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.
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.
Defines fiber-wise linear differential operators on vector bundles.
problem No specific problem stated; focuses on definition and equivalence.
method Definition and equivalence of fiber-wise linear differential operators to derivations of line bundles.
result Equivalence of fiber-wise linear differential operators to derivations of line bundles.
Equivalence of second order differential operators in vector bundles studied.
problem Equivalence problem for second order linear differential operators in vector bundles.
method Description of rational invariants of symbols, finding connections associated with differential operators.
result Solving problems of local and global equivalency of differential operators.
Classifies and constructs intertwining differential operators between line and vector bundles over real projective space.
problem Classifying and constructing intertwining differential operators between line and vector bundles over real projective space.
method F-method for classification and construction of intertwining differential operators.
result Generalizes a classical result of Bol for SL(2,R) and classifies intertwining operators for SL(n,R). Diffeology explores k-forms and bundles with more information than traditional differential forms.
problem Understanding k-forms and bundles in diffeological spaces. method Developed theory of diffeological vector pseudo-bundles, including limits and colimits, and various operations.
result Sections of bundles of k-forms contain more information than differential forms. In a natural way, the local diffeomorphisms of a manifold onto itself act on the reference frame bundles of any order and on the bundles associated with them. Due to the transitivity, the invariants by diffeomorphisms of an associated bundle correspond to the real functions on the orbit space of the action of the jet g…
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.
Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract setting for differential geometry by axiomatizing key aspects of the subject which all…
Defines linear weightings for vector bundles and explores their applications.
problem Understanding and extending the concept of weightings in vector bundles.
method Constructs weighted normal bundles and deformation spaces; explains the relationship between weightings and differential operators.
result Captures the rescaled spinor bundle and related constructions.
The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.
problem Classifying and constructing differential symmetry breaking operators.
method Utilizing factorization identities and branching laws of generalized Verma modules.
result Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces are classified and constructed.
Study describes splitting and filtration of Hodge bundle on quadratic differentials.
problem Understanding the structure of Hodge bundles on quadratic differentials.
method Harder-Narasimhan filtration and splitting as direct sum of line bundles.
result Determine all Lyapunov exponents of algebraically primitive Teichmüller curves.
The caloron correspondence is a tool that gives an equivalence between principal G-bundles based over the manifold M×S1 and principal LG-bundles on M, where LG is the Fréchet Lie group of smooth loops in the Lie group G. This thesis uses the caloron correspondence to construct certain differential f…
Study non-formal pseudo-differential operators over formal ones.
problem Understanding structure of non-formal pseudo-differential operators.
method Diffeological principal bundles, smoothing connections.
result Structure of diffeological bundle of non-formal pseudo-differential operators over formal ones.
We study differential invariants of linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles over smooth manifolds with respect to groups of authomorphisms.
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.
Extends differential calculus to triole algebras.
problem No specific problem stated; focuses on extending differential calculus.
method Generalizes diolic differential calculus to triole algebras with fiber metrics.
result Established a conceptual framework for calculus on bundles with vector-valued fiber metrics.
The paper proves positivity of characteristic forms for certain vector bundles.
problem Characterizing positivity conditions for vector bundles.
method Operator theory, pushforward identities, and differential forms.
result Schur polynomials in Chern forms of Nakano and Griffiths positive vector bundles are positive as differential forms.
The paper explores dualities in differential equations and their applications in Riemannian geometry.
problem Developing comparison theorems for mixed type differential equations.
method Utilizing dualities in differential equations and inequalities, and applying them to Riemannian geometry.
result Proves Hessian and Laplacian comparison theorems under various curvature assumptions.
We study a generalized Abreu Equation in n-dimensional polytopes and prove some differential inequalities for homogeneous toric bundles.
Develops global pseudo-differential calculus on homogeneous vector bundles.
problem Global theory of subelliptic pseudo-differential operators on homogeneous vector bundles.
method Global symbolic calculus, complex functional calculus, Hörmander system of vector-fields.
result Global pseudo-differential calculus on homogeneous vector bundles.
In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beiträge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the synthetic approach to jet bundles in terms of infinitesimal spaces Dⁿ's. In our succeeding [Synthetic differential geo…
Develops differential KO-character to determine real vector bundles in multiples of 8.
problem Determining real vector bundles in multiples of 8.
method Constructs eta-invariants and differential KO-character to determine differential KO-theory.
result Eta-invariants and index invariants completely determine differential KO-theory in degree (0 mod 8).
Proof confirms preservation of projective limits in synthetic differential geometry.
problem Prove preservation of projective limits in synthetic differential geometry.
method Detailed proof using synthetic differential geometry and Cahiers topos.
result Projective limits preserved in synthetic differential geometry.
The paper contains a review on the general connection theory on differentiable fibre bundles. Particular attention is paid to (linear) connections on vector bundles. The (local) representations of connections in frames adapted to holonomic and arbitrary frames is considered.
