Smooth Manifold Research Articles

Invariant probability measures from pseudoholomorphic curves Ⅱ: Pseudoholomorphic curve constructions

In the previous work, we introduced a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds with pseudoholomorphic curve techniques from symplectic geometry. The technique requires existence of certain pseudoholomorphic curves satisfying some weak assumptions. In this work, we appeal to Gromov–Witten theory and Seiberg–Witten theory to construct large classes of examples where these pseudoholomorphic curves exist. Our argument uses neck stretching along with new analytical tools from Fish–Hofer's work on feral pseudoholomorphic curves.

Viscosity solutions of contact Hamilton-Jacobi equations with Hamiltonians depending periodically on unknown functions

Assume $ H = H(x, p, u) $ with $ (x, p)\in T^*M $ and $ u\in\mathbb{S} $, is smooth and satisfies Tonelli conditions in $ p $, Lipschitz continuity condition in $ u $, where $ M $ is a compact connected smooth manifold without boundary. We find a compact interval $ \mathcal C $ such that equation \begin{document}$ H(x, \partial_xu(x), u(x)) = c $\end{document} has solutions if and only if $ c\in \mathcal C $. We also study the long-time behavior of the unique viscosity solution $ u^c $ of$ \partial_t u(x, t)+H(x, \partial_xu(x, t), u(x, t)) = c, \quad u(x, 0) = \varphi(x)\in C(M, \mathbb{R}). $If $ c\in \mathcal C $, $ u^c $ is bounded by a constant independent of $ c $ and Lipschitz with respect to the argument $ x $ with a Lipschitz constant independent of $ c $ and $ \varphi $. If $ c\notin \mathcal C $, then the long-time average of $ u^c $ can be characterized by a function $ c\mapsto\rho(c) $ which admits a modulus of continuity. We obtain these results by analyzing properties of a kind of one-parameter semigroups of operators. All the aforementioned results show the fundamental difference between Hamilton-Jacobi equations with Hamiltonians $ H(x, p, u) $ and $ \bar H(x, p) $.

О конструкции канонической формы на расслоении реперов

The detailed description of the construction of the canonical form on the higher order frame bundle over an n-dimensional smooth manifold is given. In particular, it is shown that some vector space isomorphism play­ing the key role in this construction is defined correctly, i. e. it depends only on the frame of order p + 1 and does not depend on the choice of its representative, i. e. a local diffeomorphism which (p + 1)-jet is exactly this frame. This isomorphism acts from the direct sum of n-dimensional arithmetic space and the Lie algebra of the p-th order differential group to the tangent space to the p-th order frame bundle over the manifold at the p-th order frame lying “below”. The action of this isomorphism can be splitted into two its restrictions. The first one acts from the first direct sum­mand, and the second one acts from the second direct summand. It is shown that the first restriction depends only on the choice of the (p + 1)-fra­me, while the second one is closely related to fundamental vector fields and therefore does not depend of this frame at all.

Sampling for Snapshot Compressive Imaging

In this study, we compare interlaced and multiscale sampling of smooth manifolds for snapshot compressive imaging. With a particular focus on spectral, spatial, and temporal focal photographic imaging systems, we show that structured transformer networks enable the efficient integration of multiscale manifolds. In the applications considered here, transformer networks enable simpler and more target-specific sampling strategies for compressive tomography.

Weak measure expansivity of C 2 dynamics

Abstract Let f f be a C 2 {C}^{2} -diffeomorphism with Axiom A and no cycle condition on a two-dimensional smooth manifold. In this article, we prove that if f f is C 2 {C}^{2} -robustly weak measure expansive, then it is Q 2 {Q}^{2} -Anosov. Moreover, we expand the results of the C 2 {C}^{2} -diffeomorphism case into the C 2 {C}^{2} -vector field on a three-dimensional smooth manifold. Let X X be a C 2 {C}^{2} -vector field with Axiom A and no cycle condition. We prove that if X X is C 2 {C}^{2} -robustly weak measure expansive, then it is Q 2 {Q}^{2} -Anosov.

Invariants of 4–manifolds from Khovanov–Rozansky link homology

We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere.

Topological obstructions to the diagonalisation of pseudodifferential systems

Given a matrix pseudodifferential operator on a smooth manifold, one may be interested in diagonalising it by choosing eigenvectors of its principal symbol in a smooth manner. We show that diagonalisation is not always possible, on the whole cotangent bundle or even in a single fibre. We identify global and local topological obstructions to diagonalisation and examine physically meaningful examples demonstrating that all possible scenarios can occur.

Floer Homology: From Generalized Morse–Smale Dynamical Systems to Forman’s Combinatorial Vector Fields

We construct a Floer type boundary operator for generalised Morse–Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_2$$\\end{document} homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.

On Not Canonical Equivalence Between Diffeomorphism Subgroupoid And Fundemental Fibre Bundle

The main purpose of this work is to find not canonical equivalence to category and morphism of diffeomorphism from smooth manifold category with Fundemental Fibre Bundle denoted by groupoids morphism and denoted P.

Lifting Diffeomorphisms to Vector Bundles

Criteria for a diffeomorphism of a smooth manifold M to be lifted to a linear automorphism of a given real vector bundle p : V → M, are stated. Examples are included and the metric and complex vector-bundle cases are also considered.

Natural differentiable structures on statistical models and the Fisher metric

In this paper I discuss the relation between the concept of the Fisher metric and the concept of differentiability of a family of probability measures. I compare the concepts of smooth statistical manifolds, differentiable families of measures, $k$-integrable parameterized measure models, diffeological statistical models, differentiable measures, which arise in Information Geometry, mathematical statistics and measure theory, and discuss some related problems.

