Researchers find invariant Kähler metrics on tangent disk bundles of space-forms.
problem Finding invariant Kähler metrics on tangent disk bundles of space-forms.
method Defined a family of Kähler metrics invariantly on the radius r0>0 tangent disk bundle of any real space-form or its quotients. result The metrics are complete in non-negative curvature and non-complete in negative curvature cases.
We equip the whole tangent space TM to a hyperbolic manifold M (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of M extend to isometries of TM by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In t…
We show, using standard results in length spectrum rigidity and symplectic homology, that if the unit tangent bundles of two compact surfaces of negative curvature are exact symplectomorphic, then the underlying surfaces are isometric, and hence the disk bundles are symplectomorphic smoothly up to the boundaries. Motiv…
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.
Stein and Weinstein structures are described for disk cotangent bundles of surfaces.
problem Characterizing Stein and Weinstein structures on disk cotangent bundles.
method Using Legendrian handlebody diagrams and symplectic/contact mappings.
result The canonical contact structure on the unit cotangent bundle of S is obtained via surgery.
A genus one Lefschetz fibration is described for disk cotangent bundles of surfaces.
problem Describing Lefschetz fibrations on cotangent bundles.
method Explicit construction of genus one Lefschetz fibration.
result Explicit genus one open book decomposition for unit cotangent bundle.
Study on acceptable bundles on a punctured disk.
problem Understanding acceptable bundles in Simpson--Mochizuki theory.
method Detailed study and introduction of a new invariant.
result Arguments differ from Simpson and Mochizuki's.
New knot invariant derived from string topology.
problem Detecting the unknot using knot contact homology.
method String topology and Legendrian contact homology.
result Knot contact homology detects the unknot.
Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.
problem Characterizing symplectic fillings of prequantization bundles with finite capacities.
method Analysis of symplectic capacities and diffeomorphisms.
result Symplectic fillings of prequantization bundles are diffeomorphic to disk bundles under finite capacity conditions.
The study classifies manifolds that can be split into two disk bundles.
problem Understanding manifolds that can be decomposed into two disk bundles.
method Established through topological restrictions and rational ellipticity.
result Classification of manifolds up to diffeomorphism in dimensions five and six.
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 X and a suitable pair of classes in H2(X), we investigate when this pair of classes may be represented by a config…
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.
Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. Th…
In this paper, the third of its series, we prove that the sobolev spaces of L^p_k approximate solutions to the Dirichlet problem for the epsilon-Yang Mills equations on a four dimensional disk, carry a natural manifold structure (more precisely a natural structure of Banach bundle), for p(k+1)> 4. All results apply als…
Finite capacity proved for a specific manifold.
problem Calculating the capacity of a specific manifold's cotangent bundle.
method Analyzing the unit disk bundle of the cotangent bundle.
result The Hofer-Zehnder capacity is finite for a rationally inessential manifold.
No Einstein metrics found on certain double disk bundles.
problem Existence of Einstein metrics on specific manifolds.
method Phase space barrier argument to show non-existence.
result Proves non-existence of cohomogeneity one Einstein metrics.
Study metallic structures on tangent bundles with lifts and integrability.
problem Integrability and lifts of metallic structures on tangent bundles.
method Complete and horizontal lifts with respect to almost product structure, fundamental formulae establishment.
result Behavior of metallic structure cross-sections in tangent bundle TM.
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…
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 study finds infinitely many counterexamples to a generalized Double Soul Conjecture.
problem The Double Soul Conjecture for non-simply connected manifolds.
method Analysis of double disk bundles and counterexamples.
result Infinitely many counterexamples to the generalized Double Soul Conjecture.
The article investigates conditions for isomorphism of singular tangent bundles.
problem Conditions for isomorphism of singular tangent bundles.
method Logarithmic and b-tangent bundles approach to resolve singularities. result Established a Poincaré-Hopf theorem for bm-tangent bundles. 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.
Study on Ricci solitons on tangent and unit tangent bundles.
problem Characterizing Ricci solitons on tangent and unit tangent bundles.
method Analyzing pseudo-Riemannian g-natural metrics and their Ricci soliton properties. result Classification of conformal vector fields and existence of non-Einstein Ricci solitons.
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. The paper extends tangent space theory for diffeological spaces and bundles.
problem Understanding tangent spaces of diffeological bundles and spaces.
method Introducing weakly filtered and filtered diffeological spaces, extending exact sequences, and defining Hector's tangent bundle.
result Tangent bundles of filtered diffeological spaces are diffeological vector spaces.
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.
Researchers compute the cohomology of an elliptic tangent bundle.
problem Computing the cohomology of a specific Lie algebroid.
method Direct computation of cohomology.
result The cohomology of the elliptic tangent bundle is computed.