This paper exhibits equivalences of 2-stacks between certain models of S1-gerbes and differential 3-cocycles. We focus primarily on the model of Dixmier-Douady bundles, and provide an equivalence between the 2-stack of Dixmier-Douady bundles and the 2-stack of differential 3-cocycles of height 1, where the …
Transport functions for principal bundles and Morse homology with differential graded coefficients
problem Transport functions for principal bundles
method Constructing transport functions as maps from broken gradient flow lines to a topological group
result Recovering the principal bundle from the transport function
Constructs graded jet bundles for Z-graded manifolds and vector bundles.
problem Generalizing jet manifolds to Z-graded structures for differential equations.
method Directly constructs the sheaf of sections of the k-th order jet bundle of a Z-graded vector bundle.
result Establishes a graded version of Atiyah Lie algebroid.
The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The existence of bundle coordinates normal at a given point and/or along injective horizonta…
Study essential spectrum of differential operators on geometrically finite orbifolds.
problem Analyzing the essential spectrum of differential operators over specific geometric structures.
method Investigates first order and Laplace type elliptic differential operators on Riemannian vector bundles over geometrically finite orbifolds.
result Discovers properties of essential spectra for these operators.
The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan' theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin^C-structures, Dirac operators, exterior algebra bundles and Connes' differential algebras in the commutative case…
Constructs differential characters on nonlinear Graßmannians.
problem No specific problem stated; focuses on mathematical construction.
method Using a nonlinear version of the tautological bundle, a transgression map is constructed from M to nonlinear Graßmannians of submanifolds of fixed type. result Obtains prequantum circle bundles and central Lie group extensions.
Defines a bundle map for currents on manifolds using higher covariant derivatives.
problem Defining a bundle map for currents on manifolds.
method Using higher covariant derivatives on a manifold equipped with a torsion-free connection.
result The bundle of generalized Weyl algebras and its properties.
In this paper we introduce an equivariant extension of the Chern-Simons form, associated to a path of connections on a bundle over a manifold M, to the free loop space LM, and show it determines an equivalence relation on the set of connections on a bundle. We use this to define a ring, loop differential K-theory of M,…
We deal with finite dimensional differentiable manifolds. All items are concerned with are differentiable as well. The class of differentiability is C∞. A metric structure in a vector bundle E is a constant rank symmetric bilinear vector bundle homomorphism of E×E in the trivial bundle line bundle. We…
Formula for Euler characteristic of moduli spaces of Abelian differentials.
problem Computing the Euler characteristic of moduli spaces of Abelian differentials.
method Intersection theory on the smooth compactification by multi-scale differentials, Euler sequence for cotangent bundle, and tools in the Chow ring.
result Formula for the full Chern polynomial of the cotangent bundle.
Study connections on Lie groupoids and stacks using Atiyah sequences.
problem No specific problem stated; general connections on Lie groupoids and stacks.
method Construct connections using Atiyah sequences associated with transversal tangential distributions.
result Detailed study and construction of connections on Lie groupoids and stacks.
For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan-Weil equivariant forms and to Borel's equivariant integral cohomology. We sh…
Establishes a framework for stringor bundles, proving their canonical isomorphism to Stolz-Teichner's.
problem Defining and rigorously studying higher differential geometric objects like stringor bundles.
method Developed a framework of 2-Hilbert bundles, including an associated bundle construction.
result Proves the Stolz-Teichner stringor bundle is canonically isomorphic to the 2-Hilbert bundle.
Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed, including the description of a natural differential calculus over the structure …
We study S1-bundles and S1-gerbes over differentiable stacks in terms of Lie groupoids, and construct Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvature.
The study explores weightings on submanifolds and their geometric properties.
problem Understanding weightings on submanifolds and their geometric implications.
method Detailed exploration of weighted normal bundles, weighted deformation spaces, and weighted blow-ups.
result A description of weightings in terms of subbundles of higher tangent bundles, leading to new concepts for Lie algebroids and groupoids.
Researchers compute differential K-theory for moduli stacks.
problem Computing differential K-theory for moduli stacks of principal G-bundles.
method Using homotopy theory of presheaves of spaces and spectra, they formulate results in terms of invariant polynomials and representation rings.
result They successfully compute the connective differential K-theory and differential cohomology of moduli stacks.
In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of nonc…
New calculus framework for vector bundles with metrics.
problem Developing calculus for vector bundles with fiber metrics.
method Adapting differential calculus to graded commutative algebras and focusing on diole and triole algebras.
result Triole algebra provides a suitable environment for vector bundle calculus with fiber metrics.
Classifies solutions of Toda equations near singularities.
problem Classifying solutions of Toda equations near singularities.
method Analyzes meromorphic and essential singularities of r-differentials. result Classifies all solutions on C for finite sums of exponentials of polynomials. Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other discretisation schemes. There are, however, several theoretical approaches to systems…
A frame independent formulation of analytical mechanics in the Newtonian space-time is presented. The differential geometry of affine values i.e., the differential geometry in which affine bundles replace vector bundles and sections of one dimensional affine bundles replace functions on manifolds, is used. Lagrangian a…