Smooth Manifold Research Articles

Positive scalar curvature on manifolds with fibered singularities

Abstract A (compact) manifold with fibered 𝑃-singularities is a (possibly) singular pseudomanifold M Σ M_{\Sigma} with two strata: an open nonsingular stratum M ̊ \mathring{M} (a smooth open manifold) and a closed stratum β ⁢ M \beta M (a closed manifold of positive codimension), such that a tubular neighborhood of β ⁢ M \beta M is a fiber bundle with fibers each looking like the cone on a fixed closed manifold 𝑃. We discuss what it means for such an M Σ M_{\Sigma} with fibered 𝑃-singularities to admit an appropriate Riemannian metric of positive scalar curvature, and we give necessary and sufficient conditions (the necessary conditions based on suitable versions of index theory, the sufficient conditions based on surgery methods and homotopy theory) for this to happen when the singularity type 𝑃 is either Z / k \mathbb{Z}/k or S 1 S^{1} , and 𝑀 and the boundary of the tubular neighborhood of the singular stratum are simply connected and carry spin structures. Along the way, we prove some results of perhaps independent interest, concerning metrics on spin𝑐 manifolds with positive “twisted scalar curvature,” where the twisting comes from the curvature of the spin𝑐 line bundle.

Bifurcation of Piecewise Smooth Manifolds from 3D Center-Type Vector Fields

Bifurcation of Piecewise Smooth Manifolds from 3D Center-Type Vector Fields

Gelfand duality for manifolds, and vector and other bundles

In general terms, Gelfand duality refers to a correspondence between a geometric, topological, or analytical category, and an algebraic category. For example, in smooth differential geometry, Gelfand duality refers to the topological embedding of a smooth manifold in the topological dual of its algebra of smooth functions. This is generalised here in two directions. First, the topological embeddings for manifolds are generalised to the cases of real analytic and Stein manifolds, using a unified cohomological argument. Second, this type of duality is extended to vector bundles, affine bundles, and jet bundles by using suitable classes of functions, the topological duals in which the embeddings take their values.

Geometric Hodge filtered complex cobordism

We construct a geometric cycle model for a Hodge filtered extension of complex cobordism for every smooth manifold with a filtration on its de Rham complex with complex coefficients. Using a refinement of the Pontryagin–Thom construction, we construct an explicit isomorphism between our geometric model and the abstract model of Hodge filtered complex cobordism of Hopkins–Quick for every complex manifold with the Hodge filtration.

Controlling electromagnetic surface waves with conformal transformation optics

The application of transformation optics to the development of intriguing electromagnetic devices can produce weakly anisotropic or isotropic media with the assistance of quasi-conformal and/or conformal mapping, as opposed to the strongly anisotropic media produced by general mappings; however, it is typically limited to two-dimensional applications. By addressing the conformal mapping between two manifolds embedded in three-dimensional space, we demonstrate that electromagnetic surface waves can be controlled without introducing singularity and anisotropy into the device parameters. Using fruitful surface conformal parameterization methods, a near-perfect conformal mapping between smooth manifolds with arbitrary boundaries can be obtained. Illustrations of concealing and illusions, including surface Luneburg and Eaton lenses and black holes for surface waves, are provided. Our work brings the manipulation of surface waves at microwave and optical wavelengths one step closer.

Geometric learning for computational mechanics, Part III: Physics-constrained response surface of geometrically nonlinear shells

This paper presents a graph-manifold iterative algorithm to predict the configurations of geometrically exact shells subjected to external loading. The finite element solutions are first stored in a weighted graph where each graph node stores the nodal displacement and nodal director. This collection of solutions is embedded onto a low-dimensional latent space through a graph isomorphism encoder. This graph embedding step reduces the dimensionality of the nonlinear data and makes it easier for the response surface to be constructed. The decoder, in return, converts an element in the latent space back to a weighted graph that represents a finite element solution. As such, the deformed configuration of the shell can be obtained by decoding the predictions in the latent space without running extra finite element simulations. For engineering applications where the shell is often subjected to concentrated loads or a local portion of the shell structure is of particular interest, we use the solutions stored in a graph to reconstruct a smooth manifold where the balance laws are enforced to control the curvature of the shell. The resultant computer algorithm enjoys both the speed of the nonlinear dimensional reduced solver and the fidelity of the solutions at locations where it matters.