The paper classifies surfaces with pseudo-effective tangent bundles.
problem Understanding projective manifolds with pseudo-effective tangent bundles.
method Developed singular hermitian metrics on vector bundles and applied to projective manifolds.
result Projective manifolds with pseudo-effective tangent bundles admit a smooth fibration to a flat projective manifold.
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.
We propose a special deformation of the Sasaki metric on tangent and unit tangent bundle of a Hermitian locally symmetric manifold. Geodesics of this deformed metric have different projections on a base manifold for tangent or unit tangent bundle cases in contrast to usual Sasaki metric. Nevertheless, the projections o…
Higher-order tangent bundles have geometric structures compatible with their iterated bundle structure.
problem Connection towers and Sasaki metrics on higher-order tangent bundles
method Introduce the notion of a connection tower and study the geometric structures induced by such towers.
result Connection towers determine multiconnections, adapted splittings, and canonical vector bundle structures.
Study finds Scherk type surfaces as extremals for zero-curvature minimal graphs.
problem Finding extremal minimal graphs with zero Gaussian curvature at the center.
method Analyzing Scherk surfaces and their properties.
result Scherk type minimal surfaces are extremals for zero-curvature minimal graphs.
Characterizes special curves on surface tangent bundles.
problem Understanding curves on surface tangent bundles.
method Characterization of Legendre and slant curves.
result Characterizations for N-Legendre and N-slant curves.
New method constructs stable generalized complex structures.
problem Creating stable generalized complex structures.
method Developed Gompf-Thurston symplectic techniques adapted to Lie algebroids, used to construct stable structures from log-symplectic structures.
result Introduced boundary Lefschetz fibrations to obtain stable structures from genus one Lefschetz fibrations.
Study on singularities of bundle homomorphisms induced by Morin maps.
problem Characterizing singular points of bundle homomorphisms.
method Analyzing conditions for singularities induced by Morin maps, using Hamilton vector fields for contact structures.
result Characterization of singularities in bundle homomorphisms induced by Morin maps.
Homogeneous Fano manifolds with specific bundle properties proven.
problem Characterizing Fano manifolds with nef tangent bundles.
method Proving homogeneity for specific Fano manifolds.
result Any Fano manifold of coindex three with nef tangent bundle is homogeneous.
This paper develops differential bundles and fibrations in tangent categories.
problem Abstract setting for differential geometry in categories.
method Develops differential bundles and fibrations in tangent categories, considering their relation to differential objects.
result Strikingly, in display differential bundles, fibres are Cartesian differential categories.
Study positive characteristic Fano 4-folds with nef tangent bundles.
problem Positive characteristic version of the Campana-Peternell conjecture for Fano 4-folds.
method Analyzes Fano 4-folds with nef tangent bundles in positive characteristic.
result Affirmative answer for Fano 4-folds with Picard number > 1 and nef tangent bundle.
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.
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 finds representations of surface groups in SO(4,1) with specific curvature properties.
problem Finding convex-cocompact representations of surface groups with minimal map properties.
method Complex variation of Hodge structures and embedded minimal maps.
result Examples of generalized almost-Fuchsian representations not deformations of Fuchsian representations.
New surgery operation preserves monotonicity of Lagrangians.
problem Preserving monotonicity of Lagrangians in surgery operations.
method BSP surgery, wall-crossing formula for disk-potentials.
result BSP surgery can preserve monotonicity of Lagrangians.
We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundl…
New metrics found on tangent bundles of Riemannian surfaces.
problem Finding new metrics on tangent bundles.
method Invariant metrics on tangent bundles of almost-Hermitian manifolds.
result New Kählerian Ricci-flat manifolds in four dimensions.
In this paper we study a Riemanian metric on the tangent bundle T(M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to T(M) a structure of locally conformal almost Kählerian manifold. This is th…
Paper finds conditions for special geometric structures on certain spaces.
problem Existence of specific geometric structures on double disk bundles.
method Derives a sufficient condition involving geometric data from principal orbits.
result Sufficient condition for the existence of cohomogeneity one Einstein metrics.
Study metallic structures on P-Sasakian manifolds.
problem Investigate metallic structures on tangent bundles of P-Sasakian manifolds.
method Complete lift, horizontal lift, and vertical lift of a P-Sasakian structure to study metallic structures.
result Investigate integrability and parallelity of metallic structures.
Classifies contact metric spaces using hyperquadric bundles.
problem Classifying contact metric spaces with specific invariant.
method Using tangent hyperquadric bundles of Lorentzian space forms.
result Locally classify spaces with Boeckx invariant ≤ -1.