Paper constructs infinitely many tangent functors on diffeological spaces.
problem Tangent spaces in diffeological spaces are not uniquely defined.
method Introduced and constructed infinitely many non-isomorphic tangent functors.
result The choice of tangent functor is not unique outside smooth manifolds.
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.
Monochromatic Finsler metrics are shown to be generalized Berwald metrics.
problem Characterizing Finsler metrics with isomorphic tangent spaces.
method Demonstrating the existence of an affine connection preserving the Finsler function.
result Monochromatic Finsler metrics are generalized Berwald metrics.
The paper studies smooth projective varieties with strictly nef tangent bundles and their properties.
problem Understanding smooth projective varieties with strictly nef tangent bundles.
method Analyzing the tangent bundle properties and using rational connectedness.
result Smooth projective varieties with strictly nef tangent bundles are rationally connected.
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. The paper characterizes projective spaces and quadrics using strictly nef bundles.
problem Characterizing projective spaces and quadrics using geometric properties of bundles.
method Analyzing strictly nef bundles on smooth projective varieties and curves.
result Strictly nef bundles on smooth projective varieties and curves have specific geometric properties.
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
This paper has three objectives. First to recall the link between the classical Legendre-Fenschel transformation and a useful isomorphism between 1-jets of functions on a vector bundle and on its dual. As a particular consequence we obtain the classical isomorphism between the cotangent bundle of the tangent bundle $T^…
Study of tangent bundle positivity on complex projective varieties.
problem Positivity of the second exterior power of tangent bundles on smooth complex projective varieties.
method Analyzes properties of tangent bundles and uses algebraic geometry techniques.
result Proves that up to a finite cover, the Albanese map is a locally trivial fibration with nef fibers.
We study manifolds arising as spaces of sections of complex manifolds fibering over the projective line with normal bundle of each section isomorphic to several copies of O(k). Such manifolds provide a natural setting for certain integrable systems, "monopoles", which comprise the Bogomolny hierarchy and the self-dual …
Researchers create non-isomorphic holomorphic Engel structures on C4.
problem Constructing non-isomorphic holomorphic Engel structures on C4.
method Controlled curves and distributions to create Engel structures.
result Existence of uncountably many non-isomorphic holomorphic Engel structures on C4.
Study on Kähler-Einstein metrics with polynomial convergence rates.
problem Understanding convergence rates of singular Kähler-Einstein metrics.
method Analyzing non-collapsed limits and tangent cones of polarized Kähler-Einstein manifolds.
result Polynomial convergence of Kähler potentials on tangent cones.
Study defines hyper-dual spheres and ruled surfaces, proving geometric relationships.
problem Understanding geometric properties of hyper-dual spheres and ruled surfaces.
method Defined hyper-dual spheres, developed ruled surfaces, and established geometric relationships.
result Proved isomorphism between hyper-dual sphere and tangent bundle, and geometric interpretation of ruled surfaces.
In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…
A VB-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between VB-groupoids and 2-term representations up to homotopy of Lie groupoids. Under this correspondence, the tangent bundle of a Lie groupoid G corresponds to the "adjoint repr…
New construction shows VMRTs of unbendable curves can be Legendrian.
problem Characterize VMRTs of unbendable rational curves under contact structures.
method Used geometry of contact lines and symplectic geometry of distributions.
result VMRTs of Legendrian submanifolds can be realized.
The base space of a semi-universal unfolding of a hypersurface singularity carries a rich geometric structure, which was axiomatized as a CDV-structure by C. Hertling. For any CDV-structure on a Frobenius manifold M, the pull-back of the (1,0)-tangent bundle of M to the product of M by the complex line carries two natu…
In the theory of so called "Covariant Quantum Mechanics" a basic role is played by Hermitian vector fields on a complex line bundle in the frameworks of Galilei and Einstein spacetimes. In fact, it has been proved that the Lie algebra of Hermitian vector fields is naturally isomorphic to a Lie algebra of "special funct…
The abstract extends twistor construction to manifolds with generalized metrics.
problem Extending twistor construction to manifolds with generalized metrics.
method Defining generalized twistor space and finding integrability conditions for intrinsic isomorphisms.
result Existence of intrinsic isomorphisms in generalized twistor spaces.
This work defines a categorical notion of principal bundles.
problem Different definitions of principal bundles in various categories.
method Formulated in join-restriction categories, which generalize partial maps.
result Shows the tangent bundle as the product of tangent space and group object.
Study of Lagrangian dynamics on matched Lie groups.
problem Understanding dynamics on matched Lie groups.
method Isomorphic tangent bundle and Euler-Lagrange/Poincaré equations.
result Covering semi-direct product theory and explicit equations.
Paper disproves a Smith conjecture about sphere actions.
problem Smith conjecture about sphere actions with two fixed points.
method Induction of group representations to show counterexamples.
result Negative answer to Smith conjecture for specific groups.
The paper explores metrics on Lie groups and their connections to dual quaternions.
problem Understanding metrics on Lie groups and their geometric properties.
method Analyzing Cartan-Schouten metrics on perfect Lie groups and their connections to dual quaternions.
result Biinvariant metrics on perfect Lie groups are shown to be Cartan-Schouten metrics.
Lie algebras of quotient groups defined under specific conditions.
problem Conditions for Lie differentiation of quotient groups.
method Diffeological group theory, tangent structure, Lie functor instantiation.
result Lie algebra structure on quotient groups derived from Lie algebras of parent groups.
