arXiv research

Study of de Rham cohomology on non-Hausdorff manifolds.

Paper introduces a new multilinear functional for spectral triples and computes its properties.

Study shows gap between de Rham and symplectic-Bott-Chern harmonic forms for specific almost-Kähler manifolds.

The paper extends Hodge-de Rham theory to higher-dimensional Sierpinski gaskets.

Local index density of perturbed de Rham complex is invariant under certain conditions.

In the present paper, we consider the Hodge-de Rham Laplacian that acts on conformal Killing and projective Killing one-forms of a compact Riemannian manifold.

The primitive cohomology of Calabi-Yau intersections is described using a twisted de Rham complex.

A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is general…

The chiral de Rham complex of Malikov, Schechtman, and Vaintrob, is a sheaf of differential graded vertex algebras that exists on any smooth manifold $Z$, and contains the ordinary de Rham complex at weight zero. Given a closed 3-form $H$ on $Z$, we construct the twisted chiral de Rham differential $D_H$, which coincid…

Generalizes integration map to coinvariants of bounded functions.

Locally-verifiable conditions ensure exactness of spline discrete de Rham complex.

We characterize Lie group actions for which there exists, at least locally, an evaluation map that defines a cochain map from the differential complex of invariant forms on a manifold to the De Rham complex for the quotient.

Proves Mayer-Vietoris sequence for diffeological spaces using generating families.

The paper extends Hodge-de Rham and Lichnérowicz Laplacians to double forms and proves vanishing theorems.

Formulas derived for operators on forms in anti-de Sitter spaces.

Two de Rham complexes in diffeology are compared using a factor map.

Study cohomology of odd symplectic manifolds, linking to Lagrangian submanifolds and BV Laplacians.

On the slit tangent manifold of a Finsler manifold M are given the vertical and the Liouville foliations. In this paper we define some new types of vertical forms with respect to the Liouville foliation on TM^0. We define a cohomology group of TM^0 using these new forms. We prove a de Rham type theorem.

Explains Hodge theory and Kodaira embedding theorem for complex manifolds.

The paper generalizes current constructions to cohesive modules and characteristic forms.

Introduces differential forms to study inequalities between eigenvalues.

This paper extends T-duality to exotic chiral de Rham complexes.

Proves cup product homomorphism for bounded cohomology on negatively curved manifolds.

A measured solenoid is a compact laminated space endowed with a transversal measure. The De Rham $L^2$-cohomology of the solenoid is defined by using differential forms which are smooth in the leafwise directions and $L^2$ in the transversal direction. We develop the theory of harmonic forms for Riemannian measured sol…

Let M be a hypercomplex Hermitian manifold, (M,I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate (2,0)-form on (M,I). It is well known that M is hyperkaehler if and only if the …

We are interested in the spectrum of the Hodge-de Rham operator on a cyclic covering $X$ over a compact manifold $M$ of dimension $n+1$. Let $Σ$ be a hypersurface in $M$ which does not disconnect $M$ and such that $M-Σ$ is a fundamental domain of the covering. If the cohomology group $H^{n/2 (Σ)$ is trivial, we can con…

Harmonic forms and Rumin complex linked on Sasakian manifolds.

Unified method for analyzing evolving manifolds using de Rham-Hodge theory.

We exhibit the Chern-Simons forms of some characteristic classes in the simplicial de Rham complex.

The paper explores de Rham theory for singular spaces and stacks.

New Lipschitz de Rham theorem for $L_p$-cohomology.

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…

Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.

The paper studies Hodge structures on contact manifolds and their cohomology.

We consider the notion of the De Rham operator on finite-dimensional diffeological spaces such that the diffeological counterpart Λ^1(X) of the cotangent bundle, the so-called pseudo-bundle of values of differential 1-forms, has bounded dimension. The operator is defined as the composition of the Levi-Civita connection…

This note proves equivariant de Rham cohomology for quotient spaces.

We introduce a weighted de Rham operator which acts on arbitrary tensor fields by considering their structure as r-fold forms. We can thereby define associated superpotentials for all tensor fields in all dimensions and, from any of these superpotentials, we deduce in a straightforward and natural manner the existence …

The paper proves formulas and theorems for statistical de Rham Hodge operators on manifolds with boundary.

Constructs a support-preserving homotopy for differential forms with boundary decay estimates.

The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.

Simple construction of Rumin algebra for contact manifolds.

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…

In this paper we prove a useful formula for the graded commutator of the Hodge codifferential with the left wedge multiplication by a fixed $p$-form acting on the de Rham algebra of a Riemannian manifold. Our formula generalizes a formula stated by Samuel I. Goldberg for the case of 1-forms. As first examples of applic…

On any compact manifold of dimension greater than 6, we prescribe the volume and any finite part of the spectrum Hodge Laplacian acting on $p$-form for $1\leq p<\frac n2$. In particular, we prescribe the multiplicity of the first eigenvalues.

Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.

The paper explores the Rumin complex and spectral sequence on Carnot groups.

Let $(M,\mathcal{F})$ be a foliated manifold. We prove that there is a canonical isomorphism between the complex of base-like forms $Ω^*_b(M,\mathcal{F})$ of the foliation and the "De Rham complex" of the space of leaves $M/\mathcal{F}$ when considered as a "diffeological" quotient. Consequently, the two corresponding …

The paper explores similarities in even and odd-dimensional geometry.