Yes, precisely. The metric is locally defined - that's why one calls it a metric (tensor) field - it may be different at every point. There is however an invariant derived from it, called the invariant line element. I need to get my head straight after a bruising day to find a way to make this accessible. Later..... You do when possible. Manifolds with boundaries are nightmarish to deal with! Regarding what you called the "Maxwellian manifold". I never heard of such a beast, but then, I am not qualified answer your specific question. It is possible - just possible - that the Faraday field strength tensor may take the place of the metric in EM, but since it is clearly anti-symmetric, I am not sure.
Nor have I. I think in fact circuit analysis of the analog variety corresponds pretty much to a boundary value problem. One important, perhaps invariant, aspect are what's known as Kirchoff's Laws of voltage and current, namely that in any closed loop the sum of voltages is zero, and at any node the sum of currents is zero. We were told that the analysis also depends on complex frequency; the domain is \( e^{st} \), where \( s = \sigma + i\omega \). Engineers then replace i with j because i represents current. I'll dig up what my notes say about why complex frequency is used, but it has to do with the Laplace transform. Anyway, if the sine and cosine terms of a Fourier series are an orthonormal basis, what is the dual basis?
Found this post here: " ... Spectral theory, however, is not well-adapted to general relativity for another reason: the metric is not Riemannian. The associated Laplace-Beltrami operator for a Lorentzian manifold is the d'Alembert, or wave operator, which is not a signed operator like the Laplacian. So even for compact space-times the spectral theory is not well developed. Furthermore, if you are interested in a evolutionary problem, you will not want to take a Fourier transform in the time-direction. One may think that because a spatial slice has a Riemannian structure, if we assume a compact spatial slice, the Fourier transform is well behaved. Unfortunately, since the metric will also evolve, one cannot compare the eigenfunctions from one slice to the eigenfunctions of the next. Furthermore, this will also break the general covariance of the problem, as one will be forced to contend with a 3+1 splitting of the manifold. III. Further difficulties Even assuming you have some way of getting a good definition for the Fourier transform of the function on a manifold, you will still have to contend with the problems posed by vectors and tensors. The problem is that the tangent bundle for a curved manifold is not trivial: One cannot just decompose TM = M x R^n the way one does Euclidean/Minkowski space. So now you will have to adjust your Fourier transform so that it plays well with the connection / covariant derivative. Suffice to say, the task is daunting. The only progress I know in this direction is the work of Klainerman and Rodnianski" From which I can gather it's a good thing electronics engineers don't worry about gravity . . .
OK,let me draw this largely unpopular thread to a close with the so-called invariant line element. Suppose \(M\) an arbitrary manifold with a metric. This manifold may be trivial. Further suppose a point \(p\in M\) with a coordinate representation \(p=\{x^1,x^2,......,x^n\}\). Then for an arbitrary displacement \(dx^j\) (a vector, obviously), write \(ds^2=\sum_{j,k}\delta^j_k dx^jdx^k\) where one has that \(\delta^j_k=\cases{1\quad j=k\\0 \quad j \ne k}\). This obviously becomes \(ds^2 =(dx^1)^2+(dx^2)^2+.......+(dx^n)^2\) which we recognize as a generalized Pythagorean. In other words, we have a line element in a Euclidean n-space. Now write \(ds^2=\sum_{j,k}\eta_{jk}dx^jdx^k\) where \(\eta_{jk}=\cases{-1\quad j=k=1\\1 \quad j=k \ne 1\\0\quad j \ne k}\). This is the line element in (flat) Minkowski spacetime Finally write \(ds^2=\sum_{j,k}g_{jk}dx^jdx^k\) where \(g_{jk}=g_{kj}\). This is the Riemann line element.
You know, I had a look at this stuff and your presentation style just doesn't mesh with my learning style well enough to make this worth my while to pursue, QuarkHead. Thanks for trying though. I guess I'll just have to muddle along with trig, algebra, and differential calculus. Seems like I do OK with that for most of the posting I see here anyway.
I.e. Wikipedia is saying the tangent space (tangent to what?), plus the metric gives you a way to define a rest frame in what I like to think of as a velocity field. . And a nice 1 + 1 diagram: Please Register or Log in to view the hidden image! Note the red curve passing through the origin for some object which is accelerating relative to the at rest frame.
