Does parallel transport mean relative to the manifold or the ambient space? But if the former, what does that mean? I always envision parallel as being defined relative to a plane of some sort. So I'm having trouble visualizing more complex structures where you might parallel transport
@StanShunpike Have you looked at the equivalence of parallel transport and connection? Essentially, you are defining what it means to be "parallel" when you give the notion of covariant derivative - or you are defining what it means to covariantly derive when you give the notion of parallel transport
What confuses me is aren't there many kinds of connections? Whereas constant direction and magnitude are intuitive, I don't understand what the properties of a connection (for example as defined in Wald or do Carmo) has to do with commecting tangent spaces
And I just learned from Wald that the uniqueness property of the Levi-Civita is crucial because it allows us to pick out a specific directional derivative operator. Or something like that. Why wouldn't there automatically be a unique on on a smooth manifold?
@StanShunpike Heh. Well, because every bundle has its own curvature, and what you usually call "curvature of the manifold" is really "curvature of the tangent bundle with the Levi-Civita connection".
And because the tangent bundle is somewhat natural to consider for a manifold, one does not really distinguish here
@Danu Well, but, for example, a very interesting construction is localizing a ring, i.e. partially making its elements invertible w.r.t. multiplication. This fails as a nice construction horribly if you don't have associativity, and you can't even say what invertible means lacking a unit
The reason I am asking this question is because if all points in space observe recession of galaxies the same as we do from Earth, the universe would have to be infinite (or a closed sphere in 4D or something.
I know infinite space isn't a formal position of Big Bang cosmology, but is a non infi...
How can curvature of a bundle inform me about curvature of a manifold? If I am measuring properties about the bundle, is there a one-to-one correspondence between points in the bundle and points on the manifold such that I get info about the manifold from the bundle?
The absolute best book for self-study in algebra to me is E.B. Vinberg's A Course In Algebra. A Course In Algebra by E.B.Vinberg This book very rapidly became my favorite reference for algebra. Translated from the Russian by Alexander Retakh, this book by one of the world’s preeminent algebracist...
@ACuriousMind the connection is defined on the bundle. And the tangent bundle is just the disjoint union of tangent spaces. So what does that have to do with the original manifold? Like if the tangent bundle has the structure of a connection that doesn't mean the manifold does, right?
@StanShunpike The manifold cannot have the structure of a connection, because connections, by definition, live on bundles. "Disjoint union" is a characterization of the tangent bundle I've grown to dislike, because it does not carry the disjoint union topology.
I'd rather say that the tangent bundle arises naturally as the vector bundle on which the Jacobians of the coordinate transformations between the local charts of the manifold act, but for this to make sense, you'd have to internalize that bundles are defined by local patches and transitions between them.
Hmm...I see your point. I thought that definition seemed oversimplisitc somehow. Why do they have to live on bundles? Wikipedia does say that is part of the definition but doesn't say why. wikipedia mentions a space of infinitely differentiable vector fields and the connection is a map on this. Can that space not be defined on the manifold itself without the bundle?
I thought smooth manifolds were by definition infinitely differentiable.
On page 31 of Wald, he discusses defining a commutator of two vector fields "in terms of any derivative operator". What does he mean? Does he mean write the vector fields in terms of the derivative operator or the commutator?
All I'm familiar with are $\alpha, \beta, \gamma$ decays and the above is no direct disintegration.
I too feel that this is thermonuclear fusion. But maybe my lack of knowledge on this subject leaves me out of being any judge of it. So, could someone please inform me on what the mechanism behind the above is?
... oh wait! Ive found it
It's in the wiki article for fusion.
Ohk, it's $\beta^+$ decay followed by $\gamma$ decay and then fusion.
So, I guess it's still fusion.
What was my teacher smoking to have said fission!?
The problem here was, I think, a lack of insight into how that final step took place. He must have thought that the heliums combined together and split up again as opposed to a head on collision during which two protons are thrown away.