Related papers: Monistic conception of geometry
In this paper we discuss, from a historical and philosophical point of view, a variation of the meaning of the five postulates in Euclidean Geometry and we make a short reference to D. Hilberts formalism. We examine, throughout the ages,…
Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in…
We describe for any Riemannian manifold a certain infinitesimal neighbourhood of the diagonal. Semi-conformal maps are analyzed as those that preserve such neighbourhoods; harmonic maps are analyzed as those that preserve mirror image…
Geometric quantization is an attempt at using the differential-geometric ingredients of classical phase spaces regarded as symplectic manifolds in order to define a corresponding quantum theory. Generally, the process of geometric…
We consider spaces for which there is a notion of harmonicity for complex valued functions defined on them. For instance, this is the case of Riemannian manifolds on one hand, and (metric) graphs on the other hand. We observe that it is…
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.
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…
Contemporary relativity theory is restricted in two points: (1) a use of the Riemannian space-time geometry and (2) a use of inadequate (nonrelativistic) concepts. Reasons of these restrictions are analysed in [1]. Eliminating these…
Physical geometry studies mutual disposition of geometrical objects and points in space, or space-time, which is described by the distance function $ d$, or by the world function $\sigma =d^{2}/2$. One suggests a new general method of the…
The unsatisfactory status of the search for a consistent and predictive quantization of gravity is taken as motivation to study the question whether geometrical laws could be more fundamental than quantization procedures. In such an…
Riemannian geometry is a mathematical field which has been the cornerstone of revolutionary scientific discoveries such as the theory of general relativity. Despite early uses in robot design and recent applications for exploiting data with…
Perceptual geometry refers to the interdisciplinary research whose objectives focuses on study of geometry from the perspective of visual perception, and in turn, applies such geometric findings to the ecological study of vision. Perceptual…
The domains of mesh functions are strict subsets of the underlying space of continuous independent variables. Spaces of partial maps between topological spaces admit topologies which do not depend on any metric. Such topologies…
A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions,…
A generalisation of Riemannian geometry is considered, based exclusively on the minimal assumptions that the line element $ds$ is a regular function of position and direction and that the distance of every point from itself is equal to…
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in…
Synthetic Differential Geometry (SDG) is a categorical version of differential geometry based on enriching the real line with infinitesimals and weakening of classical logic to intuitionistic logic. We show that SDG provides an effective…
We define the simplest log-euclidean geometry. This geometry exposes a difficulty hidden in Hilbert's list of axioms presented in his "Grundlagen der Geometrie". The list of axioms appears to be incomplete if the foundations of geometry are…
We show that Euclidean geometry in suitably high dimension can be expressed as a theory of orthogonality of subspaces with fixed dimensions and fixed dimension of their meet.
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…