Smooth Manifold Research Articles

$C^{\ast}$-algebraic approach to the principal symbol. III

We treat the notion of principal symbol mapping on a compact smooth manifold as a \ast -homomorphism of C^{\ast} -algebras. Principal symbol mapping is built from the ground, without referring to the pseudodifferential calculus on the manifold. Our concrete approach allows us to extend Connes trace theorem for compact Riemannian manifolds.

Persistence of smooth manifolds for a non-autonomous coupled system under small random perturbations

We consider a non-autonomous coupled system (x,y), whose x coordinate satisfies a semilinear parabolic equation, and y coordinate satisfies a differential equation whose solutions do not converge too rapidly. Our aim is to study the persistence of dynamical behavior for such coupled system under a small random perturbation driven by stationary multipilcative noise. We show that for the perturbed system there exists a C1 invariant manifold S(t,ω)={(x,y)∈X×Y|x=σ(t,y,ω)} under the condition that the linear part of x equation satisfies the exponential dichotomy. Meanwhile, we observe that typically if the linear part of x equation is uniformly attracting or uniformly expanding, then the corresponding invariant manifold shares the same qualitative properties. In the end, as the perturbation tends to 0, we show that the invariant manifold and its derivative in y are approaching to those of the original system, suggesting that such structure is persistent under the small random perturbation.

Ricci Flow Under Kato-Type Curvature Lower Bound

In this work, we extend the existence theory of non-collapsed Ricci flows from point-wise curvature lower bound to Kato-type curvature lower bound. As an application, we prove that any compact three-dimensional non-collapsed strong Kato limit space is homeomorphic to a smooth manifold. Moreover, similar result also holds in higher dimension under stronger curvature condition. We also use the Ricci flow smoothing to study stability problem in scalar curvature geometry.

Flat F-theory and friends

We discuss F-theory backgrounds associated to flat torus bundles over Ricci-flat manifolds. In this setting the F-theory background can be understood as a IIB orientifold with a large radius limit described by a supersymmetric compactification of IIB supergravity on a smooth, Ricci flat, but in general non-spin geometry. When compactified on an additional circle these backgrounds are T-dual to IIA compactifications on smooth non-orientable manifolds with a Pin− structure.

A Discretize-then-Optimize Approach to PDE-Constrained Shape Optimization

We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a smooth manifold, termed the manifold of planar triangular meshes. The latter can be endowed with a complete Riemannian metric, which allows large mesh deformations without jeopardizing mesh quality; see R. Herzog and E. Loayza-Romero, Math. Comput. 92 (2022) 1-50. Nonetheless, the discrete shape optimization problem of finding optimal vertex positions does not, in general, possess a globally optimal solution. To overcome this ill-possedness, we propose to add a mesh quality penalization term to the objective function. This allows us to simultaneously render the shape optimization problem solvable, and keep track of the mesh quality. We prove the existence of a globally optimal solution for the penalized problem and establish first-order necessary optimality conditions independently of the chosen Riemannian metric. Because of the independence of the existence results of the choice of the Riemannian metric, we can numerically study the impact of different Riemannian metrics on the steepest descent method. We compare the Euclidean, elasticity, and a novel complete metric, combined with Euclidean and geodesic retractions to perform the mesh deformation.

No-go theorems for $ r $-matrices in symplectic geometry

If a triangular Lie algebra acts on a smooth manifold, it induces a Poisson bracket on it. In case this Poisson structure is actually symplectic, we show that this already implies the existence of a flat connection on any vector bundle over the manifold the Lie algebra acts on, in particular the tangent bundle. This implies, among other things, that $ \mathbb{C}P^n $ and higher genus Pretzel surfaces cannot carry symplectic structures that are induced by triangular Lie algebras.

Poisson manifold

In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics. A Poisson structure (or Poisson bracket) on a smooth manifold M {\displaystyle M} is a function { ⋅ , ⋅ } : C ∞ ( M ) × C ∞ ( M ) → C ∞ ( M ) {\displaystyle \{\cdot ,\cdot \}:{\mathcal {C}}^{\infty }(M)\times {\mathcal {C}}^{\infty }(M)\to {\mathcal {C}}^{\infty }(M)}on the vector space C ∞ ( M ) {\displaystyle {\mathcal {C}}^{\infty }(M)} of smooth functions on M {\displaystyle M}, making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra). Poisson structures on manifolds were introduced by André Lichnerowicz in 1977[1] and are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics.[2]

О скалярных компонентах канонической формы на расслоениях реперов высших порядков

