Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.
problem Characterizing equivariant vector bundles over toric manifolds.
method Analyzes topological and smooth equivariant vector bundles over toric manifolds.
result Every equivariant vector bundle is a Klyachko bundle.
Uniformizes compact Sasakian manifolds into circle bundles.
problem Deforming compact Sasakian manifolds to locally isomorphic forms.
method Criterion based on circle bundles of anti-canonical bundles over Hermitian symmetric spaces.
result Sasakian manifolds can be deformed to locally isomorphic forms.
Study compact Kähler manifolds with pseudo-effective tangent bundles.
problem Characterize compact Kähler manifolds with strongly pseudo-effective tangent bundles.
method New proofs and characterizations of properties of vector bundles.
result Only projective spaces have big tangent bundles.
The paper geometrizes N-manifolds using symmetric vector bundles.
problem Geometrizing N-manifolds with a specific symmetry.
method Equivalence between [n]-manifolds and symmetric n-fold vector bundles. result Identical cocycles between symmetric vector bundles and [n]-manifolds. Investigates flat bundles over low-dimensional manifolds and their cobordism classes.
problem The cobordism of flat bundles over low-dimensional manifolds.
method Study of flat M-bundles over low-dimensional manifolds, comparing a finite dimensional Lie group G with extDiff0(G) and localizing the holonomy. result Flat M-bundles over low-dimensional manifolds are cobordant to a flat M-bundle. Sphere bundles over 4-manifolds are trivial after looping, except for two cases.
problem Understanding the triviality of sphere bundles over 4-manifolds.
method Analyzing the splitting of sphere bundles after looping.
result The loop spaces of total manifolds of sphere bundles are homotopy equivalent, except for two special cases.
The paper extends vector bundle theory to non-Hausdorff manifolds.
problem Generalizing vector bundle theory to non-Hausdorff manifolds.
method Using Čech cohomology to classify real non-Hausdorff line bundles.
result Vector bundles over non-Hausdorff manifolds can be constructed as colimits of standard vector bundles.
This paper shows vector bundles and differential bundles are equivalent in smooth manifolds.
problem Characterizing vector bundles in smooth manifolds.
method Introducing differential bundles in a tangent category and proving equivalence with vector bundles in smooth manifolds.
result Differential bundles in smooth manifolds are equivalent to vector bundles.
Characterizes Q-manifold bundles over C-spaces.
problem Homological characterization of Q-manifolds bundles. method Homological proof of Q-manifolds bundles over C-spaces. result Provides a partial answer to Question QM22.
The paper studies geometric structures on tangent and sphere bundles over statistical manifolds.
problem Defining geometric structures on tangent and sphere bundles over statistical manifolds.
method Using a statistical structure (g,abla), the paper defines a Riemannian structure on the tangent bundle and derives expressions for various curvatures. result Basic formulas for the geometry of sphere bundles are established, and rigidity results are proved for these structures.
Extends differential geometry concepts to manifolds with super tangent bundles.
problem No specific problem stated; extending differential geometry to super tangent bundles.
method Introduces super tangent bundle and extends differential geometry concepts.
result Basic notions of differential geometry extended to manifolds with super tangent bundles.
Classifies equivariant vector bundles over toric manifolds.
problem Classifying vector bundles over toric manifolds.
method Klyachko-type classification over invariant affine charts.
result Generalizes Klyachko's classification of toric vector bundles.
Study Poisson metrics on noncompact Kähler manifolds and their Higgs bundle applications.
problem Existence of Poisson metrics on flat vector bundles over noncompact Riemannian manifolds.
method Generalization of Corlette-Donaldson-Hitchin-Simpson's nonabelian Hodge correspondence to noncompact Kähler manifolds.
result Existence of Poisson metrics on Higgs bundles over noncompact Kähler manifolds.
Study Higgs bundles and flat connections on quasi-regular Sasakian manifolds.
problem Investigate Higgs bundles and flat connections on compact Sasakian manifolds.
method Introduce quasi-regularity and regularity of vector bundles, relate to orbibundles, extend non-abelian Hodge correspondence.
result Extend non-abelian Hodge correspondence to quasi-regular Sasakian manifolds.
