Structure Of Smooth Manifold Research Articles

The Covariance Metric in the Blaschke Locus

We prove that the Blaschke locus has the structure of a finite dimensional smooth manifold away from the Teichmüller space and study its Riemannian manifold structure with respect to the covariance metric introduced by Guillarmou, Knieper and Lefeuvre in Guillarmou et al. in (Ergod Theory Dyn Syst 43:974–1022, 2021). We also identify some families of geodesics in the Blaschke locus arising from Hitchin representations for orbifolds and show that they have infinite length with respect to the covariance metric.

Target detection based on generalized Bures–Wasserstein distance

Radar target detection with fewer echo pulses in non-Gaussian clutter background is a challenging problem. In this instance, the conventional detectors using coherent accumulation are not very satisfactory. In contrast, the matrix detector based on Riemannian manifolds has shown potential on this issue since the covariance matrix of radar echo data during one coherent processing interval(CPI) has a smooth manifold structure. The Affine Invariant (AI) Riemannian distance between the cell under test (CUT) and the reference cells has been used as a statistic to achieve improved detection performance. This paper uses the Bures–Wasserstein (BW) distance and Generalized Bures–Wasserstein (GBW) distance on Riemannian manifolds as test statistics of matrix detectors, and propose relevant target detection method. Maximizing the GBW distance is formulated as an optimization problem and is solved by the Riemannian trust-region (RTR) method to achieve enhanced discrimination for target detection. Our evaluation of simulated data and measured data show that the matrix detector based on GBW distance leads to a significant performance gain over existing methods.

Manifolds of mappings on Cartesian products

Given smooth manifolds M_1,ldots , M_n (which may have a boundary or corners), a smooth manifold N modeled on locally convex spaces and alpha in ({{mathbb {N}}}_0cup {infty })^n, we consider the set C^alpha (M_1times cdots times M_n,N) of all mappings f:M_1times cdots times M_nrightarrow N which are C^alpha in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders le alpha _j in the jth variable for jin {1,ldots , n}, in local charts. We show that C^alpha (M_1times cdots times M_n,N) admits a canonical smooth manifold structure whenever each M_j is compact and N admits a local addition. The case of non-compact domains is also considered.

On the Mathematical Foundations of Causal Fermion Systems in Minkowski Space

The emergence of the concept of a causal fermion system is revisited and further investigated for the vacuum Dirac equation in Minkowski space. After a brief recap of the Dirac equation and its solution space, in order to allow for the effects of a possibly nonstandard structure of spacetime at the Planck scale, a regularization by a smooth cutoff in momentum space is introduced, and its properties are discussed. Given an ensemble of solutions, we recall the construction of a local correlation function, which realizes spacetime in terms of operators. It is shown in various situations that the local correlation function maps spacetime points to operators of maximal rank and that it is closed and homeomorphic onto its image. It is inferred that the corresponding causal fermion systems are regular and have a smooth manifold structure. The cases considered include a Dirac sea vacuum and systems involving a finite number of particles and antiparticles.

A Manifold-Based Dimension Reduction Algorithm Framework for Noisy Data Using Graph Sampling and Spectral Graph

This paper proposes a new manifold-based dimension reduction algorithm framework. It can deal with the dimension reduction problem of data with noise and give the dimension reduction results with the deviation values caused by noise interference. Commonly used manifold learning methods are sensitive to noise in the data. Mean computation, a denoising method, is an important step in data preprocessing but leads to a loss of local structural information. In addition, it is difficult to measure the accuracy of the dimension reduction of noisy data. Thus, manifold learning methods often transform the data into an approximately smooth manifold structure; however, practical data from the physical world may not meet the requirements. The proposed framework follows the idea of the localization of manifolds and uses graph sampling to determine some local anchor points from the given data. Subsequently, the specific range of localities is determined using graph spectral analysis, and the density within each local range is estimated to obtain the distribution parameters. Then, manifold-based dimension reduction with distribution parameters is established, and the deviation values in each local range are measured and further extended to all data. Thus, our proposed framework gives a measurement method for deviation caused by noise.

Weil-Petersson Teichmüller space revisited

In our previous paper [23] it was proved that a sense-preserving homeomorphism g on the unit circle S1 belongs to the Weil-Petersson class WP(S1), namely, g can be extended to a quasiconformal mapping to the unit disk whose Beltrami coefficient is square integrable in the Poincaré metric if and only if g is absolutely continuous such that log⁡g′ belongs to the Sobolev class H12. In this sequel to [23], we show that the smooth Hilbert manifold structure on WP(S1) inherited from H12 by the pullback g↦log⁡|g′| is compatible with the standard Hilbert manifold structure introduced by Takhtajan-Teo [27]. This enables us to give a fast approach to some results in our previous papers [23] and [24].