A detailed obtaining of the expressions for the scalar components of the canonical form on higher order frame bundles over a smooth manifold has been done. The canonical form on the frame bundle of order p + 1 over an n-dimensional smooth manifold is a vector-valued differential 1-form with values in the tangent space to the p-th order frame bundle over the n-di­mensional arithmetical space at the unit of the p-th order differential group. The scalar components of the canonical form are its coefficients with respect to natural basis of the tangent space. For every frame, there exists a polynomial mapping representing the frame in a given local chart on the manifold. Therefore, for any tangent vector to the frame bundle there is a first order Taylor expansion of one-parametric family of poly­nomial mappings representing the tangent vector. We obtain the formulas of the scalar components from the equations for coefficients of the two expansions for some tangent vector.

Disk-band graphs in the theory of framed tangles

A disk-band graph is a smooth compact two-dimensional manifold with boundary, partitioned into handles; the partition contains only index zero and index one handles and imitates the structure of the graph. (Index zero handles are analogues of the vertices of the graph, index one handles are analogues of the graph edges.) A disk-band graph is called spatial if it is a smooth submanifold of three-dimensional Euclidean space. A tangle is usually understood as a smooth compact one-dimensional submanifold of the standard three-dimensional ball that intersects the boundary of the ball orthogonally, only along its boundary, the intersection is contained in the equator. We call a tangle framed if it is equipped with a smooth field of normal straight lines. It is well known that there is a reduction of the problem of isotopic classification of spatial disk-band graphs to the problem of isotopic classification of framed tangles. This work focuses on the application of (abstract) disk-band graphs to study the set of isotopic classes of framed tangles.

Аналоги симметрической и плоской связностей с нетензорами кручения и кривизны

The paper is devoted to affine connection in the frame bundle associ­ated with a manifold which structure equations and derivation formulas are constructed using deformations of the exterior and ordinary differen­tials. Curvature and torsion objects of this connection are not tensors. A cha­racteristic of a curvature which is a convolution of a deformation tensor and a torsion, is considered. Torsion-free connections are not distingui­shed on the introduced manifold, even in the case of symmetric deforma­tion, a class of semi-symmetric connections is distinguished, which is an analogue of symmetric connection on an ordinary smooth manifold. It is proved that if the connection deformation tensor is symmetric or zero, then the connection is semi-symmetric. Analogues of torsion-free and cur­vature-free connections are constructed. The torsion and curvature of this connection are expressed in terms of the symmetric deformation tensor for the connection. Canonical connection is a special case of this connec­tion, it is semi-symmetric and curvature-free.

High excursion probabilities for gaussian fields won smooth manifolds

Рассматриваются гауссовские случайные поля, заданные на конечномерном гладком многообразии, при этом дисперсии достигают своего максимума также на гладком подмногообразии. Найдены точные асимптотики вероятностей выхода траекторий за высокий уровень. Показано, что приведенные условия на поведение ковариации и дисперсии неулучшаемы при использовании классического метода Пикандса двойных сумм. Рассмотрено применение полученных асимптотик к большим уклонениям гауссовских векторных процессов и приведены примеры. Работа является продолжением совместных работ автора с Э. Хашорвой, И. В. Родионовым, С. Г. Кобельковым (Фундамент. и прикл. матем., 23:1 (2020), 161-174), где рассматривались гауссовские процессы и поля на многообразиях с единственной точкой максимума дисперсии.

A cocyclic construction of $S^1$-equivariant homology and application to string topology

Given a space with a circle action, we study certain cocyclic chain complexes and prove a theorem relating cyclic homology to S^1 -equivariant homology, in the spirit of celebrated work of Jones. As an application, we describe a chain level refinement of the gravity algebra structure on the (negative) S^1 -equivariant homology of the free loop space of a closed oriented smooth manifold, based on work of Irie on chain level string topology and work of Ward on an S^1 -equivariant version of operadic Deligne’s conjecture.

