We present a boundary version of a theorem about solenoidal unit vector fields with minimum energy on a spherical domain of an odd dimensional Euclidean sphere.
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48 results for “solenoidal vector fields”
Study on vector field relations on compact manifolds.
problem Understanding vector field connections on compact manifolds.
method Integration over compact manifolds to investigate Killing, Jacobi, and solenoidal fields.
result Differences in vector field relationships between compact and noncompact manifolds.
Study on Einstein solitons with specific vector fields and their properties.
problem Characterizing Einstein solitons with gradient, solenoidal, or concircular vector fields.
method Explicitly express the function λ by gradient vector field V and deduce geometric properties under certain curvature conditions.
result Explicit expressions for λ and geometric properties of Einstein solitons.
In this paper, we define a certain "proportional volume property" for an unit vector field on a spherical domain in S3. We prove that the volume of these vector fields has an absolute minimum and this value is equal to the volume of the Hopf vector field. Some examples of such vector fields are given. We also study the…
Reconstruction formulas for geodesic X-ray transforms on negatively curved surfaces.
problem Inverting geodesic X-ray transforms on surfaces with negative curvature.
method Generalizes Pestov-Uhlmann formulas to surfaces with trapping, using Fredholm equations and new estimates for the normal operator.
result Reconstruction formulas for geodesic X-ray transforms on surfaces with negative curvature.
Study on almost Riemann solitons with gradient or torse-forming vector fields.
problem Characterizing almost Riemann solitons with specific vector fields.
method Using Bochner formula and properties of gradient and torse-forming vector fields.
result Explicit expressions for the soliton function under gradient and torse-forming conditions.
Study on solitons in deformed Kenmotsu manifolds with specific vector fields.
problem Analyzing geometric solitons in deformed Kenmotsu manifolds.
method Examined almost Riemann and Ricci solitons in a -homothetically deformed Kenmotsu manifold with specific vector fields.
result Explicitly obtained Ricci and scalar curvatures for some cases, provided a lower bound for Ricci curvature.
Characterizes winding of braided vector fields in tubular domains.
problem Understanding the topology of braided vector fields in complex domains.
method Defines field line winding as a measure of entanglement, proving its uniqueness in classifying vector field topology.
result Field line winding uniquely classifies the topology of braided vector fields.
Two new methods for injective ray transforms of tensor fields on surfaces are introduced.
problem Injective ray transforms for tensor fields on surfaces.
method Two approaches: weights varying along geodesic flow and transverse ray transforms.
result Constructive solutions for tensor tomography on simple surfaces.
We present a numerical implementation of the geodesic ray transform and its inversion over functions and solenoidal vector fields on two-dimensional Riemannian manifolds. For each problem, inversion formulas previously derived in \cite{Pestov2004,Krishnan2010} are implemented in the case of simple and some non-simple m…
Compact aspherical solenoidsmath.GT
We consider compact, aspherical solenoids obtained as the inverse limit of a system of CW~complexes and covering maps. This includes -adic solenoids, as well as the universal hyperbolic solenoid of Teichmüller theory. Using ideas from shape theory, we classify maps between such solenoids up to homotopy, and we prove…
We introduce the concept of solenoid as an abstract laminated space. We do a thorough study of solenoids, leading to the notion of ergodic and uniquely ergodic solenoids. We define generalized currents associated with immersions of oriented solenoids with a transversal measure into smooth manifolds, generalizing Ruelle…
We show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long standing inverse problem in the two-dimensional case.
A measured solenoid is a laminated space endowed with a tranversal measure invariant by holonomy, as defined in arXiv:0910.2836. A measured solenoid immersed in a smooth manifold produces a closed current (known as generalized Ruelle-Sullivan current). Uniquely ergodic solenoids are those for which there is a unique (u…
The spectrum of a solenoidmath.SP
Study the spectrum of covering solenoids from manifold sequences.
problem Understanding the spectrum of covering solenoids from manifold sequences.
method Examined the leaf-wise Laplacian on covering solenoids and related it to the spectra of individual manifolds in the sequence.
result The spectrum of the Laplacian on the covering solenoid equals the closure of the union of the spectra of the manifolds in the sequence.