Smooth manifold structure for extreme channels

A quantum channel from a system A of dimension dA to a system B of dimension dB is a completely positive trace-preserving map from complex dA × dA to dB × dB matrices, and the set of all such maps with Kraus rank r has the structure of a smooth manifold. We describe this set in two ways. First, as a quotient space of (a subset of) the rdB × dA dimensional Stiefel manifold. Second, as the set of all Choi-states of a fixed rank r. These two descriptions are topologically equivalent. This allows us to show that the set of all Choi-states corresponding to extreme channels from system A to system B of a fixed Kraus rank r is a smooth submanifold of dimension 2rdAdB−dA2−r2 of the set of all Choi-states of rank r. As an application, we derive a lower bound on the number of parameters required for a quantum circuit topology to be able to approximate all extreme channels from A to B arbitrarily well.

Discriminative Structured Dictionary Learning on Grassmann Manifolds and Its Application on Image Restoration.

Image restoration is a difficult and challenging problem in various imaging applications. However, despite of the benefits of a single overcomplete dictionary, there are still several challenges for capturing the geometric structure of image of interest. To more accurately represent the local structures of the underlying signals, we propose a new problem formulation for sparse representation with block-orthogonal constraint. There are three contributions. First, a framework for discriminative structured dictionary learning is proposed, which leads to a smooth manifold structure and quotient search spaces. Second, an alternating minimization scheme is proposed after taking both the cost function and the constraints into account. This is achieved by iteratively alternating between updating the block structure of the dictionary defined on Grassmann manifold and sparsifying the dictionary atoms automatically. Third, Riemannian conjugate gradient is considered to track local subspaces efficiently with a convergence guarantee. Extensive experiments on various datasets demonstrate that the proposed method outperforms the state-of-the-art methods on the removal of mixed Gaussian-impulse noise.

Higher Order Tangent Bundles

The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. For a Banach manifold $M$ and a natural number $k$ first we determine a smooth manifold structure on $T^kM$ which also offers a fiber bundle structure for $(\pi_k,T^kM,M)$. Then we introduce a particular lift of linear connections on $M$ to geometrize $T^kM$ as a vector bundle over $M$. More precisely based on this lifted nonlinear connection we prove that $T^kM$ admits a vector bundle structure over $M$ if and only if $M$ is endowed with a linear connection. As a consequence applying this vector bundle structure we lift Riemannian metrics and Lagrangians from $M$ to $T^kM$. Also, using the projective limit techniques, we declare a generalized Fr\'echet vector bundle structure for $T^\infty M$ over $M$.

Optimization on the Hierarchical Tucker manifold – Applications to tensor completion

In this work, we develop an optimization framework for problems whose solutions are well-approximated by Hierarchical Tucker (HT) tensors, an efficient structured tensor format based on recursive subspace factorizations. By exploiting the smooth manifold structure of these tensors, we construct standard optimization algorithms such as Steepest Descent and Conjugate Gradient for completing tensors from missing entries. Our algorithmic framework is fast and scalable to large problem sizes as we do not require SVDs on the ambient tensor space, as required by other methods. Moreover, we exploit the structure of the Gramian matrices associated with the HT format to regularize our problem, reducing overfitting for high subsampling ratios. We also find that the organization of the tensor can have a major impact on completion from realistic seismic acquisition geometries. These samplings are far from idealized randomized samplings that are usually considered in the literature but are realizable in practical scenarios. Using these algorithms, we successfully interpolate large-scale seismic data sets and demonstrate the competitive computational scaling of our algorithms as the problem sizes grow.

Deforming discontinuous subgroups of reduced Heisenberg groups

Let G=H2n+1r be the (2n+1)-dimensional reduced Heisenberg group, and let H be an arbitrary connected Lie subgroup of G. Given any discontinuous subgroup Γ⊂G for G/H, we show that resulting deformation space T(Γ,G,H) of the natural action of Γ on G/H is endowed with a smooth manifold structure and is a disjoint union of open smooth manifolds. Unlike the setting of simply connected Heisenberg groups, we show that the stability property holds and that any discrete subgroup of G is stable, following the notion of stability. On the other hand, a local (and hence global) rigidity theorem is obtained. That is, the related parameter space R(Γ,G,H) admits a rigid point if and only if Γ is finite.

