Extended de Rham theorem for manifolds with boundaries.
problem Applying de Rham decomposition to manifolds with boundaries.
method Using development of curves to extend classical theorem.
result Extended de Rham theorem for manifolds with boundaries.
In this paper, we study the uniqueness in the de Rham-Wu decomposition for pseudo-Riemannian manifolds.
Decomposes Riemannian and Lorentzian manifolds for computing holonomy groups.
problem Computing the connected holonomy group of manifolds.
method Finding parallel symmetric bilinear forms and applying linear algebra.
result Computes the connected holonomy group of Riemannian and Lorentzian manifolds.
We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riemannian or a warped product. We…
The paper studies similarity structures on manifolds and their de Rham decompositions.
problem Understanding similarity structures and their de Rham decompositions on manifolds.
method Analyzes similarity structures on universal covers and uses de Rham decomposition.
result Shows that for compact manifolds, the universal cover has at most two de Rham factors.
We show that the de Rham theorem, interpreted as the isomorphism between distributional de Rham cohomology and simplicial homology in the dual dimension for a simplicial decomposition of a compact oriented manifold, is a straightforward consequence of elementary properties of currents. The explicit construction of this…
Researchers compute de Rham cohomology of geodesic flow foliations on hyperbolic surfaces.
problem Answering a problem posed by Haefliger and Li about geodesic flow foliations.
method Unitary representation theory of PSL(2, R) and Hodge decompositions of de Rham complexes.
result Computed de Rham cohomology of weak stable foliations for various coefficients.
Geodesic completeness proven for certain Lorentzian spaces.
problem Geodesic completeness of compact Lorentzian locally symmetric spaces.
method Using recent and earlier results on Lorentzian and de Rham-Wu decompositions.
result Geodesically complete if Lorentzian factor is of Cahen-Wallach type or maximal flat factor is one-dimensional and time-like.
Study elliptic isometries on a matrix manifold with specific metrics.
problem Differential-geometric properties of fixed point loci.
method Explicit description and De Rham decomposition of fixed point loci.
result Explicit description and De Rham decomposition of fixed point loci.
Study the structure of Kähler foliations with negative Ricci curvature.
problem Characterize the structure of Kähler foliations with negative Ricci curvature.
method Prove a de Rham type theorem decomposition on the leaf space.
result Characterize each factor in the decomposition of the leaf space.
In the present paper, we give a necessary and sufficient condition for a Riemannian manifold (M,g) to have a reducible action of a hyperbolic analogue of the holonomy group. This condition amounts to a decomposition of (M,g) as a warped product of a special form, in analogy to the classical de Rham decomposition th…
The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a nove…
Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V the V-valued total de Rham complex of G. We describe the G-equivariant de Rham coh…
Unified mathematical theory for analyzing biomolecular geometry and flexibility.
problem Lack of a unified mathematical theory for analyzing biomolecular geometry and flexibility.
method Introducing de Rham-Hodge theory, Helmholtz-Hodge decomposition, and discrete exterior calculus.
result Unified framework for predicting macromolecular flexibility and natural modes.
Study of Ricci-Yamabe solitons on Walker 3-manifolds.
problem Characterizing Ricci-Yamabe solitons on Walker 3-manifolds.
method Using Hodge decomposition of De-Rham, the soliton field is found from the potential function.
result Classification of all Ricci-Yamabe and gradient Ricci-Yamabe solitons in a Walker 3-manifold.
Unified view of geometries with parallel skew torsion via submersions.
problem No de Rham decomposition for geometries with torsion.
method Developed and unified submersion constructions for geometries with parallel skew torsion.
result Completed and extended classification of irreducible geometries with parallel skew torsion.
Explains Hodge theory and Kodaira embedding theorem for complex manifolds.
problem Understanding complex manifold properties and their geometric implications.
method Expository review of harmonic forms, Hodge theory, and Kodaira embedding theorem.
result Establishes connections between de Rham cohomology, Dolbeault cohomology, and projective varieties.
Two superholonomy theories are shown equivalent.
problem Equivalence of two superholonomy theories.
method Comparison Theorem and de Rham-Wu decomposition theorem.
result Holonomy theories are equivalent.
Two de Rham complexes in diffeology are compared using a factor map.
problem Comparing two de Rham complexes in diffeology.
method Using a factor map to connect the two de Rham complexes and Čech--de Rham spectral sequence.
result Singular de Rham cohomology of irrational torus is isomorphic to tensor product of original de Rham cohomology and exterior algebra.
