Related papers: Key developments in geometry in the 19th Century

We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the…

The concept of number and its generalization has played a central role in the development of mathematics over many centuries and many civilizations. Noteworthy milestones in this long and arduous process were the developments of the real…

We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential…

This article explores the overall geometric manner in which human beings make sense of the world around them by means of their physical theories; in particular, in what are nowadays called pregeometric pictures of Nature. In these, the…

We give detailed exposition of modern differential geometry from global coordinate independent point of view as well as local coordinate description suited for actual computations. In introduction, we consider Euclidean spaces and different…

In this note, we use the disformal transformation to induce a geometry from the manifold which is originally Riemannian. The new geometry obtained here can be considered as a generalization of Weyl integrable geometry. Based on these…

We present an overview of recent developments concerning modifications of the geometry of space-time to describe various physical processes of interactions among classical and quantum configurations. We concentrate in two main lines of…

We show how the birth of perspective painting in the Italian Renaissance led to a new way of interpreting space that resulted in the creation of projective geometry. Unlike other works on this subject, we explicitly show how the craft of…

Riemannian metric learning is an emerging field in machine learning, unlocking new ways to encode complex data structures beyond traditional distance metric learning. While classical approaches rely on global distances in Euclidean space,…

We introduce the notion of Fermi flow for hypersurfaces in Riemannian manifolds. It turns out that this is a powerful tool to study the geometry of distance surfaces about a given initial hypersurface. Some of the results in this paper are…

We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree $n$ associated to any given oriented Riemannian manifold $M$ of dimension $n+1$.…

Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…

This talk reviews recent developments in the field of analytical Feynman integral calculations. The central theme is the geometry associated to a given Feynman integral. In the simplest case this is a complex curve of genus zero (aka the…

An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended to complex manifolds.…

This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic…

In its most general form, the recognition problem in Riemannian geometry asks for the identification of an unknown Riemannian manifold via measurements of metric invariants on the manifold. We introduce a new infinite sequence of…

Over the past few years there has been considerable progress in the structural understanding of special Colombeau algebras. We present some of the main trends in this development: non-smooth differential geometry, locally convex theory of…

We briefly review a few aspects of the development of differential geometry which may be considered as being influenced by Einstein's general relativity. We focus on how Einstein's quest for a complete geometrization of matter and…

[1] investigates advanced connotations of Hardy and Rellich-type inequalities on complete noncompact Riemannian manifolds, delving on deriving inequalities that incorporate poignant weight functions. These inequalities prolongate classical…

We introduce an alternative formalization of curved spaces in which the concept of a pointwise affine space, as defined here, replaces that of a manifold. New or modified definitions of familiar notions from differential geometry such as…