Embeds Higson compactification into adelic solenoids.
problem Embedding Higson compactification into solenoids.
method Induces isomorphism of 1-dimensional cohomology.
result Higson compactification embeds into adelic solenoids.
A measured solenoid is a compact laminated space endowed with a transversal measure. The De Rham -cohomology of the solenoid is defined by using differential forms which are smooth in the leafwise directions and in the transversal direction. We develop the theory of harmonic forms for Riemannian measured sol…
When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give…
Study of fundamental groups of knotted solenoid complements in 3D sphere.
problem Determining fundamental groups of knotted solenoid complements.
method Using canonical sequence of knot groups and embedding up to mirror reflection.
result Fundamental groups of knotted solenoid complements are solely determined by a sequence of knot groups and embedding up to mirror reflection.
The paper explores low-dimensional solenoidal manifolds and their properties.
problem Characterizing and understanding solenoidal manifolds of dimensions 1, 2, and 3.
method Survey and new results about solenoidal manifolds, using theorems of A. Clark and S. Hurder.
result Topologically homogeneous, compact solenoidal manifolds are McCord solenoids and behave like laminated versions of compact manifolds.
Study proves solenoidal injectivity for tensor fields on curved manifolds with low regularity.
problem Injectivity for tensor fields on negatively curved manifolds with low regularity metrics.
method Pestov energy estimates for transport equation on non-smooth unit sphere bundle, keeping track of regularity, and using functions with more vertical than horizontal regularity.
result Proves solenoidal injectivity for tensor fields on simple Riemannian manifolds with metrics and non-positive sectional curvature.
Injectivity of X-ray transform on solenoidal tensors proven for certain manifolds.
problem Injectivity of X-ray transform on solenoidal tensors on manifolds with hyperbolic trapped set.
method Proof of equivalence principle concerning injectivity and surjectivity of operators on symmetric solenoidal tensors.
result Injectivity of X-ray transform on solenoidal tensors of any order for surfaces with hyperbolic trapped set.
Solenoids are ``inverse limits'' of the circle, and the classical knot theory is the theory of tame embeddings of the circle into the 3-space. We give some general study, including certain classification results, of tame embeddings of solenoids into the 3-space as the ``inverse limits'' of the tame embeddings of the ci…
Geodesic ray transform links solenoidal injectivity to invariant distributions.
problem Injectivity of geodesic ray transform on symmetric tensors.
method Equivalence principle between solenoidal injectivity and invariant distributions.
result Existence of invariant distributions for geodesic ray transform.
Proves magnetic gap-labelling conjecture for specific tori.
problem Magnetic spectral gap-labelling conjecture for principal solenoidal tori.
method Proves conjecture in all dimensions for principal solenoidal tori.
result Proves magnetic spectral gap-labelling conjecture.
Mostow rigidity proven for special geometric manifolds.
problem Proving rigidity for specific geometric structures.
method Analyzing foliated bundles over closed hyperbolic manifolds with invariant measures.
result Mostow rigidity theorem extended to skew solenoidal manifolds.
Abstract commensurators linked to topological models of solenoids.
problem Understanding abstract commensurators through topological models.
method Relating homotopy equivalences of full solenoids to abstract commensurators.
result Isomorphism between homotopy equivalences and abstract commensurators in specific cases.
We develop the intersection theory associated to immersed, oriented and mea- sured solenoids, which were introduced in arXiv:0910.2836.
We define generalized currents associated with immersions of abstract oriented solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geo…
A quick proof of Bing's theorem indicated by the title is given. The proof also concludes Gumerov's result on covering degrees of solenoids.
Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? T…
New proof for solenoidal laminations with symmetric leaves in dimensions other than 4.
problem Characterizing Riemannian solenoidal laminations with symmetric leaves.
method Proving homeomorphism to inverse limits of finite covers of compact locally-symmetric manifolds.
result Each lamination is homeomorphic to a specific inverse limit structure.