Develops intrinsic curved cosets for Cartan geometries.
problem Defines curved cosets for arbitrary Cartan geometries.
method Defines intrinsic holonomy group and curved cosets.
result Curved cosets retain characteristics of homogeneous counterparts and behave well under automorphisms.
The paper defines coeffective cohomologies for K-contact and Sasakian manifolds and derives inequalities and decomposition theorems.
problem Defining and studying coeffective cohomologies for K-contact and Sasakian manifolds.
method Introducing coeffective de Rham, Dolbeault, and Bott-Chern cohomologies for basic forms on K-contact and Sasakian manifolds and proving Hodge decomposition theorems.
result Inequalities relating coeffective basic cohomology numbers to classical basic Betti numbers for finite type manifolds, and Hodge decomposition theorems for coeffective basic de Rham cohomology.
Introduces a new Hodge theory using vector fields on manifolds.
problem Developing a new Hodge theory for manifolds with vector fields.
method Defines a vector field induced Hodge L2-inner product, codifferential, and Laplacian. result Established de Rham-Hodge theory for closed and boundary manifolds.
Study of de Rham cohomology on non-Hausdorff manifolds.
problem De Rham cohomology on non-Hausdorff manifolds.
method Careful discussion of non-Hausdorff differential forms, Mayer-Vietoris sequences.
result Proved de Rham's Theorem and Gauss-Bonnet theorem for non-Hausdorff manifolds, including counterterms.
Classifies 4D metric Lie algebras with parallel skew-symmetric tensors.
problem Classifying 4D metric Lie algebras with parallel skew-symmetric tensors.
method Complete classification up to isometric isomorphism and scaling.
result Classification of 4D metric Lie algebras with parallel skew-symmetric tensors.
New proof of Wolpert's Magic Formula using de Rham cohomology.
problem Proving Wolpert's Magic Formula for Teichmüller space.
method Using de Rham cohomology and Goldman's symplectic form.
result Derive a new proof of Wolpert's Magic Formula.
The paper explores de Rham theory for singular spaces and stacks.
problem Identifying de Rham theory for singular differentiable spaces.
method Identifying two potential answers and studying them, including the exterior algebra of the cotangent complex and de Rham stacks.
result There exists a version of the de Rham theorem for singular differentiable spaces with almost no restrictions.
New Lipschitz de Rham theorem for Lp-cohomology.
problem Developing a new de Rham theorem for Lp-cohomology. method Regularization procedure in Lipschitz de Rham calculus applied to metric simplicial complexes.
result Established Lipschitz de Rham theorem for Lp-cohomology. The paper proves formulas and theorems for statistical de Rham Hodge operators on manifolds with boundary.
problem Formulating Lichnerowicz type formulas and Kastler-Kalau-Walze theorems for statistical de Rham Hodge operators.
method Developed Lichnerowicz type formulas and proved Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with boundary.
result Proved Lichnerowicz type formulas and Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with boundary.
The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.
problem Analyzing perturbations of de Rham Hodge operators on manifolds with boundaries.
method Lichnerowicz type formulas and Kastler-Kalau-Walze type theorems for 4D and 6D compact manifolds with or without boundaries.
result Proves Kastler-Kalau-Walze type theorems for perturbations of de Rham Hodge operators on 4D and 6D manifolds with or without boundaries.
We prove an analogue of the de Rham theorem for the extended L^2-cohomology introduced by M. Farber. This is done by establishing that the de Rham complex over a compact closed manifold with coefficients in a flat Hilbert bundle E of A-modules over a finite von Neumann algebra A is chain-homotopy equivalent (with bound…
We find a cocycle for the Euler class in simplicial de Rham complex.
problem Finding a cocycle for the Euler class in simplicial de Rham complex.
method Exhibiting a cocycle in the simplicial de Rham complex.
result Constructing a Lie algebra cocycle on Lso(4) as an application. The paper studies reducibility properties in Sasakian geometry, classifying certain contact structures and extremal metrics.
problem Reducibility properties in Sasakian geometry, focusing on contact structures and extremal metrics.
method Developed the Sasaki version of the de Rham Decomposition Theorem, introduced cone reducible concept, and classified Sasakian structures.
result Classified all Sasakian structures up to contact isotopy on S3 bundles over a Riemann surface of genus greater than zero, and showed extremal Sasaki metrics split in the toric case. Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.
problem Triviality of de Rham homomorphism kernel and non-increasing monotonicity of parameters.
method Regularization in Lipschitz de Rham calculus on metric simplicial complexes with bounded geometry.
result Explicit specification of non-trivial cohomology classes for a sequence of parameters.
