In this paper, we study the uniqueness in the de Rham-Wu decomposition for pseudo-Riemannian manifolds.
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In this paper, we extend the classical de Rham decomposition theorem to the case of Riemannian manifolds with boundary by using the trick of development of curves.
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…
It is explained how to find the de~Rham decomposition of a Riemannian manifold and the Wu decomposition of a Lorentzian manifold. For that it is enough to find parallel symmetric bilinear forms on the manifold, and do some linear algebra. This result will allow to compute the connected holonomy group of an arbitrary Ri…
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 to have a reducible action of a hyperbolic analogue of the holonomy group. This condition amounts to a decomposition of 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…
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.
A similarity structure on a connected manifold M is a Riemannian metric on its universal cover such that the fundamental group of M acts by similarities. If the manifold M is compact, we show that the universal cover admits a de Rham decomposition with at most two factors, one of which is Euclidean. Very recently, afte…
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 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.
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.
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.
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 -inner product, codifferential, and Laplacian.
result Established de Rham-Hodge theory for closed and boundary manifolds.
New Lipschitz de Rham theorem for -cohomology.
problem Developing a new de Rham theorem for -cohomology.
method Regularization procedure in Lipschitz de Rham calculus applied to metric simplicial complexes.
result Established Lipschitz de Rham theorem for -cohomology.
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.
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.
Recent years have witnessed a trend that advanced mathematical tools, such as algebraic topology, differential geometry, graph theory, and partial differential equations, have been developed for describing biological macromolecules. These tools have considerably strengthened our ability to understand the molecular mech…
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…
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 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 -cohomology and simplicial -cohomology.
result Isomorphism between de Rham -cohomology and simplicial -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.
In this paper we define coeffective de Rham cohomology for basic forms on a --contact or Sasakian manifold and we discuss its relation with usually basic cohomology of . When is of finite type (for instance it is compact) several inequalities relating some basic coeffective numbers to classical basic Bett…
The Burde--de Rham theorem is extended to finitely presented pro- groups with specific conditions.
problem Extending the Burde--de Rham theorem to pro- 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- groups under specified conditions.
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.
Statistical de Rham Hodge operators, spectral Einstein functionals and the noncommutative residuemath.DG
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.
Twisted de Rham complex for toric Calabi-Yau complete intersections and flat -manifold structuresmath.AG
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.
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.
Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.
problem Understanding Hodge-de Rham numbers for almost complex 4-manifolds.
method Introduced and studied Hodge-de Rham numbers, extending properties from complex surfaces.
result All Hodge-de Rham numbers for compact almost complex 4-manifolds are determined by the cohomology, except for one (the irregularity).
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.
The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It …
A De Rham model for string topology based on the theory of iterated integrals is presented.
GEOMANCER learns manifold factors without supervision.
problem Learning to factorize Lie group orbits from data.
method Subspace diffusion estimation of invariant subspaces.
result Unsupervised disentanglement of complex manifolds possible.
The article studies spinor and tensor fields on curved spaces, deriving formulas and spectra.
problem Understanding spinor and tensor fields on curved spaces.
method Weitzenböck-type formulas, explicit factorization of Laplace operator, representation theory.
result Explicit factorization of the Laplace operator and spectra calculation on constant curvature spaces.
We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S_3, the dihedral group D_4 and the quaternion group Q. Poincare' duality holds in every case, and under some assumptions (essentially the existence of a top form) we find that it must hold…
We prove a vanishing theorem for the twisted de Rham cohomology of a compact manifold.
In this article we study compact K\ahler manifolds satisfying a certain nonnegativity condition on the bisectional curvature. Under this condition, we show that the scalar curvature is nonnegative and that the first Chern class is positive assuming local irreducibility. We also obtain a partial classification of possib…