Ricci flow smooths locally collapsing manifolds with controlled curvature.
problem Locally collapsing manifolds with controlled Ricci curvature.
method Ricci flow for a definite period of time, detecting collapsing infranil fiber bundles.
result Topological conditions detect collapsing infranil fiber bundles.
The paper proves properties of non-collapsed RCD spaces with bounded covering geometry.
problem Characterizing properties of non-collapsed RCD spaces.
method Analyzing local covering geometry and applying Gromov's almost flat manifold theorem.
result RCD spaces with bounded covering geometry are biHölder homeomorphic to infranil-manifolds.
Introduces Darboux-Lie derivative for fiber bundles.
problem None explicitly stated; focuses on introducing a new derivative.
method Study of Darboux-Lie derivative for fiber-bundle maps.
result Properties of Darboux-Lie derivative for fiber bundles.
Investigates conditions for GKM fiber bundles and realizability of fiber bundles of GKM graphs.
problem Conditions for GKM fiber bundles and realizability of fiber bundles of GKM graphs.
method Analysis of GKM graphs and fiber bundles, counterexamples, and classification of twist automorphisms.
result Realizability of fiber bundles of GKM graphs depends on the twist automorphism and can be decided in terms of the classification.
New examples show not all homology fiber bundles are topological.
problem Disprove conjecture about homology fiber bundles.
method Construct flat, projective morphisms that are Z-homology fiber bundles. result Disprove conjecture about homology fiber bundles.
New exotic 4-manifolds found with fiber bundles.
problem Finding non-diffeomorphic exotic 4-manifolds.
method Smooth fiber bundles over the circle.
result Exotic 4-manifolds with non-simply-connected fiber.
The paper proves non-existence of positive scalar curvature on certain fiber bundles.
problem The existence of positive scalar curvature metrics on fiber bundles.
method Analyzing fiber bundles over specific manifolds with incompressible or homotopically nontrivial fibers.
result Non-existence of PSC metrics on certain fiber bundles under specific conditions.
The notion of a (stably) decomposable fiber bundle is introduced. In low dimensions, for torus fiber bundles over a circle the notion translates into a property of elements of the special linear group of integral matrices. We give a complete characterization of the stably decomposable torus fiber bundle of fiber-dimens…
The paper constructs trisections for fiber bundles over a circle.
problem Building trisections for fiber bundles over the circle.
method Using sutured Heegaard splittings and diagrams, the paper provides an algorithm to construct relative trisection diagrams.
result Explicit construction of relative trisection diagrams for fiber bundles over the circle.
Study simplicial volume in fiber bundles with connected groups.
problem Understanding simplicial volume in fiber bundles.
method Investigated fiber bundles with connected structure groups.
result Simplicial volume of total space matches trivial bundle under certain conditions.
Study characterizes martingales on fiber bundles for harmonic map analysis.
problem Characterizing martingales on fiber bundles for harmonic maps.
method Investigated ablaP-martingales on principal fiber bundles P(M,G) with a projectable connection. result New characterizations of harmonic maps established.
Study shows simplicial volume of certain fiber bundles is zero.
problem Understanding simplicial volume in fiber bundles with nonpositive curvature.
method Proved simplicial volume is zero for specific fiber bundles.
result Simplicial volume of certain fiber bundles is zero.
Introduces generalized principal bundles and connections, linking them to standard gauge theories.
problem Generalized principal bundles and connections in field theories.
method Local coordinate transformation laws and horizontal lifts.
result Generalized principal connections are associated to Lie group fiber bundle connections.
Base of fibered correspondence is arbitrary correspondence. Fibered correspondence is interesting when we consider relationship between different bundles. However composition of fibered correspondences may not always be defined. Reduced fibered correspondence is defined only between fibers over the same point of base. …
Kontsevich's classes distinguish smooth structures on fiber bundles.
problem Distinguishing smooth structures on fiber bundles.
method Using Kontsevich's characteristic classes and real blow-up construction.
result Kontsevich's classes are determined by the topology of the 2-point configuration space bundle.
