Smooth functions | Maps of manifolds | Differential geometry | Connection (mathematics)

In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space Rn by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space. On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a metric tensor then there is a natural choice of affine connection, called the Levi-Civita connection. The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties (linearity and the Leibniz rule). This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle. A choice of affine connection is also equivalent to a notion of parallel transport, which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the frame bundle. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group or as a principal connection on the frame bundle. The main invariants of an affine connection are its torsion and its curvature. The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection. Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry; the main differences are encapsulated in the curvature of the connection. (Wikipedia).

Affine Springer fibers and representation theory - Cheng-Chiang Tsai

Short talk by postdoctoral members Topic: Affine Springer fibers and representation theory Speaker: Cheng-Chiang Tsai, Member, School of Mathematics For more videos, visit http://video.ias.edu

From playlist Mathematics

Tensor Calculus Episode 10 | Is the Affine Connection a Tensor?

In todays video I look at the transformation properties of the affine connection coefficients to see if they transform as tensor components. This series is based off "Tensor Calculus for Physics" by Dwight Neuenschwander which can be found at: https://www.amazon.com/gp/product/1421415658/

From playlist New To Tensors? Start Here

Novel Algebraic Operations for Affine Geometry | Algebraic Calculus One | Wild Egg

We introduce some novel conventions to help us set up the foundations of affine geometry. We learn about differences of points, sums of points and vectors, affine combinations and vector proportions. And then use these to state a number of important results from affine geometry, including

From playlist Algebraic Calculus One from Wild Egg

Affine geometry and barycentric coordinates | WildTrig: Intro to Rational Trigonometry

Affine geometry is the geometry of parallel lines. Using parallelism, we show how to construct a ruled line, how to find the midpoint of a segment, and divide a segment into a given ratio. We connect this to Archimedes law of the lever, and then extend to barycentric coordinates with respe

From playlist WildTrig: Intro to Rational Trigonometry

Affine and mod-affine varieties in arithmetic geometry. - Charles - Workshop 2 - CEB T2 2019

François Charles (Université Paris-Sud) / 24.06.2019 Affine and mod-affine varieties in arithmetic geometry. We will explain how studying arithmetic versions of affine schemes and their bira- tional modifications leads to a generalization to arbitrary schemes of both Fekete’s theorem on

From playlist 2019 - T2 - Reinventing rational points

What are affine transformations?

Algorithm Archive: https://www.algorithm-archive.org/contents/affine_transformations/affine_transformations.html Github sponsors (Patreon for code): https://github.com/sponsors/leios Patreon: https://www.patreon.com/leiosos Twitch: https://www.twitch.tv/leioslabs Discord: https://discor

From playlist Algorithm Archive

Affine Transformations — Topic 27 of Machine Learning Foundations

In this video we use hands-on code demos in NumPy to carry out affine transformations, a particular type of matrix transformation that may adjust angles or distances between vectors, but preserves parallelism. These operations can transform the target tensor in a variety of ways including

From playlist Linear Algebra for Machine Learning

algebraic geometry 5 Affine space and the Zariski topology

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the definition of affine space and its Zariski topology.

From playlist Algebraic geometry I: Varieties

What are the Angle Relationships for Parallel Lines and a Transversal

👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i

From playlist Parallel Lines and a Transversal

Holomorphic rigid geometric structures on compact manifolds by Sorin Dumitrescu

Higgs bundles URL: http://www.icts.res.in/program/hb2016 DATES: Monday 21 Mar, 2016 - Friday 01 Apr, 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore DESCRIPTION: Higgs bundles arise as solutions to noncompact analog of the Yang-Mills equation. Hitchin showed that irreducible solutio

From playlist Higgs Bundles

Mad Max: Affine spline insights into deep learning - Richard Baraniuk, Rice University

This workshop - organised under the auspices of the Isaac Newton Institute on “Approximation, sampling and compression in data science” — brings together leading researchers in the general fields of mathematics, statistics, computer science and engineering. About the event The workshop ai

From playlist Mathematics of data: Structured representations for sensing, approximation and learning

From Cohomology to Derived Functors by Suresh Nayak

PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics

From playlist Dualities in Topology and Algebra (Online)

Affine Combinations and Barycentric Coords | Algebraic Calculus One | Wild Egg

In this video we show how affine combinations and barycentric coordinates express mathematically what Archimedes' Law of the Lever captures in terms of the centre of mass of a triangle. We examine both the one dimensional case of a segment, as well as the more general two dimensional case

From playlist Algebraic Calculus One from Wild Egg

Dynamics on character varieties - William Goldman

Character Varieties, Dynamics and Arithmetic Topic: Dynamics on character varieties Speaker: William Goldman Affiliation: University of Maryland; Member, School of Mathematics Date: November 17, 2021 In these two talks, I will describe how the classification of locally homogeneous geomet

From playlist Mathematics

Approximation with deep networks - Remi Gribonval, Inria

This workshop - organised under the auspices of the Isaac Newton Institute on “Approximation, sampling and compression in data science” — brings together leading researchers in the general fields of mathematics, statistics, computer science and engineering. About the event The workshop ai

From playlist Mathematics of data: Structured representations for sensing, approximation and learning

Rod Gover - Geometric Compactification, Cartan holonomy, and asymptotics

Conformal compactification has long been recognised as an effective geometric framework for relating conformal geometry, and associated field theories « at infinity », to the asymptotic phenomena of an interior (pseudo‐)‐Riemannian geometry of one higher dimension. It provides an effective

From playlist Ecole d'été 2014 - Analyse asymptotique en relativité générale

Modular Perverse Sheaves on the affine Flag Variety - Laura Rider

Virtual Workshop on Recent Developments in Geometric Representation Theory Topic: Modular Perverse Sheaves on the affine Flag Variety Speaker: Laura Rider Affiliation: University of Georgia Date: November 16, 2020 For more video please visit http://video.ias.edu

From playlist Virtual Workshop on Recent Developments in Geometric Representation Theory

algebraic geometry 17 Affine and projective varieties

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the relation between affine and projective varieties, with some examples such as a cubic curve and the twisted cubic.

From playlist Algebraic geometry I: Varieties

Lecture 2A: What is a "Mesh?" (Discrete Differential Geometry)

Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg

From playlist Discrete Differential Geometry - CMU 15-458/858