Proof shows volume equals integral points for certain manifolds.
problem Counting integral points on affine manifolds.
method Rational Ehrhart theory and Fourier analysis.
result Volume equals number of integral points for integral-integral affine manifolds.
Integrable LCK manifolds characterized as Kähler Lie algebras.
problem Characterizing LCK manifolds with integrable anti-Lee forms.
method Examining LCK manifolds with integrable anti-Lee forms and applying to Lie algebras.
result Unimodular integrable LCK Lie algebras are Kähler Lie algebras with specific derivations.
Proves a theorem for normal distributions on manifolds with boundary.
problem Normal distributions on manifolds with boundary require a new approach to integration.
method Introduces neat integral manifolds with boundary and conditions for integrability.
result Conditions for integrability expressed in terms of adapted collars and integrability on interior and boundary.
Integrates rough geometric forms on manifolds.
problem Integrating rough forms on complex manifolds.
method Combines Whitney's geometric integration and sewing approaches.
result Introduced distributional k-forms for integration.
New integrable deformations for topological hierarchies from Frobenius manifolds.
problem Integrable deformations of topological hierarchies from Frobenius manifolds.
method Construction of integrable deformations with polynomial tau-structures.
result Conjecture of universal object for Riemann--Hopf hierarchy.
The article constructs stochastic integration in Riemannian manifolds.
problem No specific problem stated; focuses on the construction of stochastic integration.
method Functional-analytic approach to stochastic integration in Riemannian manifolds.
result There are infinitely many stochastic integrals, and they are related by a simple formula.
Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…
Method calculates function integrals on complex manifolds.
problem Integrating functions on complex manifolds.
method Digital representation and calculation method.
result Integral calculation on compact manifolds.
Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal …
Sharp spectral gap estimates on manifolds with integral curvature bounds.
problem Proving spectral gap estimates on manifolds with integral curvature bounds.
method Generalizing previous results to include integral curvature bounds.
result Confirms a conjecture about spectral gap estimates on manifolds with integral curvature bounds.
Extends integrability to cosymplectic manifolds.
problem Integrability of Hamiltonian systems on cosymplectic manifolds.
method Extended Arnold-Liouville and noncommutative integrability to cosymplectic manifolds, proved a variant of non-commutative integrability for specific fields, constructed action-angle variables.
result Variant of non-commutative integrability for evaluation and Reeb vector fields on cosymplectic manifolds.
The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T∗Q∖Q. If the geodesic flow is toric integrable, the cosphere bundle admit…
The study proves conditions for Hermitian metrics on compact almost complex manifolds.
problem Conditions for Hermitian metrics on compact almost complex manifolds.
method Analyzes compact almost complex manifolds with Hermitian metrics and integral conditions involving ∂-harmonic (0,1)-forms. result The integral condition is automatically satisfied for strongly Gauduchon metrics, and equivalent to being strongly Gauduchon for integrable almost complex structures.
Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.
problem Integral of scalar curvature on manifolds with a pole.
method Asymptotic scaling invariant integral of scalar curvature equals a term determined by asymptotic volume ratio.
result Integral of scalar curvature equals a volume ratio term.
We show that various notions of integrability for Poisson brackets are all equivalent, and we give the precise obstructions to integrating Poisson manifolds. We describe the integration as a symplectic quotient, in the spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we express th…
The paper proves integral formulas for manifolds with multiple orthogonal distributions.
problem Understanding geometric properties of manifolds with multiple orthogonal distributions.
method Develops integral formulas for Riemannian manifolds with k>2 orthogonal complementary distributions. result Generalizes known formulas for k=2 and applies to manifold splitting and immersions. Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
Researchers compute Connes-Chamseddine cycle on 6D manifolds using noncommutative integral.
problem Computing the Connes-Chamseddine cycle for 6D manifolds.
method Using noncommutative integral on 6D manifolds, they compute the cycle.
result The Connes-Chamseddine cycle on 6D manifolds is computed.
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.
Develops integrators for Hamiltonian systems in Jacobi manifolds.
problem Modeling conservative systems with dissipative and thermodynamic phenomena.
method Constructs structure-preserving integrators for Hamiltonian systems in Jacobi manifolds.
result Proposes a numerical integration technique compatible with Jacobi dynamics.
Maps between certain Lipschitz manifolds are isometries if they preserve volume.
problem Volume preservation and isometry conditions for Lipschitz manifolds.
method Volume-preserving 1-Lipschitz maps from integral currents onto infinitesimally Euclidean Lipschitz manifolds.
result Volume-preserving maps are isometries under given conditions.
A symplectic integration of a Poisson manifold (M,Λ) is a symplectic groupoid (Γ,η) which realizes the given Poisson manifold, i.e. such that the space of units Γ0 with the induced Poisson structure Λ0 is isomorphic to (M,Λ). This notion was introduced by A. Weinstein in order to quantize Poisson manifolds …
Perfect pairing for tropical cycles on integral affine manifolds.
problem Computing period integrals and versality of Calabi-Yau degenerations.
method Introducing a cap product pairing and using simplicial methods for constructible sheaves.
result The pairing is perfect in degree one for symplectic singularities.
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's pri…
Sharp bound on scalar curvature integral in 3-manifolds.
problem Bounding the integral of scalar curvature on 3-manifolds.
method Geodesic ball analysis with nonnegative Ricci curvature.
result Integral of scalar curvature is bounded by 8πR for large radii. RIG extends IG to Riemannian manifolds for explainable AI.
problem Lack of explainability in AI models.
method Extension of Integrated Gradients to Riemannian manifolds.
result RIG restricts to IG in Euclidean space.
