Hi,
I've been watching lectures from XylyXylyX on YouTube. I believe they are really great !
One doubt about the introduction of Covariant Derivative. At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a...
I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors.
What people usually do is
take the covariant derivative of the covector acting on a vector, the result being a scalar
Invoke a product rule to...
Hello.
The curve y = x2 is a parabola that looks like this:
I have a shape Square that looks like this:
What I am noticing is that if I consider the equation y = x2 and also the shape Square, I find that there is no connection between them but the equation y = x2 is pronounced as x-square...
Hello!
I have learned Riemannian Geometry, so the only connection I have ever worked with is the Levi-Civita connection(covariant derivative of metric tensor vanishes and the Chrystoffel symbols are symmetric).
When performing a parallel transport with the L-C connection, angles and lengths are...
Hello!!
Since connections in general do not require that we have a Riemannian manifold, but only a smooth manifold, I find it kind of weird that the only examples of connections that I find in the internet are those which use the Levi-Civita connection.
So, I wanted to know of any examples of...
I'm currently in a GR class and have come across the notion of parallel transport, and I've searched and searched the last few days to try and understand it but I just can't seem to wrap my head around it, so I'm hoping someone here can clarify for me.
The way I picture parallel transport is...
Homework Statement
Show that the metric connection transforms like a connection
Homework Equations
The metric connection is
Γ^{a}_{bc} = \frac{1}{2} g^{ad} ( ∂_{b} g_{dc} + ∂_{c} g_{db} - ∂_{d} g_{bc} )
And of course, in the context of Einstein's GR, we have a symmetric connection,
Γ^{a}_{bc}...
I have question about the proof of Cartan's structure equation in the context of pincipal bundle in Nakahara's book. The attached image is taken from the book.
To show that the curvature two-form ##\Omega ## satisfies
##\Omega (X,Y) = {d_P}\omega (X,Y) + [\omega (X),\omega (Y)]##
The author...
I have been reading Nakahara's book "Geometry, Topology & Physics" with the aim of teaching myself some differential geometry. Unfortunately I've gotten a little stuck on the notion of a connection and how it relates to the covariant derivative.
As I understand it a connection ##\nabla...
Hi, some one know the expression of the affine connection Γ in terms of tetrad formalism? I would like also some references if it's possible, i found a hit but i think that is wrong... please help me it's so important!
I have a question about the directional derivative of the Ricci scalar along a Killing Vector Field. What conditions are necessary on the connection such that K^\alpha \nabla_\alpha R=0. Is the Levi-Civita connection necessary?
I'm not sure about it but I believe since the Lie derivative is...
Hello everyone :)
I came across a strange situation today when I was listening to my music with my headphones on. When I pressed play to a song, the instrumental intro started off fairly normally (although the audio sounded a little "thinner" than usual, as if compressed), and then, when...
Homework Statement
[/B]
(a) Find christoffel symbols and ricci tensor
(b) Find the transformation to the usual flat space form ## g_{\mu v} = diag (-1,1,1,1)##.
Homework Equations
The Attempt at a Solution
Part(a)
[/B]
I have found the metric to be ## g_{tt} = g^{tt} = -1, g_{xt} =...
"Everyone" knows what an affine connection on a smooth manifold is a.k.a. covariant derivative. My questions are:
i) Why are those connections called affine?
ii) Is there a mathematical object that 'connects neighboring tangent spaces', that could be termed a 'non-affine connection'?
Dear Readers,
I'm facing a pretty interesting challenge, which is: "How to connect multiple smartphone mic's in the quickest and simplest way?"
Here is a use case. I go into a meeting with a group of random (strangers) that all have smartphones, Android, iOS, Windows. Now I want to record this...