We exhibit several transformations of surfaces in R^4. First, one that takes a flat surface and gets a surface with flat normal bundle; then, one that takes a surface with flat normal bundle and gets a flat surface; finally, a one-parameter family of transformations on a flat surface with flat normal bundle and gives a…
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48 results for “flat bundles”
Criterion for flat circle bundles using intrinsically harmonic forms.
problem Characterizing flat circle bundles.
method Criterion based on intrinsic harmonicity of a specific form.
result Flatness of a principal circle bundle is equivalent to intrinsic harmonicity of a certain form.
The paper examines the limit of harmonic flow on flat vector bundles.
problem Understanding the limiting behavior of harmonic flow on flat complex vector bundles.
method Analyzes the harmonic flow and proves the limit is isomorphic to a graded flat complex vector bundle.
result The limit of the harmonic flow on flat complex vector bundles is isomorphic to a graded flat complex vector bundle.
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 -bundles over low-dimensional manifolds, comparing a finite dimensional Lie group with and localizing the holonomy.
result Flat -bundles over low-dimensional manifolds are cobordant to a flat -bundle.
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.
Establishes uniform Hörmander estimates for flat line bundles on Kähler manifolds.
problem Estimating -operators for flat line bundles.
method Uniform -estimates for -operators on Kähler manifolds.
result Recovers Ueda's lemma for compact Kähler manifolds and generalizes to Ricci-flat manifolds.
Local index theory and the Riemann-Roch-Grothendieck theorem for complex flat vector bundles IImath.DG
Proves a theorem for complex flat vector bundles using differential forms.
problem No specific problem stated; focuses on proving a theorem.
method Uses differential forms to prove the Riemann-Roch-Grothendieck theorem.
result Proves the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles.
We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in must locate in some , from which we characterize Clifford torus as the only non-equatorial homogeneous minimal surface in with flat normal…
Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.
problem Understanding projective flat holomorphic vector bundles over Riemann surfaces.
method Assigning Wronskian line bundles to vector bundles and interpreting Abel's identity.
result Abel's identity is the first Chern class of the Wronskian line bundle.
Develops method to create non-Abelian Ricci-flat graphs via bundles.
problem Creating non-Abelian Ricci-flat graphs.
method Develops systematic way via graph bundles with constraints.
result Non-trivial graph bundles are not isomorphic to product of base and fiber.
Generalizes Novikov conjecture results to infinite-dimensional bundles.
problem Proving special cases of the Strong Novikov Conjecture.
method Introduces asymptotically flat Fredholm bundles and proves index theorem.
result Relates index of asymptotic Fredholm bundle to asymptotic index of representation.
In this paper we introduce the notion of almost flatness for (stably) relative bundles on a pair of topological spaces and investigate basic properties of it. First, we show that almost flatness of topological and smooth sense are equivalent. This provides a construction of an almost flat stably relative bundle by usin…
We show that a flat principal bundle with compact connected structure group and its adjoint bundles of Lie groups have the same cohomology as the trivial bundle, which is done by proving they satisfy the condition for the Leray-Hirsch theorem. This information has been used to construct a cohomology class of the adjoin…
Planes are the only calibrated submanifolds with flat normal bundles.
problem Characterizing submanifolds with specific geometric properties.
method Using constant-coefficient differential forms and parallel calibrations.
result Calibrated submanifolds with flat normal bundles are planes.
Study on immersions with flat normal bundle in curved spaces.
problem Behavior of isometric immersions with negative curvature.
method Investigation of second fundamental form growth in space forms.
result Second fundamental form grows exponentially if normal bundle is flat.
New equivalence found for flat vector bundles without extra conditions.
problem Flat vector bundles over compact Riemannian manifolds.
method Extended Corlette and Donaldson's result to arbitrary vector bundles.
result Equivalence of harmonic metrics and semi-simpleness for arbitrary vector bundles.
Study describes moduli spaces of flat bundles on Sasakian manifolds.
problem Understanding moduli spaces of flat bundles on Sasakian manifolds.
method Shows moduli space of simple flat bundles is a union of spaces with fixed basic structures.
result Detailed description of non-abelian Hodge correspondence on compact Sasakian manifolds.
The paper proves conditions for Kähler-Einstein metrics on certain bundles.
problem Conditions for the existence of Kähler-Einstein metrics on unit sphere bundles.
method Analyzes curvature conditions and Ricci eigenvalues of Kähler manifolds.
result Conditions for obstruction flatness and existence of Kähler-Einstein metrics.
Involutory Hopf group-coalgebras provide new invariants for 4-manifold bundles.
problem Developing invariants for flat bundles over 4-manifolds.
method Utilizing Hopf -triplets and colored trisection diagrams.
result Involutory Hopf -triplets yield well-defined invariants of -colored trisection diagrams.
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.
Study of symplectically flat connections and their functionals on smooth manifolds.
problem Understanding symplectically flat connections and their functionals on smooth manifolds.
method Extend symplectically flat connections to -flat connections, introduce functionals with zeroes as symplectically flat connections, study critical points of these functionals, describe characteristic classes of -flat bundles.
result Novel geometric flows and characteristic classes of -flat bundles are described.