Generalized complex structures on certain torus bundles are explored.
problem Exploring generalized complex structures on specific torus bundles.
method Analyzing principal torus bundles over complex manifolds with even dimensional fibers and characteristic class of type (1,1).
result Generalized complex structures on these bundles are equivalent to products of complex and symplectic structures in tubular neighborhoods of fibers.
Defines fiber-wise linear differential operators on vector bundles.
problem No specific problem stated; focuses on definition and equivalence.
method Definition and equivalence of fiber-wise linear differential operators to derivations of line bundles.
result Equivalence of fiber-wise linear differential operators to derivations of line bundles.
A nonassociative generalization of the principal fiber bundles with a smooth loop mapping on the fiber is presented. Our approach allows us to construct a new kind of gauge theories that involve higher ''nonassociative'' symmetries.
The study classifies Kähler threefolds with special fiber bundles.
problem Classifying Kähler threefolds with specific fiber bundles.
method Generalized from projective case, using fiber bundles over the circle and étale covers.
result Proves Kotschick's conjecture in dimension 3.
Let P→M be a principal bundle. Consider a sequence of metrics on P obtained by re-scaling the fibers to points. The Gromov-Hausdorff limit of the tangent bundles over these principal bundles with their Sasaki metric is seen herein to be a locally trivial fiber bundle containing the tangent space to the base as a…
The paper studies mapping class groups of nontrivial S2 fiber bundles.
problem Analyzing the mapping class groups of nontrivial S2 fiber bundles. method Using generalizations of Dax invariants for embedded surfaces in 4-manifolds.
result Surjective homomorphisms from MCG(X) and MCG(X′) to Z∞ are shown. A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theor…
The study examines connections and their curvatures on different types of bundles.
problem Understanding connections and curvatures on various bundle types.
method Analysis of connections and curvatures on fiber, principal, and vector smooth bundles.
result Investigations into the relationships between connections and curvatures on different bundle types.
This text explains how fiber bundle structure is fundamental for classical physics.
problem None explicitly stated, but implied as understanding fiber bundles is crucial for physics.
method Explains the fiber bundle structure and its universality for physics laws.
result Fiber bundle structure is fundamental for classical physics laws.
We define contact fiber bundles and investigate conditions for the existence of contact structures on the total space of such a bundle. The results are analogous to minimal coupling in symplectic geometry. The two applications are construction of K-contact manifolds generalizing Yamazaki's fiber join construction and a…
Surjectivity of Cannon-Thurston map proven for metric graph bundles.
problem Proving surjectivity of Cannon-Thurston map in metric graph bundles.
method Generalized Mj-Sardar's result to include more types of fibers.
result Continuous extension map between boundaries is surjective.
In this note we show that if a compact Kahler manifold with trivial canonical bundle is the total space of a holomorphic fibration without singular fibers, then the fibration is a holomorphic fiber bundle. In the algebraic case, the fibration becomes trivial after a finite base change.
We prove a stability result for volume forms on fiber bundles with compact base and noncompact fibers. This generalizes the classical results of Moser and Greene--Shiohama, and recent work by the authors.
In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each n≥3 we construct 4-manifolds E admitting at least n distinct fiberings pi:E→Σgi as a surface bundle over a surface with base and fiber both closed surfaces of negative Eule…
FiberNet integrates geometry into machine learning for clearer classification.
problem Lack of interpretability in traditional deep learning.
method Reformulates classification as geometric optimization on fiber bundles, introducing learnable Riemannian metrics and variational prototype optimization.
result Clear geometric interpretability and efficiency in classification.
The paper proves conditions for self-covering manifolds to be fiber bundles over a circle.
problem Conditions for self-covering manifolds to be fiber bundles over a circle.
method Developed a criterion for finite generation of modules over a commutative Noetherian ring and proved conditions for fiber bundles.
result Proved that certain self-covering manifolds with free abelian fundamental group are fiber bundles over a circle.
