相关论文: Riemannian Geometry: Definitions, Pictures, and Re…
These notes on Riemannian geometry use the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It has more problems and omits the background material. It starts with the definition of Riemannian…
Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing…
A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing…
The Riemannian geometry is one of the main theoretical pieces in Modern Mathematics and Physics. The study of Riemann Geometry in the relevant literature is performed by using a well defined analytical path. Usually it starts from the…
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…
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…
This document contains a description of physics entirely based on a geometric presentation: all of the theory is described giving only a pseudo-riemannian manifold (M, g) of dimension n > 5 for which the g tensor is, in studied domains,…
Newtonian, Lagrangian, and Hamiltonian dynamical systems are well formalized mathematically. They give rise to geometric structures describing motion of a point in smooth manifolds. Riemannian metric is a different geometric structure…
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…
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.…
Motivated by a paper of Zirnbauer, we develop a theory of Riemannian supermanifolds up to a definition of Riemannian symmetric superspaces. Various fundamental concepts needed for the study of these spaces both from the Riemannian and the…
Some geometric structures with associated Riemannian metrics have been considered in the book.
A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step towards the construction of a consistent noncommutative gravitational theory. Historically, as well, Riemannian geometry was recognized to be the…
This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra…
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,…
Graphs are ubiquitous, and learning on graphs has become a cornerstone in artificial intelligence and data mining communities. Unlike pixel grids in images or sequential structures in language, graphs exhibit a typical non-Euclidean…
Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations…
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.
I introduce a new geometrical approach to thermo--statistical mechanics. Here I highlight the main physical ideas, and how do they translate into geometrical language. I contrast the present approach with previous…
A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new T-geometric one are considered. T-geometry is built only on the basis of information included in the metric (distance between two…