Narrow Results

It is shown that higher degree exact differential forms on compact Riemannian $n$-manifolds possess continuous primitives whose uniform norm is controlled by their $L^n$ norm. A contact sub-Riemannian analogue is proven, with differential forms replaced with Rumin differential forms.

For domains in ℝⁿ we construct the Bergman kernel on the diagonal using solutions of the Dirichlet problem. Starting from this, in a natural way we obtain an algebra 𝔸_n of dimension n(n - 1)/2 + 1 over ℝ and a class of holomorphic functions valued in 𝔸_n. Of course 𝔸₂ is the field of complex numbers, and it turns out that 𝔸₃ is the algebra of quaternions, whereas for n ≥ 4, 𝔸_n is non-associative. Holomorphic functions can be written as f + ω, where f is a (real-valued) function and ω a differential 2-form such that d* ω = df and dω = 0. We investigate the main properties of the obtained objects, especially from the analytic point of view.

This paper deals with the summability of conjugate Laplace series in any dimension. In particular, a M. Riesz type inequality is proved and criteria for the convergence of relevant Cesàro means in $L^p$-norm are given. The Fourier character of conjugate Laplace series is investigated too.

Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin${}^c$ manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin${}^c$ case the result of Kowalski stating that, every totally umbilical hypersurface of an Einstein manifold of dimension greater or equal to $3$ is of constant mean curvature. As an application, we prove that there are no extrinsic hypersheres in complete Riemannian $\mathrm{\text{Spin}}$ manifolds of non-constant sectional curvature carrying a parallel, Killing or imaginary Killing spinor.

We introduce a method of constructing once punctured Riemann surfaces by cutting the complex plane along "line segments" and pasting by "parallel transformations". The advantage of this construction is to give a good visualization of the deformation of complex structures of Riemann surfaces. In fact, given a positive integer g, there appears a family of once punctured Riemann surfaces of genus g which is complete and effectively parametrized at any point. Our construction naturally gives each of the resulting surfaces what we call a Lagrangian lattice Λ, a certain subgroup of the first homology. Furthermore Λ and the puncture determine an Abelian differential ω_Λ of the second kind on the Riemann surface. Using Λ and ω_Λ we consider the Kodaira–Spencer maps and some extension of the family to obtain any once punctured Riemann surface with a Lagrangian lattice. In particular we describe the moduli space of once punctured elliptic curves with Lagrangian lattices.

We construct realizations of the generators of the κ-Minkowski space and κ-Poincaré algebra as formal power series in the h-adic extension of the Weyl algebra. The Hopf algebra structure of the κ-Poincaré algebra related to different realizations is given. We construct realizations of the exterior derivative and one-forms, and define a differential calculus on κ-Minkowski space which is compatible with the action of the Lorentz algebra. In contrast to the conventional bicovariant calculus, the space of one-forms has the same dimension as the κ-Minkowski space.

In a recent work we derived the kinematic Hamiltonian and primary constraints of the new general relativity class of teleparallel gravity theories and showed that these theories can be grouped in 9 classes, based on the presence or absence of primary constraints in their Hamiltonian. Here we demonstrate an alternative approach towards this result, by using differential forms instead of tensor components throughout the calculation. We prove that also this alternative derivation yields the same results and show how they are related to each other.

We describe all operators on scalar finite element spaces which appear as the restriction of a second-order (linear) differential operator. More precisely we provide a family of isomorphisms between this space of discrete differential operators and products of some exotic finite element spaces. This provides a unified framework for the Regge calculus of numerical relativity and the Nédélec edge elements of computational electromagnetism.

Given a cellular complex, we construct spaces of differential forms which form a complex under the exterior derivative, which is isomorphic to the cochain complex of the cellular complex. The construction applies in particular to subsets of Euclidean space divided into polyhedra, for which it provides, for each k, a space of k-forms with a basis indexed by the set of k-dimensional cells. In the framework of mimetic finite differences, the construction provides a conforming reconstruction operator. The construction requires auxiliary spaces of differential forms on each cell, for which we provide two examples. When the cells are simplexes, the construction can be used to recover the standard mixed finite element spaces also called Whitney forms. We can also recover the dual finite elements previously constructed by A. Buffa and the author on the barycentric refinement of a two-dimensional mesh.