Study on triviality of tangent and generalized tangent bundles of manifolds.
problem Triviality of tangent and generalized tangent bundles of manifolds.
method Analyzing relations between tangent bundle TM and generalized tangent bundle TM=TM⊕T∗M of manifolds. result The generalized tangent bundle of a parallelizable manifold is trivial, but the converse is not always true.
Projective manifolds with specific bundles are isomorphic to simpler spaces.
problem Characterizing projective manifolds with tangent bundles containing strictly nef subsheaves.
method Analyzing the structure of the tangent bundle and using properties of strictly nef subsheaves.
result Projective manifolds with the described bundles are isomorphic to projective bundles over hyperbolic manifolds or projective spaces.
Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.
problem Yau-Tian-Donaldson conjecture for toric manifolds and bundles.
method Filtration approach to prove the conjecture.
result Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.
Study on stable vector bundles over Gauduchon manifolds.
problem Existence and stability of vector bundles over Gauduchon manifolds.
method Uhlenbeck--Yau's continuity method for approximate Hermitian--Einstein structures.
result Equivalence of semi-stability and existence of Hermitian--Einstein structures.
Study proves Hodge symmetry on Oeljeklaus-Toma manifolds with line bundles.
problem Hodge symmetry on complex manifolds with line bundles.
method Analyzes Dolbeault cohomology of Oeljeklaus-Toma manifolds with holomorphic line bundles.
result Proves Hodge symmetry and vanishing/non-vanishing of Dolbeault cohomology.
Spinor bundle constructed on loop space for string manifolds.
problem Constructing spinor bundles for string manifolds.
method Using ideas from Stolz and Teichner, a spinor bundle and fusion product are defined for a string manifold.
result Existence of spinor bundle with fusion product on a manifold X if and only if X admits a string structure.
In this paper, we introduce the concept of principal bundles on statistical manifolds. After necessary preliminaries on information geometry and principal bundles on manifolds, we study the α-structure of frame bundles over statistical manifolds with respect to α-connections, by giving geometric structures. The man…
We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex 3-folds of the form P(T∗X) whose tangent bundles are nef. Moreover, we show that if X is a Fano manifold such t…
Real vector bundles are determined by their Dirac indices on specific spin manifolds.
problem Determining real vector bundles using Dirac indices.
method Mapping spin or spinh manifolds into a compact smooth manifold and using Dirac indices. result Real vector bundles are uniquely determined by their Dirac indices on prescribed manifolds.
Study the structure of linear bundle morphisms between vector bundles.
problem Understanding the structure of linear bundle morphisms between vector bundles.
method Analyzing the set of smooth linear bundle morphisms between fibre bundles.
result Detailed structure of linear bundle morphisms between vector bundles.
Holomorphic vector bundles on Hopf manifolds admit flat connections.
problem Understanding flat connections on holomorphic vector bundles over Hopf manifolds.
method Defining resonant and non-resonant Mall bundles, proving the existence of flat connections on non-resonant bundles, and applying the Poincare-Dulac theorem.
result Non-resonant Hopf manifolds are linearizable, generalizing Kodaira's result.
In this paper, we introduce the notions of α-Hermitian-Einstein metric and α-stability for I±-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi-Hitchin correspondence for I±-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such vector bundles include…
Study geodesics on a modified cotangent bundle over Kählerian manifolds.
problem Investigate geodesics on a modified cotangent bundle.
method Introduced Berger-type deformed Sasaki metric, investigated Levi-Civita connections, and studied geodesics.
result Geodesic properties on modified cotangent bundles.
Study on Gauduchon manifolds finds metrics for projectively flat bundles.
problem Existence of Hermitian-Poisson metrics on projectively flat bundles.
method Heat flow techniques and continuity methods.
result Established a correspondence between Hermitian-Poisson metrics and semi-simplicity.
The paper proves conditions for minimal compact Kähler manifolds with vanishing second Chern class.
problem Conditions for minimal compact Kähler manifolds with vanishing second Chern class.
method Study of the abundance conjecture and associated Iitaka fibrations.
result For a minimal compact Kähler manifold, the second Chern class vanishes if and only if the cotangent bundle is nef and the canonical bundle has numerical dimension 0 or 1.
