Related papers: Axiomatic Differential Geometry III-1
The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H.…
A frame independent formulation of analytical mechanics in the Newtonian space-time is presented The differential geometry of affine values i.e., the differential geometry in which affine bundles replace vector bundles and sections of one…
We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds…
As the fourth paper of our series of papers concerned with axiomatic differential geometry, this paper is devoted to the general Jacobi identity supporting the Jacobi identity of vector fields. The general Jacobi identity can be regarded as…
We present a differential-geometric formulation of automatic differentiation (AD) based on jet functors and Weil algebras. In this framework, forward- and reverse-mode differentiation arise naturally as pushforward and cotangent pullback,…
Let M be a smooth manifold, A a local algebra in sense of Andr\'e Weil, M^{A} the manifold of near points on M of kind A and X(M^{A}) the module of vector fields on M^{A}. We give a new definition of vector fields on M^{A} and we show that…
Our main interest in this paper is chiefly concerned with the conditions characterizing \textit{orthogonal and symplectic abstract differential geometries}. A detailed account about the sheaf-theoretic version of the \textit{symplectic…
It is shown that, given any finite dimensional, split basic algebra $\Lambda = K\Gamma/I$ (where $\Gamma$ is a quiver and $I$ an admissible ideal in the path algebra $K \Gamma$), there is a finite list of affine algebraic varieties, the…
The paper is a survey of some results about Weil algebras applicable in differential geometry, especially in some classification questions on bundles of generalized velocities and contact elements. Mainly, a number of claims concerning a…
We introduce an axiomatic theory of spherical diagrams as a tool to study certain combinatorial properties of polyhedra in $\mathbb R^3$, which are of central interest in the context of Art Gallery problems for polyhedra and other…
We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding…
In this article we present pictorially the foundation of differential geometry which is a crucial tool for multiple areas of physics, notably general and special relativity, but also mechanics, thermodynamics and solving differential…
Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials…
The ``geometry'', in the sense of the classical differential geometry of smooth manifolds (CDG), is put under scrutiny from the point of view of Abstract Differential Geometry (ADG), along with resulting, thereby, potential physical…
Sheaf theoretically based Abstract Differential Geometry incorporates and generalizes all the classical differential geometry. Here, we undertake to partially explore the implications of Abstract Differential Geometry to classical…
In this thesis three topics on the model theory of partial differential fields are considered: the generalized Galois theory for partial differential fields, geometric axioms for the theory of partial differentially closed fields, and the…
Using the method of moving frames we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants is…
Generalized differential cohomology theories, in particular differential K-theory (often called "smooth K-theory"), are becoming an important tool in differential geometry and in mathematical physics. In this survey, we describe the…
We advocate an account of dualities between physical theories: the basic idea is that dual theories are isomorphic representations of a common core. We defend and illustrate this account, which we call a Schema, in relation to symmetries.…
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the…