Study de Rham theory for cubical manifolds and quandles.
problem Rational homotopy type of classifying spaces of smooth quandles.
method Show de Rham theory for cubical manifolds and study secondary characteristic classes.
result Secondary characteristic classes produce cocycles of quandles.
New cochain algebra for diffeological spaces connects de Rham and singular cohomologies.
problem Incompatibility of de Rham and singular cohomologies in diffeology.
method Introduces a new singular de Rham complex and proves it quasi-isomorphic to the original de Rham complex for manifolds and spaces with singularities.
result The new cochain complex resolves the incompatibility issue in diffeology.
Study describes global sections of chiral de Rham complex on compact Ricci-flat Kähler manifolds.
problem None explicitly stated; focuses on description of global sections.
method Complete description of vertex algebra of global sections.
result Complete description of vertex algebra of global sections of chiral de Rham complex.
Study mapping class group action on de Rham quasimorphisms, finding no fixed points.
problem Action of mapping class group on de Rham quasimorphisms.
method Examined the action of mapping class group on de Rham classes in bounded cohomology of a hyperbolic surface.
result No fixed points in the action of mapping class group on de Rham quasimorphisms.
De Rham theorem extended to Orlicz cohomology.
problem Extending de Rham's theorem to a broader class of cohomology.
method Proving isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. result Isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. Authors prove de Rham cohomology of Poisson and Jacobi manifolds is trivial.
problem Understanding algebraic structures on de Rham cohomology of Poisson and Jacobi manifolds.
method Using DG operads and quasi-isomorphisms, they show the de Rham cohomology structure is trivial.
result The de Rham cohomology of Poisson and Jacobi manifolds has no higher structure beyond commutativity.
New definition of de Rham spaces changes quotient structure.
problem Traditional de Rham spaces defined using nilpotents.
method Uses infinity nilpotent elements for de Rham spaces.
result De Rham spaces are quotients by germs of diagonals.
Study proves formulas linking curvature to de Rham cohomology of foliations.
problem Understanding de Rham cohomology of totally geodesic foliations.
method Proved transverse Weitzenböck identities for horizontal Laplacians.
result Nullity theorems for de Rham cohomology under curvature positivity.
The Burde--de Rham theorem is extended to finitely presented pro-p groups with specific conditions.
problem Extending the Burde--de Rham theorem to pro-p groups with certain constraints. method Assumption of total degrees of relators being 0, concrete examples, and cohomological interpretations.
result The theorem is extended to finitely presented pro-p groups under specified conditions. Promotes spectral functionals to noncommutative fields and proves a theorem.
problem Spectral functionals on noncommutative fields and manifolds with boundary.
method Carries out promotion to spectral functionals, associates with noncommutative residue, and proves theorem.
result Proves Dabrowski-Sitarz-Zalecki type theorem for statistical de Rham Hodge operators on manifolds with boundary.
The primitive cohomology of Calabi-Yau intersections is described using a twisted de Rham complex.
problem Describing the primitive cohomology of Calabi-Yau intersections.
method Using a twisted de Rham complex and formal flat F-manifold structures.
result Constructs formal flat F-manifold structures on the primitive cohomology of Calabi-Yau intersections.
This paper extends de Rham theory of smooth manifolds to exploded manifolds. Included are versions of Stokes' theorem, De Rham cohomology, Poincare duality, and integration along the fiber. The resulting cohomology theory is used to define Gromov Witten invariants of exploded manifolds in a separate paper.
Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.
problem Computing spectral functions for nonminimal de Rham-Hodge operators.
method Definitions and computations of spectral triple and functional.
result Computed spectral Einstein functional for even-dimensional compact manifolds.
Study calculates global sections on complex curves.
problem Global sections of chiral de Rham complexes on complex curves.
method Calculation on closed complex curves with genus g ≥ 2.
result Space of global sections determined.
The Hodge theorem connects cohomology groups on compact Kähler manifolds.
problem Establishing a relationship between cohomology groups on compact Kähler manifolds.
method Proving the Hodge decomposition theorem on compact d-Kähler manifolds.
result Hodge decomposition theorem on compact d-Kähler manifolds.