Some families of H-valued vector fields with calculable Lie brackets are given. These provide examples of vector fields on path spaces with a divergence and we show that versions of Bismut type formulae for forms on a compact Riemannian manifold arise as projections of the infinite dimensional theory.

In this paper we define Maxwell's equations in the setting of the intrinsic complex of differential forms in Carnot groups introduced by M. Rumin. It turns out that these equations are higher-order equations in the horizontal derivatives. In addition, when looking for a vector potential, we have to deal with a new class of higher-order evolution equations that replace usual wave equations of the Euclidean setting and that are no more hyperbolic. We prove equivalence of these equations with the "geometric equations" defined in the intrinsic complex, as well as existence and properties of solutions.

Grothendieck has proved that each class in the de Rham cohomology of a smooth complex affine variety can be represented by a differential form with polynomial coefficients. We prove a single exponential bound on the degrees of these polynomials for varieties of arbitrary dimension. More precisely, we show that the $p$th de Rham cohomology of a smooth affine variety of dimension $m$ and degree $D$ can be represented by differential forms of degree ${(pD)}^{{𝒪}(pm)}$. This result is relevant for the algorithmic computation of the cohomology, but is also motivated by questions in the theory of ordinary differential equations related to the infinitesimal Hilbert 16th problem.

We study the de Rham cohomology of a class of spaces with singularities which are called stratifolds. Such spaces occur in mathematical and theoretical physics. We generalize two classical results from the analysis of smooth manifolds, with outstanding applications in physics, to the class of stratifolds. The first one is Stokes theorem, the second one is the de Rham theorem which states that the de Rham cohomology of a stratifold is isomorphic to its singular cohomology with coefficients in ℝ. In fact, we give an explicit geometric construction of this isomorphism, given by integrating forms over stratifolds.

We present an account of negative differential forms within a natural algebraic framework of differential graded algebras, and explain their relationship with forms on path spaces.

It is shown how a system of differential forms can reproduce the complete set of differential equations generated by an SO(m) matrix Lax pair. By selecting the elements in the given matrices appropriately, examples of integrable nonlinear equations can be produced. The SO(3) case is discussed in detail, then extended to an m - 1 dimensional manifold immersed in Euclidean space.

We show how to lift a Riemannian metric and almost symplectic form on a manifold to a Riemannian structure on a canonically associated supermanifold known as the antitangent or shifted tangent bundle. We view this construction as a generalization of Sasaki’s construction of a Riemannian metric on the tangent bundle of a Riemannian manifold.

This paper focuses on the Einstein–Cartan theory, an extension of general relativity that incorporates a torsion tensor into spacetime. The differential form technique is employed to analyze the Einstein–Cartan theory, which replaces tensors with tetrads. A tetrad formalism, specifically the Newman–Penrose–Jogia–Griffiths formalism, is used to study the field equations. Also, the energy–momentum tensor is determined, considering a Weyssenhoff fluid with anisotropic matter. The spin density is derived in terms of the redshift function. Additionally, our findings demonstrate that the radial sound speed within the wormhole throat exceeds the speed of light, suggesting the existence of superluminal matter, while the tangential sound speed indicates near-light-speed propagation, particularly within the throat, emphasizing the significance of exotic matter in comprehending wormhole properties. The results also extend to examining the energy conditions at the throat of a Morris–Thorne wormhole, shedding light on the properties of wormholes within the context of the Einstein–Cartan theory, including the energy conditions at the throat.

The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincaré duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically flat link bundle. In this paper, we use differential forms on manifolds with corners to generalize the intersection space cohomology theory to a class of 3-strata spaces with flatness assumptions for the link bundles. We prove Poincaré duality over complementary perversities for the cohomology groups. To do so, we investigate fiber bundles on manifolds with boundary. At the end, we give examples for the application of the theory.

The papers introduces a new complex of differential forms which provides a fine resolution for the sheaf of regular functions in two quaternionic variables and the sheaf of monogenic functions in two vector variables. The paper announces some applications of this complex to the construction and interpretation of sheaves of quaternionic and Clifford hyperfunctions as equivalence classes of such differential forms.

Here we discuss a limit of the gravitational constant G goes to zero for the Einstein gravity. It is convenient to use for this a tetradic approach (using tetrads and spin-connection) and the Palatini formalism. Also some external field B is entered, which is 2-form. In this limit an equation for the spin-connection is obtained. Such an equation was obtained earlier in the MacDowell-Mansouri-Stelle-West gravity.