In our previous papers [Far East Journal of Mathematical Sciences, 35 (2009), 211-223] and [International Journal of Pure and Applied Mathematics, 60 (2010), 15-24] we have developed the theory of Weil prolongation, Weil exponentiability and microlinearity for Frolicher spaces. In this paper we will relativize it so as…
For a closed manifold M, let Fib(M) be the number of distinct fiberings of M as a fiber bundle with fiber a closed surface. In this paper we give the first computation of Fib(M) where 1<Fib(M)<∞ but M is not a product. In particular, we prove Fib(M)=2 for the Atiyah-Kodaira manifold and any fi…
We investigate when the idempotent barycenter map restricted to the points with no-trivial fibers is a trivial bundle with the fiber Hilbert cube.
Study shows how certain curved bundles reduce their structure group.
problem Understanding how curvature affects bundle structure.
method Analyzes bundles with fibers as real projective spaces and positive curvature metrics.
result Structure group of bundles reduces when fibers have specific curvature properties.
We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifold…
We study the Teichmüller space of negatively curved metrics on a high dimensional manifold, with applications to bundles with negatively curved fibers.
Mathematical theory of super fiber bundles and connections developed.
problem Modeling anticommuting fermionic fields in mathematical physics.
method Detailed introduction to super fiber bundles, relative supermanifolds, and connections; construction of parallel transport map.
result Construction and comparison of parallel transport map with other methods in the literature.
New gradient Ricci solitons found via fiber bundles.
problem Existence of gradient Ricci solitons.
method Warped metrics and fiber bundles.
result Necessary and sufficient conditions for existence.
Proves 3-manifold groups uniquely identify hyperbolic bundles.
problem Identifying hyperbolic 3-manifold groups from their finite quotients.
method Upgraded Liu's result to detect fiber type via profinite completion.
result Proves hyperbolic bundles are distinguished by their profinite completions.
Bundle Networks explore many-to-one maps using fiber bundles and local trivializations.
problem Exploring and sampling from fibers of many-to-one maps in machine learning.
method Introducing Bundle Networks based on fiber bundles and local trivializations, using invertible components.
result Exploring fibers of many-to-one maps becomes natural in Bundle Networks.
Fiber bundles over hyperbolic manifolds with Anosov representations.
problem Understanding the topology of fiber bundles associated with Anosov representations.
method Analyzing the structure of fiber bundles over specific hyperbolic manifolds.
result Fiber bundles over certain hyperbolic manifolds are always smooth fiber bundles.
Theorems prove upper bounds for foliations on closed Alexandrov spaces.
problem Upper bounds for the dimension of isometry groups of metric foliations.
method Proving theorems about isometries and foliations on closed Alexandrov spaces.
result Sharp upper bounds for the dimension of the isometry group of metric foliations.
Counterexample disproves conjecture on flat metrics and fiber bundles.
problem Conjecture about flat metrics and fiber bundles on manifolds.
method Study of transversely flat Riemannian foliations.
result Found a counterexample to the conjecture.
New method for decomposing manifolds into 1-handlebodies.
problem Decomposing manifolds into 1-handlebodies.
method Multisections of surface bundles and fiber bundles over the circle.
result Multisections exist for surface bundles and fiber bundles over the circle.
A locally conformally symplectic (LCS) form is an almost symplectic form ω such that a closed one-form θ exists with dω=θ∧ω. A fiber bundle with LCS fiber (F,ω,θ) is called LCS if the transition maps are diffeomorphisms of F preserving ω (and hence θ). In this paper, we find conditions for the total…
Study of semi-principal bundles using group actions and wreath products.
problem Understanding bundles with fibers as free G-spaces. method Defining semi-principal bundles, bases, and frame bundles; using wreath products and functors.
result Semi-principal bundles can be retracted to principal bundles, preserving parallel transport.
A principal toric bundle M is a complex manifold equipped with a free holomorphic action of a compact complex torus T. Such a manifold is fibered over M/T, with fiber T. We discuss the notion of positivity in fiber bundles and define positive toric bundles. Given an irreducible complex subvariety X⊂M o…