The paper defines a new structure on tangent sphere bundles and characterizes their properties.
problem Characterizing properties of tangent sphere bundles with contact pseudo-metric structures.
method Introduced a contact pseudo-metric structure on TεM and proved manifold properties based on constant sectional curvature. result The tangent sphere bundle TεM is (κ,μ)-contact pseudo-metric manifold if and only if the manifold M has constant sectional curvature. Finite vector bundles over complex manifolds are trivializable via finite covers.
problem Understanding when holomorphic vector bundles over compact complex manifolds are trivializable.
method Introducing finite bundles and using finite étale covers to trivialize holomorphic vector bundles.
result Holomorphic vector bundles over compact complex manifolds are finite if and only if they admit a flat holomorphic connection with finite monodromy.
Local generalization of frame bundles using a weakened Maurer-Cartan equation.
problem Characterizing the local structure of frame bundles with a weakened Maurer-Cartan equation.
method Introducing generalised frame bundles and studying their properties.
result Generalised frame bundles can have singular underlying manifolds.
Study the Albanese map for Kähler manifolds with nef anticanonical bundle.
problem Characterize the structure of the Albanese map for Kähler manifolds with nef anticanonical bundle.
method Analyze two cases: general fiber is Calabi-Yau or projective space. Provide proofs for both cases.
result For the case where the general fiber is Calabi-Yau, the manifold itself must be Calabi-Yau.
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 method constructs tractor conformal bundles for spacelike submanifolds in Lorentzian manifolds.
problem Characterize conditions for a tractor conformal bundle to be standard and normal.
method Introduce a canonical construction of a tractor conformal bundle and characterize conditions for it to be standard and normal.
result Characterizes conditions for a tractor conformal bundle to be standard and normal.
On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's r…
Eta invariant computed for circle bundles over Fano manifolds.
problem Computing eta invariant for circle bundles over Fano manifolds.
method Using spin-c Dirac operator and adiabatic limit.
result Eta invariant computed for arbitrary adiabatic parameters.
Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…
Global Pestov identity proved on frame bundle and related fibrations.
problem Global Pestov identity on frame bundles and fibrations.
method Global Pestov identity on frame bundles and fibrations.
result Global Pestov identity on frame bundles and fibrations.
Develops SGH bundles and theories for GC manifolds.
problem No specific problem stated; focuses on new bundle theory.
method Introduces SGH bundles, develops cohomology, and establishes theories.
result Established a Chern-Weil theory and Hodge theory for SGH bundles.
The study realizes symmetric spaces as cotangent bundles and finds nonnegative curvature examples.
problem Understanding the geometry of symmetric spaces and their associated vector bundles.
method Realizing symmetric spaces as cotangent bundles of flag manifolds and constructing vector bundles.
result Examples of vector bundles over simply connected manifolds with nonnegative curvature but not nonnegative sectional curvature.
In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic (0,q)-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we e…
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tange…
Study geodesics and F-geodesics on tangent bundles over para-Kähler-Norden manifolds.
problem Investigate geodesics and F-geodesics on tangent bundles.
method Investigate geodesics and F-geodesics on tangent bundles and φ-unit tangent bundles equipped with φ-Sasaki metric over para-Kähler-Norden manifolds.
result Investigate and analyze geodesics and F-geodesics on tangent bundles.
Investigates J-equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of J-equation on holomorphic vector bundles. method Introduces and studies J-equation, provides algebraic and numerical criteria. result Provides an algebraic condition (asymptotic J-stability) and a numerical criterion for vortex bundles. Constructs infinitely many examples of large manifolds with circle bundles of positive scalar curvature.
problem Existence of circle bundles over large manifolds with positive scalar curvature metrics.
method Symplectic geometry techniques.
result Infinitely many examples of macroscopically large manifolds with circle bundles of positive scalar curvature.
In this paper we investigate what kind of manifolds arise as the total spaces of iterated S1-bundles. A real Bott tower studied in \cite{CMO}, \cite{KM} and \cite{KN} is an example of an iterated S1-bundle. We show that the total space of an iterated S1-bundle is homeomorphic to an infra-nilmanifold. A real Bo…
Any leafwise connection on a fibre bundle over a foliated manifold is proved to come from a connection on this fibre bundle.