A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach-Lie group acting smoothly…
We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M, we say E is N-rectifiable if it is the Lipschitz image of a positive measure subset of N. First, we discuss the implications of N-rectifiability, where N is a Carnot group (n…
The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space T of oriented affine lines in R3 with the tangent bundle of S2. Thus, the round metric on S2 induces a Kähler structure on T which turns out to h…
This paper explores differential and sector forms in tangent categories, finding rich structures and connections.
problem Understanding differential and sector forms in tangent categories.
method Investigates differential and sector forms in tangent categories, developing new equational presentations and structures.
result Sector forms in tangent categories form a symmetric cosimplicial object, with a subcomplex isomorphic to the de Rham complex of differential forms.
Rectifiability shown for sub-Riemannian manifolds with Carnot tangent structure.
problem Understanding rectifiability in sub-Riemannian manifolds with specific tangent properties.
method Analyzing nilpotentization and embedding properties of sub-Riemannian manifolds into Carnot groups.
result Sub-Riemannian manifolds are countably rectifiable under certain conditions.
Logarithmic connections on complex manifolds with trivial tangent bundle.
problem Finding logarithmic connections on complex manifolds with specific properties.
method Analyzing holomorphic Cartan geometries and their connections.
result Logarithmic connections preserve holomorphic Cartan geometries.
Isomorphism found between filtered calculus and crossed products.
problem Tackles isomorphism in filtered calculus and crossed products.
method Uses natural R-action and structure result for C*-algebra of graded nilpotent Lie groups.
result Found isomorphism between kernel of tangent groupoid and crossed product.
New combinatorial structures for Teichmüller spaces with Thurston's metric are explored.
problem Understanding the combinatorial structures of Teichmüller spaces with Thurston's metric.
method Analyzing the unit tangent and cotangent spheres of Teichmüller space, proving formulas for dimensions and codimensions of faces.
result The combinatorial structure of unit spheres in Teichmüller spaces is independent of the underlying point and is isomorphic to the extended mapping class group.
Let X be the Gromov-Hausdorff limit of a sequence of pointed complete Kähler manifolds (Min,pi) satisfying Ric(Mi)≥−(n−1) and the volume is noncollapsed. We prove that, there exists a Lie group isomorphic to R, acting isometrically, on the tangent cone at each point of X. Moreover, the actio…
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.
The most important examples of a double vector bundle are provided by iterated tangent and cotangent functors: TTM, TT^*M, T^*TM, and T^*T^*M. We introduce the notions of the dual double vector bundle and the dual double vector bundle morphism. Theorems on canonical isomorphisms are formulated and proved. Several examp…
Paper analyzes soft tree ensembles using NTK, finding only leaf count matters.
problem Understanding impact of various tree architectures in ensemble learning.
method Formulated and analyzed Neural Tangent Kernel (NTK) for soft tree ensembles.
result Only the number of leaves at each depth is relevant for tree architecture in ensemble learning.
New Oliver groups confirm a conjecture about sphere actions.
problem Confirming the Smith question for finite groups acting on spheres.
method Induction of group representations to find new Oliver groups.
result Found an infinite family of Oliver groups satisfying the Laitinen Conjecture.
This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the …
Computes Picard groups of complex parallelizable manifolds.
problem Calculating Picard groups of specific compact complex manifolds.
method Analyzes tangent bundle triviality and uses lattice and Lie group properties.
result Computes Picard groups for certain compact complex parallelizable manifolds.
A well-known theorem of Kapranov states that the Atiyah class of the tangent bundle TX of a complex manifold X makes the shifted tangent bundle TX[−1] into a Lie algebra object in the derived category D(X). Moreover, he showed that there is an L∞-algebra structure on the Dolbeault resolution of TX[−1]…
We observe that the Poincare duality isomorphism for a string manifold is an isomorphism of modules over the subalgebra A(2) of the modulo 2 Steenrod algebra. In particular, the pattern of the operations Sq^1, Sq^2, and Sq^4 on the cohomology of a string manifold has a symmetry around the middle dimension. We character…
Integral geometry formulas computed for exceptional spheres.
problem Kinematic formulas for invariant valuations and curvature measures on exceptional spheres.
method Computation of kinematic formulas based on isomorphisms of algebras of valuations.
result Kinematic formulas for invariant valuations and curvature measures in S6 and S7. For the cotangent bundle T∗K of a compact Lie group K, we study the complex-time evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space L2(K) under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state tra…
Study on diffeologies on locally convex spaces and smooth multiplication of distributions.
problem Geometric characterization and smoothness of distribution multiplication.
method Investigation of canonical and c∞-diffeologies on locally convex spaces, proving geometric characterizations, and comparing diffeologies. result Established a framework for nonlinear distribution theory beyond manifolds, realizing microlocally multipliable distributions as a diffeological colimit.
The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of complex-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact R3 are canonically isomorphic. Recently we constructed the first examples where Vassiliev in…
Study of rigid body displacements in a projective space over dual numbers with geometric interpretations.
problem Understanding rigid body displacements in a novel geometric space.
method Projective differential geometry over the ring of dual numbers.
result Existence of non-straight curves with multiple osculating tangents.
Study of rational curves in complex manifolds with specific normal bundles.
problem Characterizing rational curves in complex manifolds with given normal bundles.
method Analyzing differential and projective geometric properties of rational curves and their tangents.
result Classification of rational curves into Goursat and Cartan types based on their geometric properties.
We study diffeologies on locally convex spaces and their application to smooth multiplication of distributions.
problem Constructing smooth multiplication of distributions on locally convex spaces.
method Using diffeological colimits and wavefront-set criterion.
result Proving smooth multiplication of microlocally multipliable distributions.