We discuss a recurrent geometrical method, due to Élie Cartan and von Weber ([1],[11]) enabling us to determine, step by step, the maximal integral manifolds of a not necessarily integrable nor regular Pfaffian system. The dimensions of such integral manifolds can, of course, vary from point to point but more so can va…
Paper rigorously defines Feynman graph integrals on Kähler manifolds.
problem Establishing convergence of Feynman graph integrals on Kähler manifolds.
method Using Getzler's rescaling technique, graph integrands are extended to forms with divisorial-type singularities in the compactification of configuration spaces.
result Feynman graph integrals are rigorously defined as Cauchy principal value integrals.
Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.
problem Understanding variational properties of integral invariants defined from the second fundamental form.
method Derive first variational formulae for integral invariants of degree two, show Euler-Lagrange equation for Chern-Federer energy, and provide examples of submanifolds.
result The Euler-Lagrange equation of the Chern-Federer energy functional reduces to a second order PDE.
The paper shows inequality and rigidity for manifolds with integral Ricci curvature.
problem Analyzing structures of manifolds with integral Ricci curvature.
method Using segment inequality and similar methods as in \cite{CC1}, derive almost rigidity structure results.
result Sharp Hölder continuity result holds in the limit space of manifolds with integral Ricci curvature bound.
We study n-dimensional Kähler manifolds whose geodesic flows possess n first integrals in involution that are fibrewise hermitian forms and simultaneously normalizable. Under some mild assumption, one can associate with such a manifold an n-dimensional commutative Lie algebra of infinitesimal automorphisms. This,…
We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated s…
Constructs integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
problem Integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
method Constructs a bihamiltonian integrable hierarchy of hydrodynamic type.
result Integrable hierarchy possesses Virasoro symmetries and a tau structure.
The linking integral is an invariant of the link-type of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This result is then generalized to show that in certain circumstances the linking integ…
Invariants of 3-manifolds using modified Hopf G-coalgebra.
problem Constructing invariants for 3-manifolds.
method Purely Hopf G-coalgebra construction with modified integral.
result New invariants for 3-manifolds.
We show that non-elliptic prime 3-manifolds satisfy integral approximation for the simplicial volume, i.e., that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial volume and tools from ergodic theory.
Affine manifolds are called integral if there is an atlas such that all transition maps are affine transformations with integer matrices of linear parts. In this paper we describe all complete integral affine structures on compact three-dimensional manifolds up to a finite-sheeted covering. Also a complete list of inte…
In a previous paper, we proved a number of optimal rigidity results for Riemannian manifolds of dimension greater than four whose curvature satisfy an integral pinching. In this article, we use the same integral Bochner technique to extend the results in dimension three. Then, by using the classification of closed thre…
Study compares isoperimetric profiles on manifolds with integral Ricci curvature bounds.
problem Comparing isoperimetric profiles on manifolds with integral Ricci curvature bounds.
method Extending previous work, the study uses integral bounds on Ricci curvature to prove comparison results for isoperimetric profile functions.
result Comparison results for the Isoperimetric profile function in manifolds with integral bounds on Ricci curvature.
We prove that a holomorphic Lie algebroid is integrable if, and only if, its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic-Fernandes do also apply in the holomorphic context without any modification. As a consequence we give another proof of the following theorem: a holomorphic…
Generalized Huber's theorem for specific manifold curvature types.
problem Finite point conformal compactification on manifolds with certain curvature integrability.
method Generalization of Huber's theorem to higher dimensions with $L^ rac{n}{2}$ integrable Ricci curvatures.
result Validated finite point conformal compactification theorem for new class of manifolds.
Synthetic scalar curvature defined via Gaussian integrals, applied to manifolds and flows.
problem Defining scalar curvature for non-smooth spaces and flows.
method Gaussian integral approach to scalar curvature, applied to manifolds and flows.
result Characterizes Ricci flows as minimal super-Ricci flows.
For a Hamiltonian, proper and free action of a Lie group G on a Dirac manifold (M,L), with a regular moment map μ:M→g∗, the manifolds M/G, μ−1(0) and μ−1(0)/G all have natural induced Dirac structures. If (M,L) is an integrable Dirac structure, we show that M/G is always integrable,…
The paper studies Ricci flow with finite curvature integrals on manifolds.
problem Finite curvature integrals on closed manifolds.
method Ricci flow with integral curvature bounds.
result The flow converges to a smooth manifold except for orbifold singularities.
The article proves integral formulas for foliated sub-Riemannian manifolds.
problem Integral formulas for foliated sub-Riemannian manifolds.
method Proved a series of integral formulae involving mean curvatures, Newton transformations, and curvature tensor.
result Generalized known integral formulas for codimension-one foliations.
New framework for logarithmically divergent integrals on manifolds with corners.
problem Logarithmically divergent integrals on manifolds with corners.
method Introduces new geometric framework and morphisms in logarithmic geometry.
result Functorial characterization of regularized integration.
The based loop space homology of a special family of homogeneous spaces, flag manifolds of connected compact Lie groups is studied. First, the rational homology of the based loop space on a complete flag manifold is calculated together with its Pontrjagin structure. Second, it is shown that the integral homology of the…
Smooth approximation of integral cycles in manifolds.
problem Approximating integral cycles in Riemannian manifolds.
method Approximation of integral cycles by smooth submanifolds with controlled area and singularities.
result Integral cycles can be approximated by smooth submanifolds with controlled area and singularities.