Paper constructs estimates for flat vector bundles and generalizes Prékopa's theorem.
problem Constructing estimates for flat vector bundles.
method Using Hörmander's -estimate for the operator on a flat vector bundle over a -convex Riemannian manifold.
result Generalizes Prékopa's theorem in convex analysis.
The Corlette-Donaldson-Hitchin-Simpson's correspondence states that, on a compact Kähler manifold , there is a one-to-one correspondence between the moduli space of semisimple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this paper, we extend thi…
Classifies special submanifolds with specific curvature properties.
problem Classifying submanifolds with constant Moebius curvature and flat normal bundle.
method Analyzes isometric immersions with constant Moebius curvature and flat normal bundle.
result Classifies submanifolds with these curvature properties.
We study rank flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.
The paper studies Kähler-Einstein metrics on circle bundles and their obstruction flatness.
problem Understanding Kähler-Einstein metrics on circle bundles and their smoothness properties.
method Analyzing the obstruction flatness of hypersurfaces arising as unit circle bundles over Kähler manifolds.
result Complete Kähler-Einstein metrics on disk bundles are possible under certain conditions.
We show that there exist flat surface bundles with closed leaves having non-trivial normal bundles. This leads us to compute the Abelianisation of surface diffeomorphism groups with marked points. We also extend a formula of Tsuboi that expresses the Euler class of a flat circle bundle in terms of the Calabi invariant …
Constructs diffeological moduli stacks for Higgs and flat bundles on Kähler manifolds
problem Establishing an equivalence between diffeological substacks of Higgs and flat bundles
method Using diffeological moduli stacks
result Shows equivalence of categories between semistable Higgs bundles and flat bundles
Equivalence proven between two torsion invariants for flat vector bundles.
problem Equivalence of Igusa-Klein and Bismut-Lott torsion invariants for flat vector bundles.
method Reduction to trivial flat line bundles using Artin's induction theorem.
result Igusa-Klein and Bismut-Lott torsion invariants are equivalent for flat vector bundles with finite holonomy.
Higgs bundles and flat connections over compact Sasakian manifolds, II: quasi-regular bundlesmath.DG
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 Higgs sections and flat sections for nonlinear harmonic bundles.
problem Equivalence of Higgs and flat sections for nonlinear harmonic bundles.
method Analyze harmonic vector bundles, generalize to sub-fibrations and morphisms.
result Vanishing of a degree obstruction for general nonlinear harmonic bundles.
Introduces Legendre bundle for dually flat manifolds and quantum field theories.
problem Understanding duality in geometric structures and quantum field theories.
method Introduces Legendre bundle and para-Kähler structure.
result Exponential families and Hessian QFTs are realizations of the Legendre bundle.
Let be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric and a covariant constant volume form. Let be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat princip…
This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the…
We consider a proper flat fibration with real base and complex fibers. First we construct odd characteristic classes for such fibrations by a method that generalizes constructions of Bismut-Lott. Then we consider the direct image of a fiberwise holomorphic vector bundle, which is a flat vector bundle on the base. We gi…
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.
We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact Kähler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they…
New examples show not all homology fiber bundles are topological.
problem Disprove conjecture about homology fiber bundles.
method Construct flat, projective morphisms that are -homology fiber bundles.
result Disprove conjecture about homology fiber bundles.
We construct a Fourier--Mukai transform for smooth complex vector bundles over a torus bundle the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles with a flat partial unitary connection, that is families or deformations of flat …
Study relates Finsler structures to Clifford bundles for flat metrics.
problem Relating Finsler structures to Clifford bundles for flat metrics.
method Examines extensions of Clifford bundles and Finsler type structures for flat metrics.
result Triangle map exists between Finsler structures constructed from metrics and 1-forms.
Study families of flat connections with nilpotent Higgs fields, showing similar monodromy to regular Higgs bundles.
problem Investigate -families of flat connections with nilpotent Higgs fields.
method Analyze families of flat connections including real twistor lines and conformal limits, deducing monodromy similarities.
result Traces of holonomies are asymptotically exponential in rational powers of the parameter of the family.
We shall prove a semi-negative curvature property for a manifold with a flat admissible Higgs bundle.
Proves a special type of submanifolds in a curved space.
problem Characterizing submanifolds with specific properties in a curved space.
method Uses the properties of flat normal bundle and parallel mean curvature to prove the submanifolds are warped products.
result Einstein submanifolds with flat normal bundle and parallel mean curvature are warped product of isometric immersions.
Characterizes curvature positivity for Riemannian metrics on flat vector bundles.
problem Characterizing Nakano positivity of Riemannian flat vector bundles.
method Using solvability of the equation with specific estimates and inspired by recent works on Hermitian holomorphic vector bundles.
result Alternative proof of matrix-valued Prekopa's theorem.
Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.
problem Understanding foliations with numerically flat tangent bundles on Kähler manifolds.
method Analyzing the structure of foliations on compact Kähler manifolds, extending earlier results.
result Smooth foliations with numerically flat tangent bundles induce a decomposition of the ambient manifold's tangent bundle.
I prove the Bloch conjecture:all secondary characteristic classes of flat bundles over complex projective varietes are torsion, except the first.
An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geom…
We prove that any conformally flat submanifold with flat normal bundle in a conformally flat Riemannian manifold is locally holonomic, that is, admits a principal coordinate system. As one of the consequences of this fact, it is shown that the Ribaucour transformation can be used to construct an associated large family…