Spherical T-duality for iterated sphere bundles
problem T-duality for iterated sphere bundles
method Repackaging cohomological data into Massey products
result Found T-dual iterated sphere bundles associated to Massey products
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48 results for “sphere bundles”
Study spherical T-duality and Massey products in iterated sphere bundles.
problem Understanding spherical T-duality and Massey products in iterated sphere bundles.
method Analyzing Gysin sequences and Massey products to find T-dual iterated sphere bundles.
result For certain iterated sphere bundles, spherical T-duality can be represented by Massey products.
We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from the diffeomorphism group of the sphere to the orthogonal group. This result implies the existence of many smooth n-sphere bundles over a k-sphere that do not supp…
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 study finds quasi-Einstein metrics on sphere bundles.
problem Finding quasi-Einstein metrics on specific types of manifolds.
method Adapting Hall's work, the study explores quasi-Einstein metrics on sphere bundles over Fano Kaehler-Einstein manifolds and their blow-downs.
result The discovery of quasi-Einstein metrics on sphere bundles.
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 and proved manifold properties based on constant sectional curvature.
result The tangent sphere bundle is -contact pseudo-metric manifold if and only if the manifold has constant sectional curvature.
Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
problem Characterizing manifolds that are both rational homology spheres and double disk bundles.
method Analyzing the structure of manifolds as unions of disk bundles and using properties of rational homology and cohomology.
result Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
We construct a metrical framed structure on the tensor bundle of a Riemannian manifold equipped with a Cheeger-Gromoll type metric and by restricting this structure to the tensor sphere bundle, we obtain an almost metrical paracontact structure on the tensor sphere bundle. Moreover, we show that the tensor sphere bundl…
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 , 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.
Study affinely transverse foliations in sphere bundles, finding bounds and vanishing conditions.
problem Understanding affine transverse foliations in sphere bundles and their properties.
method Provided upper bounds for the Euler number and a new proof for vanishing conditions under amenable fundamental group.
result Upper bounds and vanishing conditions for the Euler number of sphere bundles.
Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying all invariant contact metric structures on tangent sphere bundles of compact rank-one symmetric spaces.
method Explicitly obtained all structures, distinguishing K-contact, Sasakian, and 3-Sasakian structures.
result There is a unique Sasakian-Einstein metric on tangent sphere bundles of spheres and real projective spaces.
Researchers classify differential operators between 3-sphere and 2-sphere bundles.
problem Classifying differential symmetry breaking operators between 3-sphere and 2-sphere bundles.
method Constructing and classifying all differential symmetry breaking operators D_{λ,ν}^m.
result Necessary and sufficient conditions for the existence of these operators.
Using the higher analytic torsion form of Bismut and Lott we construct a characteristic class for smooth sphere bundles. We calculate this class in the case where the sphere bundle comes from a complex vector bundle. Related to these characteristic classes we define nontrivial continuous group cohomology classes of the…
Unique symplectic fillings of odd spheres' cotangent bundles proven.
problem Uniqueness of symplectic fillings for odd-dimensional spheres' cotangent bundles.
method Proof of uniqueness up to diffeomorphism.
result Unique symplectically aspherical fillings of odd spheres' cotangent bundles.
Generalizes Hopf degree theorem to nontrivial bundles.
problem Classifying maps from manifolds to spheres.
method Generalization of Hopf degree theorem to nontrivial bundles.
result Classifies sections of nontrivial -sphere bundles.
Paper constructs a cohomology class related to McDuff's secondary class, proving it transgresses to the Euler class of foliated sphere bundles.
problem Finding higher-dimensional analogs of the Calabi invariant and its transgression to the Euler class.
method Constructing a cohomology class of volume-preserving diffeomorphisms and proving transgression to the Euler class of foliated sphere bundles.
result The cohomology class transgresses to the Euler class of foliated sphere bundles.
In this paper, we construct round fold maps or stable fold maps with concentric singular value sets introduced by the author on smooth bundles over spheres or bundles over more general manifolds. The class of round fold maps includes special generic maps on spheres and such maps have been constructed on smooth bundles …
We construct a co-dimension completely non-holonomic sub-bundle on the Gromoll-Meyer exotic sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7…
Classifies smooth manifolds homotopy equivalent to sphere products
problem Classifying smooth manifolds homotopy equivalent to sphere products
method Using normal-invariant map and explicit families of manifolds
result Classifies smooth manifolds up to almost diffeomorphism
We consider double plumbings of two disk bundles over spheres. We calculate the Heegaard-Floer homology with its absolute grading of the boundary of such a plumbing. Given a closed smooth 4-manifold and a suitable pair of classes in , we investigate when this pair of classes may be represented by a config…
Constructs hyperbolic affine spheres and Calabi-Yau metrics.
problem Constructing non-isometric Calabi-Yau metrics.
method Non-Abelian Hodge theory for parabolic Higgs bundles.
result Answers a question about Calabi-Yau metrics.
In dimensions congruent to 1 modulo 4, we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere. More precisely, we prove that such an exotic sphere cannot embed as a Lagrangian in the cotangent bundle of…
Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
method Characterization through invariant Riemannian metrics and Killing vector fields.
result A family of invariant contact metric structures is obtained on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two.
New -instantons found on 3-sphere's spinor bundle.
problem Classifying and constructing -instantons.
method Used -symmetries and asymptotically conical, co-homogeneity one -metric.
result Found new examples of -instantons with obstructed deformations.
We show, using two different approaches, that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian…
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.
