Related papers: Key developments in geometry in the 19th Century
The development of mathematics has been characterized by the increasing interconnectivity of seemingly separate disciplines. Such interplay has been facilitated by a massive development in formalism; category theory has provided a common…
This article presents a novel mathematical formalism for advanced manifold--metric pairs, enhancing the frameworks of geometry and topology. We construct various D-dimensional manifolds and their associated metric spaces using functional…
A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated differential form formalism developed in math.DG/0605113 and an application to general relativity is given.
We prove a number of results on the geometry associated to the solutions of evolution equations given by first-order differential operators on manifolds. In particular, we consider distance functions associated to a first-order operator,…
This paper is the third in a series dedicated to the fundamentals of sub-Riemannian geometry and its implications in Lie groups theory: "Sub-Riemannian geometry and Lie groups. Part I", math.MG/0210189, available at…
This article is the introductory part of authors PhD thesis. The article presents a new coordinate invariant definition of quasiregular and quasiconformal mappings on Riemannian manifolds that generalizes the definition of quasiregular…
We consider surfaces embedded in a Riemannian manifold of arbitrary dimension and prove that many aspects of their differential geometry can be expressed in terms of a Poisson algebraic structure on the space of smooth functions of the…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of…
Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative…
It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential…
It is known that some equations of differential geometry are derived from variational principle in form of Euler-Lagrange equations. The equations of geodesic flow in Riemannian geometry is an example. Conversely, having Lagrangian…
On a geometrical view, the conception of map geometries are introduced, which is a nice model of the Smarandache geometries, also new kind of and more general intrinsic geometry of surface. Results convinced one that map geometries are…
We present alternative postulates for Euclidean geometry whose merit is that they lead to a new class of invariants and associated geometries for real finite-dimensional unital associative algebras.
One of the major developments of twentieth century physics has been the gradual recognition that a common feature of the known fundamental interactions is their gauge structure. In this article the authors review the early history of gauge…
The main aim of this article is to investigate the geometric structures admitting by the G\"{o}del spacetime which produces a new class of semi-Riemannian manifolds (see Theorem 4.1 and Theorem 4.5). We also consider some extension of…
Infra-nilmanifold endomorphisms were introduced in the late sixties. They play a very crucial role in dynamics, especially when studying expanding maps and Anosov diffeomorphisms. However, in this note we will explain that the two main…
One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this…
The statement of the Gauss-Bonnet theorem brings up an unexpected form of reflexivity (major concept of philosophy of mathematics), so that geometry contemplates itself in it. It is therefore the revolutionary and multifaceted concept of…
The geometric evolution equations provide new ways to address a variety of non-linear problems in Riemannian geometry, and, at the same time, they enjoy numerous physical applications, most notably within the renormalization group analysis…