What is Connection: Definition and 516 Discussions
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a coordinate frame, and so is not a tensorial object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them. In physics, connection forms are also used broadly in the context of gauge theory, through the gauge covariant derivative.
A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because under a change of basis, the connection form transforms in a manner that involves the exterior derivative of the transition functions, in much the same way as the Christoffel symbols for the Levi-Civita connection. The main tensorial invariant of a connection form is its curvature form. In the presence of a solder form identifying the vector bundle with the tangent bundle, there is an additional invariant: the torsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of a fiber bundle with a structure group.
I was watching the Elegant Universe on PBS, and the show was talking about the string theory. there was one thing that i didnt really get though. It was the relation between strings and branes. I think i heard that if you were to add energy to a string, it gets bigger, and is stretched into a...
I had an idea the other day that if I could set up a wireless connection from my laptop to my pc which is connected to the internet via cable I could work on my laptop down the road (in a pub or park depending on the weather) I guess it would involve some protocol like bluetooth has anybody done...
Last December, I noticed that the December issue of "National Geographic" magazine had a feature story (30+ pages long) on samurai. I immediately suspected that this was an advertisement/PR for Tom Cruise's latest movie "The Last Samurai," probably paid by him or his camp.
What do you think?
Does existentialism have to do with idealism? I mean, in the sense that nothing exists outside of the minds' awareness of it? Sort of like Berkeley's idea? Do they have anything to do with each other?
I've been studying the rule of 72 lately, which gives the time in years that it takes for an investment to double based on the annual percentage rate of growth.I have tried to compare it's formula t=72/x (x being the number in front of the percent sign) with the exact value for the doubling time...
A good read. This gets a little more serious though. Please see what follows the mythology below.
http://www.vanhunks.com/tokoloshe1.html
This is the story that I was actually reviewing when I ran across the brown, hairy, one-butts with a lisp.
Formally known as The Zimbabwe School...
An incredibly loose thought, which i do not care to think about any further, is expressed here...
The Bible fames light alot... It characterizes light as a life giving component and a gift.
Light, physically, is possibly the most fundamental component of our universe.
a connection...
This has been bothering me for a long time. This thread (for those who will argue) will also go under the particle physics section too.
Ok, in our world we live in, we have our basic particles (strings, quarks, gluons, electrons, whatever) these particles held together by forces make up our...
how can the God connection be developed if you don’t have one?
it is not difficult. start by consciously having that as a goal. try the following prayer: God that i be with you in conscious contact, that i feel your presence in my life. then observe obstacles you have in place against feeling...
Members of Bush's Cabinet, all from big oil and energy and mining? You bet.
http://news.bbc.co.uk/1/low/world/americas/1138009.stm
http://www.opensecrets.org/bush/cabinet.asp
In the Physics forum, jby started a LQG thread which at one point included some discussion of Connections. It might be helpful to have a thread focused on the connection which is taking over the central role from the metric for some purposes.
GR used to be the study of the metric on a smooth...
My first post, about rotation groups..
A result about rotation groups.
To me this seems clear, simple and very intuitive, but in all the papers and books I've read on the subject I have never seen it presented. Maybe some of you have seen it, or maybe it is new. It is very simple to state...
Here's the idea. We can help each other study for our qualifier exams by posting qualifer-type questions along with the solutions in this thread. I think this thread will also help those studying for the physics GRE... or even those just wanting to get good at solving exam-type physics...