The holonomy inverse problem

Let (M,g) be a smooth Anosov Riemannian manifold and \mathcal{C}^\sharp the set of its primitive closed geodesics. Given a Hermitian vector bundle \mathcal{E} equipped with a unitary connection \nabla^{\mathcal{E}} , we define \mathcal{T}^\sharp(\mathcal{E}, \nabla^{\mathcal{E}}) as the sequence of traces of holonomies of \nabla^{\mathcal{E}} along elements of \mathcal{C}^\sharp . This descends to a homomorphism on the additive moduli space \mathbb{A} of connections up to gauge \mathcal{T}^\sharp: (\mathbb{A}, \oplus) \to \ell^\infty(\mathcal{C}^\sharp) , which we call the primitive trace map . It is the restriction of the well-known Wilson loop operator to primitive closed geodesics. The main theorem of this paper shows that the primitive trace map \mathcal{T}^\sharp is locally injective near generic points of \mathbb{A} when \dim(M) \geq 3 . We obtain global results in some particular cases: flat bundles, direct sums of line bundles, and general bundles in negative curvature under a spectral assumption which is satisfied in particular for connections with small curvature. As a consequence of the main theorem, we also derive a spectral rigidity result for the connection Laplacian. The proofs are based on two new ingredients: a Livšic-type theorem in hyperbolic dynamical systems showing that the cohomology class of a unitary cocycle is determined by its trace along closed primitive orbits, and a theorem relating the local geometry of \mathbb{A} to the Pollicott–Ruelle resonance near zero of a certain natural transport operator.

Intrinsic Geometric Structure of Subcartesian Spaces

Every subset S of a Cartesian space Rd, endowed with differential structure C∞(S) generated by restrictions to S of functions in C∞(Rd), has a canonical partition M(S) by manifolds, which are orbits of the family X(S) of all derivations of C∞(S) that generate local one-parameter groups of local diffeomorphisms of S. This partition satisfies the frontier condition, Whitney’s conditions A and B. If M(S) is locally finite, then it satisfies all definitions of stratification of S. This result extends to Hausdorff locally Euclidean differential spaces. The partition M(S) of a subcartesian space S by smooth manifolds provides a measure for the applicability of differential geometric methods to the study of the geometry of S. If all manifolds in M(S) are single points, we cannot expect differential geometry to be an effective tool in the study of S. On the other extreme, if M(S) contains only one manifold M, then the subcartesian space S is a manifold, S=M, and it is a natural domain for differential geometric techniques.

Maps of degree 1, Lusternik–Schnirelmann category, and critical points

Let Crit M denote the minimal number of critical points (not necessarily non-degenerate) on a closed smooth manifold M. We are interested in the evaluation of Crit. It is worth noting that we do not know yet whether Crit M is a homotopy invariant of M. This makes the research of Crit a challenging problem.In particular, we pose the following question: given a map f:M→N of degree 1 of closed manifolds, is it true that CritM≥CritN? We prove that this holds in dimension 3 or less. Some high dimension examples are considered. Note also that an affirmative answer to the question implies the homotopy invariance of Crit; this simple observation is a good motivation for the research.

O razreshimosti lineynykh differentsial'nykh operatorov na vektornykh rassloeniyakh nad mnogoobraziem

Necessary and sufficient condition is established for the closedness of the rangeor surjectivity of a differential operator acting on smooth sections of vector bundles. For connectednoncompact manifolds it is shown that these conditions are derived from the regularityconditions and the unique continuation property of solutions. An application of these results toelliptic operators (more precisely, to operators with a surjective principal symbol) with analyticcoefficients, to second-order elliptic operators on line bundles with a real leading part, and to theHodge–Laplace–de Rham operator is given. It is shown that the top de Rham (respectively, Dolbeault)cohomology group on a connected noncompact smooth (respectively, complex-analytic)manifold vanishes. For elliptic operators, we prove that solvability in smooth sections impliessolvability in generalized sections.

Bumpy metrics theorem for geodesic nets

Stationary geodesic networks are the analogs of closed geodesics whose domain is a graph instead of a circle. We prove that for a Baire-generic Riemannian metric on a smooth manifold [Formula: see text], all connected embedded stationary geodesic nets are nondegenerate.

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.

The Mapping Class Group of Manifolds Which Are Like Projective Planes

Abstract In this paper, we compute the mapping class group of simply-connected closed smooth manifolds $M$ with integral homology $H_{*}(M) \cong {\mathbb {Z}} \oplus {\mathbb {Z}} \oplus {\mathbb {Z}}$ provided that $\dim M \ne 4$.

Algebraic Structures on Smooth Vector Fields

The aim of this work is to investigate some algebraic structures of objects which are defined and related to a manifold. Consider L to be a smooth manifold and Γ∞(TL) to be the module of smooth vector fields over the ring of smooth functions C∞(L). We prove that the module Γ∞(TL) is projective and finitely generated, but it is not semisimple. Therefore, it has a proper socle and nonzero Jacobson radical. Furthermore, we prove that this module is reflexive by showing that it is isomorphic to its bidual. Additionally, we investigate the structure of the Lie algebra of smooth vector fields. We give some questions and open problems at the end of the paper. We believe that our results are important because they link two different disciplines in modern pure mathematics.