WHEN IS THE DEFORMATION SPACE $\mathscr{T}(\Gamma, H_{2n+1}, H)$ A SMOOTH MANIFOLD?

Let G = H2n + 1 be the 2n + 1-dimensional Heisenberg group and H be a connected Lie subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/H, a precise union of open sets of the resulting deformation space [Formula: see text] of the natural action of Γ on G/H is derived since the paper of Kobayshi and Nasrin [Deformation of Properly discontinuous action of ℤk and ℝk+1, Internat. J. Math.17 (2006) 1175–1190]. We determine in this paper when exactly this space is endowed with a smooth manifold structure. Such a result is only known when the Clifford–Klein form Γ\G/H is compact and Γ is abelian. When Γ is not abelian or H meets the center of G, the parameter and deformation spaces are shown to be semi-algebraic and equipped with a smooth manifold structure. In the case where Γ is abelian and H does not meet the center of G, then [Formula: see text] splits into finitely many semi-algebraic smooth manifolds and fails to be a Hausdorff space whenever Γ is not maximal, but admits a manifold structure otherwise. In any case, it is shown that [Formula: see text] admits an open smooth manifold as its dense subset. Furthermore, a sufficient and necessary condition for the global stability of all these deformations to hold is established.

On the geometry of stable discontinuous subgroups acting on threadlike homogeneous spaces

Following the notion of stability introduced by T. Kobayashi and S. Nasrin in [14], we show in the context of a threadlike Lie group G that any non-Abelian discrete subgroup is stable. One consequence is that any resulting deformation space ℐ(Γ,G,H) is a Hausdorff space, where Γ acts on the threadlike homogeneous space G/H as a discontinuous subgroup. Whenever k = rank(Γ) > 3, this space is also shown to be endowed with a smooth manifold structure. But if k = 3, then ℐ(Γ,G,H) admits a smooth manifold structure as its open dense subset. These phenomena are strongly linked to the features of adjoint orbits of the basis group G on the parameter space ℛ(Γ,G,H) (which is semi-algebraic in this case) and specifically to their dimensions, as it will be seen throughout the paper. This also allows to provide a proof of the Local Rigidity Conjecture in this setup.

General equilibrium without utility functions: how far to go?

How far can we go in weakening the assumptions of the general equilibrium model? Existence of equilibrium, structural stability and finiteness of equilibria of regular economies, genericity of regular economies and an index formula for the equilibria of regular economies have been known not to require transitivity and completeness of consumers’ preferences. We show in this paper that if consumers’ non-ordered preferences satisfy a mild version of convexity already considered in the literature, then the following properties are also satisfied: (1) the smooth manifold structure and the diffeomorphism of the equilibrium manifold with a Euclidean space; (2) the diffeomorphism of the set of no-trade equilibria with a Euclidean space; (3) the openness and genericity of the set of regular equilibria as a subset of the equilibrium manifold; (4) for small trade vectors, the uniqueness, regularity and stability of equilibrium for two version of tatonnement; (5) the pathconnectedness of the sets of stable equilibria.

General Equilibrium Without Utility Functions: How Far to Go?

How far can we go in weakening the assumptions of the general equilibrium model? Existence of equilibrium, structural stability and finiteness of equilibria of regular economies, genericity of regular economies and an index formula for the equilibria of regular economies have been known not to require transitivity and completeness of consumers’ preferences. We show in this paper that if consumers’ non-ordered preferences satisfy a mild version of convexity already considered in the literature, then the following properties are also satisfied: 1) the smooth manifold structure and the diffeomorphism of the equilibrium manifold with a Euclidean space; 2) the diffeomorphism of the set of no-trade equilibria with a Euclidean space; 3) the openness and genericity of the set of regular equilibria as a subset of the equilibrium manifold; 4) for small trade vectors, the uniqueness, regularity and stability of equilibrium for two version of tatonnement; 5) the pathconnectedness of the sets of stable equilibria.

