arXiv research

Evolutionary forms, as well as exterior forms, are skew-symmetric differential forms. But in contrast to the exterior forms, the basis of evolutionary forms is deforming manifolds (with unclosed metric forms). Such forms possess a peculiarity, namely, the closed inexact exterior forms are obtained from that. The closur…

A brief introduction to exterior differential systems for graduate students familiar with manifolds and differential forms. For complete files, see https://github.com/Ben-McKay/introduction-to-exterior-differential-systems

Extends Young integral to Hölder differential forms in arbitrary dimensions.

Survey revisits vector calculus results using exterior derivative and provides a new formulation of Stokes' theorem.

A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector spaces of finite and infinite dimension, fractional differential form spaces. The def…

A new discrete calculus for bundle-valued forms is proposed and validated.

We prove that the only natural operations between differential forms are those obtained using linear combinations, the exterior product and the exterior differential. Our result generalises work by Palais and Freed-Hopkins. As an application, we also deduce a theorem, originally due to Kolar, that determines those natu…

The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the properties and specific features of the differential-geometrical structures. The…

New expressions for Nijenhuis tensor squares found.

Extends differential geometry concepts to manifolds with super tangent bundles.

Characterizes when differential forms have weak exterior derivatives based on limiting behavior of integration over simplices.

Diffeology explores $k$-forms and bundles with more information than traditional differential forms.

It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity t…

We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allow…

We prove that the only natural differential operations between holomorphic forms on a complex manifold are those obtained using linear combinations, the exterior product and the exterior differential. In order to accomplish this task we first develop the basics of the theory of natural holomorphic bundles over a fixed …

In this paper, as the second in our series of papers on differential geometry of microlinear Frolicher spaces, we study differenital forms. The principal result is that the exterior differentiation is uniquely determined geometrically, just as grad (ient), div (ergence) and rot (ation) are uniquely determined geometric…

The L 1-Sobolev inequality states that the L n/(n--1)-norm of a compactly supported function on Euclidean n-space is controlled by the L 1-norm of its gradient. The generalization to differential forms (due to Lanzani & Stein and Bourgain & Brezis) is recent, and states that a the L n/(n--1)-norm of a compactly support…

The paper addresses the expansion of Berezinian and super exterior powers, revealing new insights into supertraces.

New integration theory on topological spaces, including fractals.

This work further develops the properties of fractional differential forms. In particular, finite dimensional subspaces of fractional form spaces are considered. An inner product, Hodge dual, and covariant derivative are defined. Coordinate transformation rules for integral order forms are also computed. Matrix order f…

We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, i…

New method solves differential equations on manifolds, with applications in physics.

We characterize all natural linear operations between spaces of differential forms on contact manifolds. Our main theorem says roughly that such operations are built from some algebraic operators which we introduce and the exterior derivative.

We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold $(M,g)$ and the $L_{q,p}$-cohomology of that manifold. The $L_{q,p}$-cohomology of $(M,g)$ is defined to be the quotient of the space of closed differential forms in $L^p(M)$ modulo the exact forms which are exterior diff…

We use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, first-order Lagrangian functionals and their associated Euler-Lagrange PDEs, subject to contact transformations. The first chapter contains an introduction of the classical Poincare-Cartan form in the context of EDS, follow…

The generalization of the n-dimensional cube, an n-dimensional chain, the exterior derivative and the integral of a differential n-form on it are introduced and investigated. The analogue of Stokes theorem for the differential space is given.

Geometrically reformulates elasticity theory using exterior calculus.

A discrete (finite-difference) analogue of differential forms is considered, defined on simplicial complexes, including triangulations of continuous manifolds. Various operations are explicitly defined on these forms, including exterior derivative and exterior product. The latter one is non-associative. Instead, as ant…

The theory of harmonic symmetric bilinear forms on a Riemannian manifold is an analogue of the theory of harmonic exterior differential forms on this manifold. To show this, we must consider every symmetric bilinear form on a Riemannian manifold as a one-form with values in the cotangent bundle of this manifold. In thi…

Geometric theory of integration developed in SDG.

Rust library solves complex equations on abstract simplicial complexes.

Building on the Utiyama principle we formulate an approach to Lagrangian field theory in which exterior covariant differentials of vector-valued forms replace partial derivatives, in the sense that they take up the role played by the latter in the usual jet bundle formulation. Actually a natural Lagrangian can be writt…

Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension $n$, arbitrary polynomial degree $r$, and arbitrary differential form degree $k$. The study of finite …

This paper has been withdrawn by the author due to its main result being included in cond-mat/0403309 by the same author.

Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of these cases one can define a module of measurable 1-forms and a first-order exteri…

Extends exterior diff. sys. to Lie algebroids with examples.

We derive a numerical method for Darcy flow, hence also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedr…

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…

The purpose of this paper is to study the shapes and stabilities of bio-membranes within the framework of exterior differential forms. After a brief review of the current status in theoretical and experimental studies on the shapes of bio-membranes, a geometric scheme is proposed to discuss the shape equation of closed…

We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic probl…

Differentiable spaces derived from Lie group actions have vector fields and forms.

We extend the validity of a Gromov's dimension comparison estimate for topological hypersurfaces to sufficiently large classes of rectifiable sets, arising from Sobolev mappings. Our tools are a suitably weak exterior differentiation for pullback differential forms and a new low rank property for Sobolev mappings.

The geometric product, defined by Graf on the space of differential forms, endows the sections of the exterior bundle by a structure that is necessary to construct a Clifford algebra. The Graf product is introduced and revisited with a suitable underlying framework that naturally encompasses a coframe in the cotangent …

A spectral approach to building the exterior calculus in manifold learning problems is developed. The spectral approach is shown to converge to the true exterior calculus in the limit of large data. Simultaneously, the spectral approach decouples the memory requirements from the amount of data points and ambient space …

Exterior differential forms with values in the (Kostant's) symplectic spinor bundle on a manifold with a given metaplectic structure are decomposed into invariant subspaces. Projections to these invariant subspaces of a covariant derivative associated to a torsion-free symplectic connection are described.

We construct radial fundamental solutions for the differential form Laplacian on negatively curved symmetric spaces. At least one of these Green's functions also yields a Biot-Savart Opearator, i.e. a right inverse of the exterior differential on closed forms with image in the kernel of the codifferential. Any Biot-Sav…

We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second ord…

We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on finding harmonic cochains cohomologous to a given cocycle. Amongst other …