Schwarzschild Spacetimes: Topology

This paper is devoted to the geometric theory of a Schwarzschild spacetime, a basic objective in applications of the classical general relativity theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group SO(3) and a Schwarzschild metric, an SO(3)-invariant metric field, satisfying the Einstein equations. We prove the existence of and find all Schwarzschild metrics on two topologically non-equivalent manifolds, R×(R3∖{(0,0,0)}) and S1×(R3∖{(0,0,0)}). The method includes a classification of SO(3)-invariant, time-translation invariant and time-reflection invariant metrics on R×(R3∖{(0,0,0)}) and a winding mapping of the real line R onto the circle S1. The resulting family of Schwarzschild metrics is parametrized by an arbitrary function and two real parameters, the integration constants. For any Schwarzschild metric, one of the parameters determines a submanifold, where the metric is not defined, the Schwarzschild sphere. In particular, the family admits a global metric whose Schwarzschild sphere is empty. These results transfer to S1×(R3∖{(0,0,0)}) by the winding mapping. All our assertions are derived independently of the signature of the Schwarzschild metric; the signature can be chosen as an independent axiom.

Characteristic numbers, Jiang subgroup and non-positive curvature

By refining a result of Farrell and in terms of the Jiang subgroups of self-maps of smooth manifolds, we obtain a sufficient condition for the vanishing of the signature and the Hirzebruch’s $$\chi _y$$ -genus. Along this line we show that the $$\chi _y$$ -genus of a non-positively curved compact Kähler manifold vanishes when the center of its fundamental group is non-trivial, which realizes an expectation of Farrell’s in the Kähler setting. Moreover, in the latter case all the Chern numbers vanish whenever its complex dimension is no more than 4, which also provides some evidence towards a complex version’s Hopf Conjecture proposed by the author and Zheng.

Lipschitz Carnot-Carathéodory Structures and their Limits

In this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not boundedly compact and the convergence is not uniform on compact sets. We discuss several examples in which our convergence result can be applied. Among them, we prove a subFinsler Mitchell’s Theorem with continuously varying norms, and a general convergence result for Carnot-Carathéodory distances associated to subspaces and norms on the Lie algebra of a connected Lie group.

On the general part of the perturbed Vafa-Witten moduli spaces on 4-manifolds

In this paper we consider the Vafa-Witten equations on closed, oriented and smooth 4-manifolds, and construct a set of perturbation terms to establish the transversality of the perturbed Vafa-Witten equations at the general part of the solutions. Then we show that for a generic choice of the perturbation terms, this part of the moduli space for the structure group SU(2) or SO(3) on a closed 4-manifold is a smooth manifold of dimension zero.

Kernel-Based Methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds

In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and ghost point diffusion maps (GPDM), to solve the time-dependent advection-diffusion PDE on unknown smooth manifolds without and with boundaries. The core idea is to directly approximate the spatial components of the differential operator on the manifold with a local integral operator and combine it with the standard implicit time difference scheme. When the manifold has a boundary, a simplified version of the GPDM approach is used to overcome the bias of the integral approximation near the boundary. The Monte-Carlo discretization of the integral operator over the point cloud data gives rise to a mesh-free formulation that is natural for randomly distributed points, even when the manifold is embedded in high-dimensional ambient space. Here, we establish the convergence of the proposed solver on appropriate topologies, depending on the distribution of point cloud data and boundary type. We provide numerical results to validate the convergence results on various examples that involve simple geometry and an unknown manifold.

AN OBSERVATION ON THE DIRICHLET PROBLEM AT INFINITY IN RIEMANNIAN CONES

AbstractIn this short paper, we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below). This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied to smooth Riemannian manifolds with a special type of metrics, which generalize the class of metrics with rotational symmetry, we obtain generalizations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODEs.

A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations

Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely on Euclidean geometry, but in the last years new approaches based on Riemannian geometry have been developed. Motivated by some open problems, we study a particular sequence of maps between manifolds, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the structures induced through pullbacks on the other manifolds of the sequence and on some related quotients. In particular, we show that the pullbacks of the final Riemannian metric to any manifolds of the sequence is a degenerate Riemannian metric inducing a structure of pseudometric space. We prove that the Kolmogorov quotient of this pseudometric space yields a smooth manifold, which is the base space of a particular vertical bundle. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between manifolds implementing neural networks of practical interest and we present some applications of the geometric framework we introduced in the first part of the paper.

The standard cohomology of regular Courant algebroids

For any regular Courant algebroid E over a smooth manifold M with characteristic distribution F and ample Lie algebroid AE, we prove that there exists a canonical homological vector field on the graded manifold AE[1]⊕(TM/F)⁎[2] such that the resulting dg manifold ME, which we call the minimal model of the Courant algebroid E, encodes all cohomological information of E. Indeed, the standard cohomology of E can be identified with the cohomology of the function space on ME, which can be computed by a Hodge-to-de Rham type spectral sequence. We apply this result to generalized exact Courant algebroids and those arising from regular Lie algebroids.

Critical Metrics and Curvature of Metrics with Unit Volume or Unit Area of the Boundary

Given a smooth compact manifold with boundary, we study variational properties of the volume functional and of the area functional of the boundary, restricted to the space of the Riemannian metrics with prescribed curvature. We obtain a sufficient and necessary condition for a metric to be a critical point. As a by-product, a very natural analogue of V-statics metrics is obtained. In the second part, using the Yamabe invariant in the boundary setting, we solve the Kazdan–Warner–Kobayashi problem in a compact manifold with boundary. For several cases, depending on the sign of the Yamabe invariant, we give sufficient and necessary condition for a smooth function to be the scalar or mean curvature of a metric with constraint on the volume or area of the boundary