ガロア第一論文と乗数イデアル他関連資料スレ2

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792: １３２人目の素数さん [] 2023/03/27(月) 12:02:51.50 ID:ZryxA1Gf

>>190
ありがとう
tractorが、不勉強で初見だな

https://arxiv.org/pdf/1412.7559
[Submitted on 23 Dec 2014 (v1), last revised 1 Aug 2015 (this version, v2)]
An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity
Sean Curry, A. Rod Gover

Abstract. The following are expanded lecture notes for the course of eight one
hour lectures given by the second author at the 2014 summer school Asymptotic
Analysis in General Relativity held in Grenoble by the Institut Fourier. The
first four lectures deal with conformal geometry and the conformal tractor calculus, taking as primary motivation the search for conformally invariant tensors
and diffrerential operators. The final four lectures apply the conformal tractor
calculus to the study of conformally compactified geometries, motivated by the
conformal treatment of infinity in general relativity.

Contents
0. Introduction 2
0.1. Notation and conventions 4
1. Lecture 1: Riemannian invariants and invariant operators 6
1.1. Ricci calculus and Weyl’s invariant theory 7
1.2. Invariant operators, and analysis 8
2. Lecture 2: Conformal transformations and conformal covariance 9
2.1. Conformal Transformations 9

P4
Also left out in these notes is any discussion of conformal spin geometry. In this
case there is again a canonical tractor calculus, known as spin tractor calculus or
local twistor calculus, which is a refinement of the usual conformal tractor calculus
in the same way that spinor calculus is a refinement of the usual tensor calculus
on pseudo-Riemannian spin manifolds. The interested reader is referred to [4, 50].

http://rio2016.5ch.net/test/read.cgi/math/1677671318/792

795: １３２人目の素数さん [] 2023/03/27(月) 17:07:19.22 ID:ZryxA1Gf

>>792
>tractor

"tractor calculus math"で検索すると下記ヒット
２件貼る
”This is completely analogous to the more familiar tensor calculus as it has come to dominate (pseudo)-Riemannian geometry. Indeed, there is a very close link between the tractor calculus on a conformal manifold and the tensor calculus on the corresponding ambient metric [5].”か

１）
https://www.semanticscholar.org/paper/COMPUTING-WITH-THE-TRACTOR-CALCULUS-IN-CONFORMAL-Case/11d1538ffa445bcec400c1890c527e3a68825d8d
Corpus ID: 37351971
COMPUTING WITH THE TRACTOR CALCULUS IN CONFORMAL GEOMETRY
Jeffrey S. Case Published 2011
http://www.personal.psu.edu/jqc5026/notes/tractor.pdf
COMPUTING WITH THE TRACTOR CALCULUS IN CONFORMAL GEOMETRY
JEFFREY S. CASE Date: September 23, 2011.
1. Introduction
The tractor calculus is an efficient and powerful tool for working in conformal geometry.
In the sense used here, the tractor calculus provides a systematic method for studying conformal geometry using a distinguished family of vector bundles, the
so-called tractor bundles, together with a distinguished connection.
By construction, these bundles are intrinsically conformally invariant, and thus are particularly well-suited to problems in conformal geometry.
This is completely analogous to the more familiar tensor calculus as it has come to dominate (pseudo)-Riemannian geometry. Indeed, there is a very close link between the tractor calculus on a conformal manifold and the tensor calculus on the corresponding ambient metric [5].

つづく

http://rio2016.5ch.net/test/read.cgi/math/1677671318/795

796: １３２人目の素数さん [] 2023/03/27(月) 17:07:58.57 ID:ZryxA1Gf

>>795
つづき

２）
（2時間もの動画）
https://www.youtube.com/watch?v=xvHPud67fdc
Rod Gover - An introduction to conformal geometry and tractor calculus (Part 2)
Institut Fourier   2015/06/01
After recalling some features (and the value of) the invariant ≪ Ricci calculus ≫ of pseudo-‐Riemannian geometry, we look at conformal rescaling from an elementary perspective. The idea of conformal covariance is visited and some covariant/invariant equations from physics are recovered in this framework. Motivated by the need to develop a more effective approach to such problems we are led into the idea of conformal geometry and a conformally invariant calculus; this ≪ tractor calculus ≫ is then developed explicitly.
We will discuss how to calculate using this, and touch on applications to the construction of conformal invariants and conformally invariant differential operators. The second part of the course is concerned with the application of conformal geometry and tractor calculus for the treatment of conformal compactification and the geometry of conformal infinity. The link with Friedrich’s conformal field equations will be made. As part of this part we also dedicate some time to the general problem of treating hypersurfaces in a conformal manifold, and in particular arrive at a conformal Gauss equation. Finally we show how these tools may be applied to treat aspects of the asymptotic analysis of boundary problems on conformally compact manifolds.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1677671318/796

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