optimization in the PDE Framework: Formulations for the Manifold of Diffeomorphisms

We consider the problem of optimization of cost functionals on the infinite dimensional manifold of diffeomorphisms. We present a new class of optimization methods, valid for any optimization problem setup on the space of diffeomorphisms by generalizing Nesterov accelerated optimization to the manifold of diffeomorphisms. While our framework is general for infinite dimensional manifolds, we specifically treat the case of diffeomorphisms, motivated by optical flow problems in computer vision. This is accomplished by building on a recent variational approach to a general class of accelerated optimization methods by Wibisono, Wilson, and Jordan [Proc. Natl. Acad. Sci. USA, 113 (2016), pp. E7351--E7358], which applies in finite dimensions. We generalize that approach to infinite dimensional manifolds. We derive the surprisingly simple continuum evolution equations, which are partial differential equations, for accelerated gradient descent, and relate them to simple mechanical principles from fluid mechanics. Our approach has natural connections to the optimal mass transport problem. This is because one can think of our approach as an evolution of an infinite number of particles endowed with mass (represented with a mass density) that moves in an energy landscape. The mass evolves with the optimization variable and endows the particles with dynamics. This is different from the finite dimensional case where only a single particle moves and hence the dynamics does not depend on the mass. We derive the theory, compute the PDEs for accelerated optimization, and illustrate the behavior of these new accelerated optimization schemes.

Orbital decompositions and integrable pseudosymmetries of control systems

Local decompositions of nonlinear dynamical control systems are investigated. We use orbital equivalences to transform systems into decomposed forms. In the language of infinite jets, every control system is interpreted as an infinite dimensional manifold. Each orbital decomposition of the system defines an invariant distribution on this manifold. A local basis of this distribution is an integrable pseudosymmetry of the system. An algorithm for constructing orbital decompositions is given and demonstrated by the example of the Kapitsa pendulum system. It is also shown how to use orbital decompositions to plan trajectories.

The principle of least action in the space of Kähler potentials

Given a compact Kahler manifold, the space $\mathcal H$ of its (relative) Kahler potentials is an infinite dimensional Frechet manifold, on which Mabuchi and Semmes have introduced a natural connection $\nabla$. We study certain Lagrangians on $T\mathcal H$, in particular Finsler metrics, that are parallel with respect to the connection. We show that geodesics of $\nabla$ are paths of least action, and prove a certain convexity property of the least action. This generalizes earlier results of Calabi, Chen, and Darvas.