Using alternating Heegaard diagrams, we construct some 3-manifolds which admit diffeomorphisms such that the non-wandering sets of the diffeomorphisms are composed of Smale-Williams solenoid attractors and repellers, an interesting example is the truncated-cube space. In addition, we prove that if the nonwandering set …
Ergodic solenoidal homologymath.DG
We define generalized currents associated with immersions of abstract solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De…
We extend Schwartzman theory beyond dimension 1 and provide a unified treatment of Ruelle-Sullivan and Schwartzman theories via Birkhoff's ergodic theorem for the class of immersions of solenoids with a trapping region.
Study geodesic X-ray transform on curved spaces, proving injectivity for decaying functions.
problem Injectivity of geodesic X-ray transform on curved spaces.
method Proving injectivity for decaying functions and tensor fields of any order.
result Injectivity of the geodesic X-ray transform for functions and tensor fields of any order on Cartan-Hadamard manifolds.
The space of non-singular flows on any given solenoid is shown to contain a generic subset consisting of flows that are not almost periodic. Whether this result carries over to Hamiltonian flows remains an open question.
Study classifies 3D superflows with icosahedral symmetry.
problem Classifying 3D superflows with specific symmetry groups.
method Analyzes projective flows with rational vector fields and specific symmetry groups.
result Identifies all 3D superflows with icosahedral symmetry.
Earthquakes and Thurston's boundary for the Teichmüller space of the universal hyperbolic solenoidmath.CV
A measured laminations on the universal hyperbolic solenoid is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on theuniversal hyperbolic solenoid is uniquely determined by a measured lamination on ; it is a leafwise earthquake with…
We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, i…
Previous work of the author has developed coordinates on bundles over the classical Teichmueller spaces of punctured surfaces and on the space of cosets of the Moebius group in the group of orientation-preserving homeomorphisms of the circle, and this work is surveyed here. Joint work with Dragomir Saric is also sketch…
On Radon transforms on torimath.DG
We show injectivity of the X-ray transform and the -plane Radon transform for distributions on the -torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvière. We also show solenoidal injectivity of the X-ray transform on the -torus for tensor fields of any order, allowing the tensor…
Study geodesic X-ray transforms on hyperbolic surfaces, proposing new reconstruction methods.
problem Inverting geodesic X-ray transforms for symmetric tensor fields on asymptotically hyperbolic surfaces.
method Developed a decomposition theorem for m-tensor fields, used Guillemin-Kazhdan operators and 0-calculus, and provided explicit reconstruction methods.
result Explicit reconstruction methods for even tensor fields from their X-ray transform or normal operator.
Study shows certain manifolds uniquely determined by X-ray data.
problem Determining manifolds from X-ray data.
method Injectivity of X-ray transform over symmetric solenoidal 2-tensors.
result Smooth compact manifolds with strictly convex boundary, no conjugate points, and hyperbolic trapped set are locally marked boundary rigid.
In this paper we find smooth embeddings of solenoids in smooth foliations. We show that if a smooth foliation F of a manifold M contains a compact leaf L with H^1(L;R)= 0 and if the foliation is a product foliation in some saturated open neighbourhood U of L, then there exists a foliation F' on M which is C^1-close to …
A remark about critical sets in math.GT
Abstract: Necessary condition for critical sets in 3D space.
problem Identifying critical sets in 3D space.
method Provided a necessary condition for closed subsets to be critical sets of smooth functions.
result Examples like the Whitehead continuum and p-adic solenoid are not critical sets.
We give a presentation for the baseleaf preserving mapping class group of the punctured solenoid . The generators for our presentation were introduced previously, and several relations among them were derived. In addition, we show that has no non-trivial central elements. Our main tool is a new com…
Study characterizes X-ray transform kernel for periodic slabs and related manifolds.
problem Characterizing the kernel of X-ray transform for tensor fields on periodic slabs.
method Characterization of the kernel for -regular -tensors on .
result Kernel characterization extends to more general manifolds, including the Möbius strip.