Pro-Lie groups which are infinite-dimensional Lie groups

AbstractA pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

Supersymmetric field-theoretic models on a supermanifold

We propose the extension of some structural aspects that have successfully been applied in the development of the theory of quantum fields propagating on a general space–time manifold so as to include superfield models on a supermanifold. We only deal with the limited class of supermanifolds which admit the existence of a smooth body manifold structure. Our considerations are based on the Catenacci–Reina–Teofillatto–Bryant approach to supermanifolds. In particular, we show that the class of supermanifolds constructed by Bonora–Pasti–Tonin satisfies the criteria which guarantee that a supermanifold admits a Hausdorff body manifold. This construction is the closest to the physicist’s intuitive view of superspace as a manifold with some anticommuting coordinates, where the odd sector is topologically trivial. The paper also contains a new construction of superdistributions and useful results on the wavefront set of such objects. Moreover, a generalization of the spectral condition is formulated using the notion of the wavefront set of superdistributions, which is equivalent to the requirement that all of the component fields satisfy, on the body manifold, a microlocal spectral condition proposed by Brunetti–Fredenhagen–Köhler.

On brane solutions in M (atrix) theory

In this paper we consider brane solutions of the form G/ H in M(atrix) theory, showing the emergence of world volume coordinates for the cases where G = SU( n). We examine a particular solution with a world volume geometry of the form C P 2 × S1 in some detail and show how a smooth manifold structure emerges in the large N limit. In this limit the solution becomes static; it is not supersymmetric but is part of a supersymmetric set of configurations. Supersymmetry in small locally flat regions can be obtained, but this is not globally defined. A general group theoretic analysis of the previously known spherical brane solutions is also given.

Yamabe metrics of positive scalar curvature and conformally flat manifolds

Let CY( n, μ, R 0 be the class of compact connected smooth manifolds M of dimension n ⩾ 3 and with Yamabe metrics g of unit volume such that each ( M, g) is conformally flat and satisfies μ(M,[g]) ⩾μ 0 > 0, ∫ M|E g| n 2 dv g⩽R 0 , where [ g], μ( M,[ g]) and E g denote the conformal class of g, the Yamabe invariant of ( M,[ g]) and the traceless part of the Ricci tensor of g, respectively. In this paper, we study the boundary ACY( n, μ 0, R 0 of CY( n, μ 0, R 0) in the space of all compact metric spaces equipped with the Hausdorff distance. We shall show that an element in ACY( n, μ 0, R 0) is a compact metric space ( X, d). In particular, if ( X, d) is not a point, then it has a structure of smooth manifold outside a finite subset S , and moreover, on F\\ S there is a conformally flat metric g of positive constant scalar curvature which is compatible with the distance d.

On the origin of spacetime topology and some generalizations of quantum field theory

The research effort reported in this paper is directed, in a broad sense, towards understanding the small-scale structure of spacetime. The fundamental question that guides our discussion is `what is the physical content of spacetime topology?'. In classical physics, this question has a natural and simple answer: spacetime (as a topological space) is a book-keeping device that we invent to make the description of classical fields (the observables) easier. More precisely, if spacetime, , has sufficiently regular topology, and if sufficiently many fields exist to allow us to observe all continuous functions on X, then this collection of continuous functions uniquely determines both the set of points X and the topology on it. Naturally, however, this answer does not yield any clues into why our spacetime is observed to have a very special (i.e. smooth manifold) topology down to the smallest scales we can probe, or into whether this smooth manifold structure persists indefinitely, at length scales smaller than the smallest observed so far. To explore these queries, we are led to consider the original question in the context of quantum, rather than classical field theory. After all, in the real world physical fields (the observables) are not classical (continuous functions) but quantum operators, and the fundamental observable is not the collection of all continuous functions but the local algebra of quantum field operators. Presently the only examples of local quantum field algebras that we know how to construct rigorously (apart from some two-dimensional models) are the operator algebras of free (linear) quantum fields propagating on a smooth, globally hyperbolic spacetime. Since this class of examples is too small, we find it necessary to generalize the algebraic notion of `quantum field' in such a way that it becomes possible to talk about quantum field theory on an arbitrary (not necessarily smooth) topological space on which no notion of spacetime metric exists a priori. (One interesting offshoot of this generalization is an algebraic framework for linear quantum field theory on non-globally-hyperbolic spacetimes (e.g. spacetimes with naked singularities or closed time-like curves), which is the subject of a separate paper that appears elsewhere in this journal.) In pursuing the original problem further, we develop a still wider generalization of quantum field theory; this ultimate generalization dispenses with the fixed background topological space altogether and proposes that the fundamental spacetime observable is a lattice (or more specifically a `frame', in the sense of set theory) of closed subalgebras of an abstract algebra. Our discussion concludes with the definition and some elementary properties of these `quantum lattices' and `quantum frames'.