Scaled homology and topological entropy

In this paper, we build up a scaled homology theory, l c lc -homology, for metric spaces such that every metric space can be visually regarded as “locally contractible” with this newly-built homology. We check that l c lc -homology satisfies all Eilenberg-Steenrod axioms except the exactness axiom whereas its corresponding l c lc -cohomology satisfies exactness axiom for cohomology. This homology can relax the smooth manifold restrictions on the compact metric space such that the entropy conjecture will hold for the first l c lc -homology group.

Ordered exponential and its features in Yang–Mills effective action

In this paper, we discuss some non-trivial relations for ordered exponentials on smooth Riemannian manifolds. As an example of application, we study the dependence of the four-dimensional quantum Yang–Mills effective action on the special gauge transformation with respect to the background field. Also, we formulate some open questions about a structure of divergences for a special type of regularization in the presence of the background field formalism.

Reflexivity of the Space of Transversal Distributions

For any smooth, Hausdorff and second-countable manifold N one can define the Fréchet space mathord {mathcal {C}^{infty }}(N) of smooth functions on N and its strong dual mathcal {E}'(N) of compactly supported distributions on N. It is a standard result that the strong dual of mathcal {E}'(N) is naturally isomorphic to mathord {mathcal {C}^{infty }}(N), which implies that both mathord {mathcal {C}^{infty }}(N) and mathcal {E}'(N) are reflexive locally convex spaces. In this paper we generalise that result to the setting of transversal distributions on the total space of a surjective submersion pi :Prightarrow M. We show that the strong mathord {mathcal {C}^{infty }_{c}}(M)-dual of the space mathcal {E}'_{pi }(P) of pi -transversal distributions is naturally isomorphic to the mathord {mathcal {C}^{infty }_{c}}(M)-module mathord {mathcal {C}^{infty }}(P).

Homology Spheres Bounding Acyclic Smooth Manifolds and Symplectic Fillings

In this paper, we collect various structural results to determine when an integral homology 3-sphere bounds an acyclic smooth 4-manifold, and when this can be upgraded to a Stein manifold. In a different direction, we study whether a smooth embedding of connected sums of lens spaces in C2 can be upgraded to a Stein embedding, and we have determined that this never happens.

Determining Lamé coefficients by the elastic Dirichlet-to-Neumann map on a Riemannian manifold

For the Lamé operator with variable coefficients λ and µ on a smooth compact Riemannian manifold (M, g) with smooth boundary , we give an explicit expression for the full symbol of the elastic Dirichlet-to-Neumann map . We show that uniquely determines the partial derivatives of all orders of the Lamé coefficients λ and µ on . Moreover, for a nonempty smooth open subset , suppose that the manifold and the Lamé coefficients are real analytic up to Γ, we prove that uniquely determines the Lamé coefficients on the whole manifold .

On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds

We introduce new metric structures on a smooth manifold (called “weak” structures) that generalize the almost contact, Sasakian, cosymplectic, etc. metric structures (φ,ξ,η,g) and allow us to take a fresh look at the classical theory and find new applications. This assertion is illustrated by generalizing several well-known results. It is proved that any Sasakian structure is rigid, i.e., our weak Sasakian structure is homothetically equivalent to a Sasakian structure. It is shown that a weak almost contact structure with parallel tensor φ is a weak cosymplectic structure and an example of such a structure on the product of manifolds is given. Conditions are found under which a vector field is a weak contact vector field.

Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory

Given a pair of normally hyperbolic operators over (possibnly different) globally hyperbolic spacetimes on a given smooth manifold, the existence of a geometric isomorphism, called Møller operator, between the space of solutions is studied. This is achieved by exploiting a new equivalence relation in the space of globally hyperbolic metrics, called paracausal relation. In particular, it is shown that the Møller operator associated to a pair of paracausally related metrics and normally hyperbolic operators also intertwines the respective causal propagators of the normally hyperbolic operators and it preserves the natural symplectic forms on the space of (smooth) initial data. Finally, the Møller map is lifted to a *-isomorphism between (generally off-shell) CCR-algebras. It is shown that the Wave Front set of a Hadamard bidistribution (and of a Hadamard state in particular) is preserved by the pull-back action of this *-isomorphism.

Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems

This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct a 2-dimensional “integral affine black hole”, which is locally convex but for which a straight ray from the center can never escape.

INVARIANT FORMS OF GEODESIC, POTENTIAL, AND DISSIPATIVE SYSTEMS ON TANGENT BUNDLE OF FINITE-DIMENSIONAL MANIFOLD

As is well-known [1–3], finding a sufficient number of tensor invariants (not only the first integrals) allows you to accurately integrate a system of differential equations. For example, the presence of an invariant differential form of the phase volume makes it possible to reduce the number of required first integrals. For conservative systems, this fact is natural, but for systems with attractive or repulsive limit sets, not only some first integrals, but also the coefficients of the available invariant differential forms should, generally speaking, include functions with essentially special points (see also [4–6]). In this paper, complete sets of invariant differential forms for homogeneous systems on tangent bundles to smooth finite-dimensional manifolds are presented for the class of dynamical systems under consideration.

Error bounds for a least squares meshless finite difference method on closed manifolds

We present an error bound for a least squares version of the kernel based meshless finite difference method for elliptic differential equations on smooth compact manifolds of arbitrary dimension without boundary. In particular, we obtain sufficient conditions for the convergence of this method. Numerical examples are provided for the equation -Delta _{mathcal {M}} u + u = f on the 2- and 3-spheres, where Delta _{mathcal {M}} is the Laplace-Beltrami operator.

Three Lagrangians for the complete-active space coupled-cluster method.

Three fully variational formulations of the complete-active space coupled-cluster method are derived. The formulations include the ability to approximate the model vectors by smooth manifolds, thereby opening up the possibility for overcoming the exponential wall of scaling for model spaces of complete-active space type. In particular, model vectors of matrix-product states are considered, and it is argued that the present variational formulation allows not only favorably scaling multireference coupled-cluster calculations but also systematic correction of tailored coupled-cluster calculations and of quantum chemical density-matrix renormalization group methods, which are fast and polynomial scaling but lack the ability to properly resolve dynamical correlation at chemical accuracy. The extension of the variational formulations to the time domain is also discussed, with derivations of abstract evolution equations.

A lower bound on the quantitative version of the transversality theorem

The present paper establishes a lower estimate on the quantitative version of the transversality theorem. More precisely, given a continuous function f:[0,1]d→Rm and a smooth manifold W⊂Rm of dimension p, a sharpness result on the upper quantitative bound of the (d+p−m)-dimensional Hausdorff measure of ZWf≐{x∈[0,1]d:f(x)∈W}, which was achieved in [11], will be proved in terms of power functions.

Continuity of families of Calderón projections

We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner solutions, we prove that the orthogonalized Calderón projections of the underlying family of elliptic operators form a continuous family of projections. Hence, its images (the Cauchy data spaces) form a continuous family of closed subspaces in the relevant Sobolev spaces. We use only elementary tools and classical results: basic manipulations of operator graphs and other closed subspaces in Banach spaces, elliptic regularity, Green's formula and trace theorems for Sobolev spaces, well-posed boundary conditions, duality of spaces and operators in Hilbert space, and the interpolation theorem for operators in Sobolev spaces.

Global functional calculus, lower/upper bounds and evolution equations on manifolds with boundary

Given a smooth manifold M (with or without boundary), in this paper we establish a global functional calculus, without the standard assumption that the operators are classical pseudo-differential operators, and the Gårding inequality for global pseudo-differential operators associated with boundary value problems. The analysis that we follow is free of local coordinate systems. Applications of the Gårding inequality to the global solvability for a class of evolution problems are also considered.