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.
Local Kan conditions enable differentiation of simplicial manifolds.
problem Differentiating simplicial manifolds into Lie algebroids.
method Expanding a technique for higher Lie groupoids to simplicial manifolds.
result Derivation of a method to differentiate simplicial manifolds into higher Lie algebroids.
We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds admit a unique differential, coinciding with the usual one. Thus smooth manifolds fo…
The study confirms essential self-adjointness for certain differential operators on manifolds.
problem Essential self-adjointness of differential operators on closed manifolds.
method Analyzing the Hamiltonian flow of the symbol of differential operators.
result The conjecture that certain differential operators are essentially self-adjoint if their Hamiltonian flow is complete.
We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly …
Study natural invariants for third order nonlinear operators on 2D manifolds.
problem Equivalence problem of third order nonlinear differential operators.
method Description of rational natural differential invariants.
result Application of natural invariants to equivalence problem.
Any discrete differential manifold M (finite set endowed with an algebraic differential calculus) can be represented by appropriate polyhedron P(M). This representation demonstrates the adequacy of the calculus of discrete differential manifolds and links this approach with that based on finitary substitutes…
This paper proves equivalence between derived manifolds and differential graded manifolds.
problem Characterizing derived manifolds and their relationship to differential graded manifolds.
method Proving equivalence between the infinity categories of derived manifolds and differential graded manifolds.
result The infinity category of differential graded manifolds is equivalent to that of derived manifolds.
Study on local convergence of min-max algorithms to differential equilibria on Riemannian manifolds.
problem Solving zero-sum differential games on Riemannian manifolds.
method Analysis of two simultaneous min-max algorithms, τ-GDA and τ-SGA, to differential Stackelberg and Nash equilibria, with conditions for linear convergence and asymptotic approximation. result Established sufficient conditions for linear convergence of τ-GDA and demonstrated faster convergence of τ-SGA in some cases. Study linear differential operators on special manifolds.
problem Analyzing elliptic differential operators on specific types of manifolds.
method Examining a linear elliptic differential operator of the form Δ + V - λ on quasi-asymptotically conical manifolds.
result Established an isomorphism theorem for these operators.
New Poincaré inequality for differential forms on manifolds.
problem Developing inequalities for differential forms on manifolds.
method Proving a new Poincaré-type inequality and deriving new inequalities involving mean and scalar curvatures.
result Characterized the limiting case of a new inequality involving mean and scalar curvatures of the boundary.
Projective geometry aids in analyzing fields near compact manifolds.
problem Analyzing fields near compact manifolds.
method Developed a projective exterior differential tractor calculus.
result Analogous calculus for projectively compact manifolds.
We study differential invariants of the third order linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles on two dimensional manifolds with respect to groups of authomorphisms.
The paper proves inequalities for twisted differential forms on manifolds.
problem Proving Sobolev-type inequalities for twisted differential forms.
method Integral representations and uniform estimates for Green forms and their differentials.
result Improved L2-estimate of Hörmander on Kähler manifolds. Paper discusses directional differentiability of interval-valued functions on Riemannian manifolds.
problem Equivalence of directional differentiability of interval-valued functions and their components.
method Analyzes directional differentiability of interval-valued functions on Riemannian manifolds.
result Directional differentiability of interval-valued functions is not equivalent to the directional differentiability of their components.
New operators generalize Michelsohn's on almost Hermitian manifolds.
problem Generalizing differential operators to almost Hermitian manifolds.
method Introducing two differential operators on sections of the complex Clifford bundle over compact almost Hermitian manifolds.
result Surprising Kähler-like symmetries in the kernel of the Laplacians of these operators.
This paper deals with the notion of quadratic differential in spherical CR geometry (or more generally on strictly pseudoconvex CR manifolds). We get to this notion by studying a splitting of Rumin complex and discuss its first features such as trajectories and length. We also define several differential operators on q…
Sharp Steklov eigenvalue estimates for differential forms on manifolds.
problem Estimating the first positive eigenvalue of the Steklov eigenvalue problem for differential forms.
method Established a weighted Reilly formula for differential forms and applied it to geometric conditions.
result Sharp lower bound for the first positive eigenvalue of the Steklov eigenvalue problem on differential forms.
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.
The paper examines scalar fourth-order linear differential operators and their invariants.
problem Equivalence problem of scalar fourth-order linear differential operators.
method Investigation of differential invariants.
result Application of differential invariants to the equivalence problem.
Derives formulas for differential forms on weighted manifolds.
problem Developing formulas for differential forms on weighted manifolds.
method Derives a Reilly formula and explores its applications.
result Proves a Poincaré-type inequality and obtains new eigenvalue estimates.
The paper extends inequalities to twisted differential forms on Kähler manifolds.
problem Generalizing Sobolev-type inequalities to twisted differential forms.
method Establishing heat kernel estimates for differential forms on Kähler manifolds.
result Proves vanishing theorem and Lq,p-estimates for ∂ˉ-operator. Paper proves inequalities for forms on sub-Riemannian manifolds.
problem Establishing inequalities for differential forms on sub-Riemannian contact manifolds.
method Using structure of Rumin's complex, Sobolev-Gaffney inequality for Heisenberg groups, and geometric properties.
result Gaffney type inequality in Sobolev spaces for differential forms on sub-Riemannian contact manifolds with bounded geometry.
The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the properties and specific features of the differential-geometrical structures. The…
Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.
problem Conditions for differential operators on non-compact harmonic manifolds to have specific properties.
method Analyzing the algebra of differential operators, their commutation properties, and using geometric averages.
result Algebra of differential operators on non-compact harmonic manifolds has specific properties related to radial fundamental solutions and dense heat-semigroups.