{\it Fold maps} are fundamental tools in generalizing the theory of Morse functions and its application to studies of geometric properties of manifolds. One of the fundamental and important problems in the theory of fold maps is to construct explicit fold maps, which are often difficult. In this paper, we construct new…
Sasakian structures found on tangent sphere bundles of certain symmetric spaces.
problem Existence of Sasakian structures on tangent sphere bundles of compact rank-one symmetric spaces.
method Proof of existence using K-contact structures and induced structures from almost Hermitian structures.
result Tangent sphere bundles of compact rank-one symmetric spaces admit unique K-contact structures that are Sasakian.
Study on rank 2 Higgs bundles on 5-punctured sphere, proving conjecture in lowest degree.
problem Proving the conjecture for rank 2 Higgs bundles on a 5-punctured sphere.
method Abelianization of Higgs bundles, fiducial solutions, and analysis of Fenchel--Nielsen co-ordinates.
result Proved the lowest degree weighted pieces of the conjecture.
We describe the (complex) quaternionic geometry encoded by the embeddings of the Riemann sphere, with nonnegative normal bundles.
This paper addresses Cheeger and Gromoll's question of which vector bundles admit a complete metric of nonnegative curvature, and relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle which admits a metric of nonnegative curvature must admit a c…
The study identifies surfaces with Maslovian normal bundles.
problem Characterizing surfaces with specific geometric properties.
method Proving equivalence to round spheres, cylinders, or cones.
result Surfaces with Maslovian normal bundles are limited to specific shapes.
The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.
problem Classifying and understanding moduli spaces of metrics with specific curvature properties on sphere bundles.
method Analyzing total spaces of -bundles over and quotients of Milnor and Shimada spheres.
result The moduli space of metrics has infinitely many path components.
The paper proves non-triviality of certain classes in sphere bundle cohomology.
problem Proving non-triviality of powers of Euler and Pontryagin classes in sphere bundle cohomology.
method Using cobordism and free torus actions on manifolds, the paper constructs examples and proves non-triviality of classes.
result Powers of the Euler class and Pontryagin classes are non-trivial in the cohomology of the diffeomorphism group of odd-dimensional spheres.
Compact quotients of homogeneous spaces are studied, leading to new findings about sphere bundles.
problem Understanding compact quotients of reductive homogeneous spaces and their implications.
method Analyzing normal bundles and sphere bundles associated with these spaces, proving homotopy triviality conditions.
result Many reductive homogeneous spaces do not admit compact quotients, resolving conjectures.
The study identifies holomorphic sections on jet spaces of the Riemann sphere.
problem Holomorphic sections on jet spaces of the Riemann sphere.
method Identifying a class of holomorphic sections of line bundles.
result Identified holomorphic sections on jet spaces of the Riemann sphere.
Constructs geometric models for moduli spaces of Higgs bundles over Riemann sphere.
problem Construction of moduli spaces for Higgs bundles with specific properties.
method Elementary geometric and combinatorial techniques, focusing on orbit stability of automorphism groups.
result Explicit geometric models for moduli spaces of parabolic Higgs bundles over Riemann sphere.
New Sasaki metrics with constant scalar curvature on sphere bundles are constructed.
problem Constructing extremal Sasaki metrics with constant scalar curvature.
method Using the fiber join construction and a recent existence theorem for constant scalar curvature Sasaki metrics.
result Explicit constructions of constant scalar curvature Sasaki metrics on specific sphere bundles.
We give an elementary treatment of the existence of complete Kahler-Einstein metrics with nonpositive Einstein constant and underlying manifold diffeomorphic to the tangent bundle of the (n+1)-sphere.
We determine the curvature equations of natural metrics on tangent bundles and radius r tangent sphere bundles S_rM of a Riemannian manifold M. A family of positive scalar curvature metrics on S_rM is found, for any M with bounded sectional curvature and any chosen constant r.
A manifold has infinitely many sphere fibrations over a sphere.
problem Infinitely many fibrations over a sphere.
method Examining and finding fibrations over for lens spaces.
result Found infinitely many fibrations from to over .
On the non-existence of almost complex structures on sphere bundles over complex projective spacesmath.AT
Study shows non-existence of almost complex structures on certain sphere bundles over complex projective spaces.
problem Existence of almost complex structures on sphere bundles over complex projective spaces.
method Chern class computations and divisibility properties of characteristic classes.
result Establishes a necessary condition for the non-existence of almost complex structures on certain bundles.
Reproves results on spherical metrics using parabolic bundles.
problem Existence and uniqueness of conformal spherical metrics with prescribed angles.
method Kobayashi-Hitchin correspondence for parabolic bundles.
result Reproves Troyanov and Luo-Tian's results.
Explains historical connections between vector bundle splitting and Riemann-Hilbert problems.
problem Vector bundle splitting over the Riemann sphere.
method Historical overview and connections to other mathematical problems.
result Explains the Riemann-Hilbert-Birkhoff problems and their relation to vector bundle splitting.
The nearly Kähler structures on the 6-sphere, as a twistor bundle sections are researched. We show that for any point of twistor bundle there exists an 1-parametric family of sections, passing through the point, which give nearly Kähler structures on the round sphere. Some properties of those sections are found.
Study on group cocycles for volume-preserving diffeomorphisms.
problem Understanding group cocycles on volume-preserving diffeomorphisms.
method Constructed two types of group cocycles on the volume-preserving diffeomorphism group.
result One cocycle yields the Euler class of flat sphere bundles for the sphere.
The paper studies Sasakian geometry on sphere bundles, focusing on extremal metrics and cohomology.
problem Understanding extremal and constant scalar curvature Sasaki metrics on sphere bundles.
method Applying the fiber join construction to K-contact manifolds, focusing on integral Kähler classes.
result Found infinite families of new inequivalent cone indecomposable Sasaki contact CR structures with extremal metrics.
We characterize Willmore tori in the 4-sphere with nontrivial normal bundle as Twistor projections of elliptic curves in complex projective space or as inverted minimal tori (with planar ends) in Euclidean 4-space.