Simplified calculus for manifold operators, proving index theorems.
problem Developing calculus for manifold operators and proving index theorems.
method Introducing a simplified pseudo-differential calculus for zero-order operators on manifolds with a tangent Lie structure.
result Proving index theorems for `h-elliptic' operators on manifolds with a tangent Lie structure.
This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (…
Formalizes integral curves on Banach manifolds in Lean.
problem Existence and uniqueness of integral curves on Banach manifolds.
method Formalized differential equations on Banach spaces, then generalized to Banach manifolds.
result Established theorems for integral curves on Banach manifolds.
In this paper we show that a parallel differential form Ψ of even degree on a Riemannian manifold allows to define a natural differential both on Ω∗(M) and Ω∗(M,TM), defined via the Frölicher-Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbe…
We show that the 2-jet bundle of local Riemannian metrics on an arbitrary differentiable manifold admits a section which pointwise fulfills the curvature relation sec(g)=a for any real number a. It follows by Gromov's h-principle for open, invariant differential relations that every noncompact differentiable manifold c…
The paper proves inequalities on Riemannian manifolds using a test function method.
problem Proving differential inequalities with (p,q)-Laplacian on Riemannian manifolds. method Using a test function argument.
result Established Liouville-type theorems under manifold's geometry and potential behavior.
Variational approach to basic manifold structures.
problem Understanding basic differential geometric structures.
method Variational description of geometric structures.
result Variational formulation of manifold structures.
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.
The paper defines and analyzes abla-Sobolev spaces and operators on manifolds.
problem Defining and analyzing Sobolev spaces and differential operators on manifolds.
method Coordinate-free approach using connections, proving properties of abla-Sobolev spaces and operators. result Equivalent definitions of abla-Sobolev spaces and operators under certain conditions. The Levy Laplacian is studied on manifolds, with heat flow solutions tending to constant functionals over time.
problem Understanding the Levy Laplacian on manifolds and its heat flow behavior.
method Various definitions of the Levy Laplacian are proven equivalent. Heat flows of differential forms are used to construct solutions.
result Solutions of the heat equation with the Levy Laplacian tend to locally constant functionals over time.
Gelfand and MacPherson provided a new formula for calculating Pontrjagin classes.
problem Calculating Pontrjagin classes of differential manifolds.
method Introduced a local and combinatorial formula.
result Expanded and clarified the original formula by Gelfand and MacPherson.
Derived differential manifolds are constructed using the usual homotopy theory of simplicial rings of smooth functions. They are proved to be equivalent to derived differential manifolds of finite type, constructed using homotopy sheaves of homotopy rings (D.Spivak), thus preserving the classical cobordism ring. This r…
We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second ord…
The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.
problem Curvature-dimension conditions and related inequalities on Riemannian manifolds.
method Information-theoretic approach to study curvature-dimension condition, rigidity theorems, and entropy differential inequalities.
result Equivalence of curvature-dimension condition and entropy differential inequalities on Riemannian manifolds.
A parametric manifold is a manifold on which all tensor fields depend on an additional parameter, such as time, together with a parametric structure, namely a given (parametric) 1-form field. Such a manifold admits natural generalizations of Lie differentiation, exterior differentiation, and covariant differentiation, …
The paper studies boundedness of pseudo-differential operators on smooth manifolds.
problem Boundedness of pseudo-differential operators in Lp-Lq spaces on smooth manifolds. method Using global symbols and extending Hörmander's condition, the paper investigates Lp-boundedness, L∞-BMO estimates, and Lp-Lq boundedness for Fourier multipliers and pseudo-differential operators. result The paper proves Lp-Lq boundedness for the range 1<p≤2≤q<∞. A brief introduction to exterior differential systems for graduate students familiar with manifolds and differential forms. For complete files, see https://github.com/Ben-McKay/introduction-to-exterior-differential-systems
Derives inequality for optimal transport on manifolds.
problem Optimal transport theory on manifolds.
method Five gradients inequality for cost functions on Lie groups and Riemannian manifolds.
result Derives inequality for optimal transport on specific manifolds.
Extends exterior diff. sys. to Lie algebroids with examples.
problem Invariant inverse problem of the calculus of variations
method Extends exterior differential systems to Lie algebroids, defines integral manifolds.
result Defines integral manifolds for exterior diff. systems on Lie algebroids.
In this note we establish the large time non-negativity of the heat kernel for a class of elliptic differential operators on closed, Riemannian manifolds, and apply this result to a problem from conformal differential geometry.
We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold (M,g) and the Lq,p-cohomology of that manifold. The Lq,p-cohomology of (M,g) is defined to be the quotient of the space of closed differential forms in Lp(M) modulo the exact forms which are exterior diff…
The paper quantizes Kähler manifolds using differential operators.
problem Quantizing classical observables on Kähler manifolds as differential operators.
method Constructing higher-order differential operators using Fedosov-type constructions and proving asymptotic equivalence to Berezin-Toeplitz operators.
result Holomorphic differential operators are precisely those that arise as Berezin-Toeplitz operators for quantizable functions.
Conformal-DP improves differential privacy on manifold data by calibrating perturbations based on local densities.
problem Lack of density-awareness in existing differential privacy mechanisms for manifold data leads to biased and suboptimal privacy-utility trade-offs.
method Proposes Conformal-DP, a density-aware differential privacy mechanism using conformal transformations to calibrate perturbations based on local densities.
result Demonstrates improved privacy-utility trade-off in heterogeneous data distribution settings compared to state-of-the-art mechanisms.