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An isotropic 2D oscillator, when taken into action-angle variables by doing a canonical transformation of the Hamiltonian, $$ H(q_1, p_1, q_2, p_2 ) = \frac{q_1^2}{2m} + \frac{kq_1^2}{2} + \frac{q_2^2}{2m} + \frac{kq_2^2}{2} $$ will yield two constants of the motion, the actions , and two angles running from 0 to $2\pi$ which generates a torus. I'm not ...


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Every single comment and answer was very useful. I think i have found an answer in an old paper by Clark Glymour (Minnesota studies in philosophy of science, volume III, pp.50-60): "It has recently been noted (Ellis, 1971; Dautcourt, 1971; Ellis and Sciama, 1972; Glymour, 1972; Trautman, 1965) that in some general relativistic cosmologies various global ...


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As has been discussed in many questions around here (e.g. here), relativity tells us only about local properties and behavior of a space-time. There are some exceptions when we make global assumptions - if we have a space of globally and strictly constant positive curvature, non-trivial topology is imminent because the space has to be the 3-sphere ...


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I'm not going to provide a full answer here, because I don't know the answer, but I want to give some statements that illustrate quite nicely the kind of problems one would face when determining topology of anything: We know spacetime is a manifold. That means, locally, it looks just like $\mathbb{R}^4$. That's already a bummer. We can't do jack at one ...


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Consider a non-relativistic massless particle with charge $q$ on a 2D torus $$\tag{1} x ~\sim~ x + L_x , \qquad y ~\sim~ y + L_y, $$ in a constant non-zero magnetic field $B$ along the $z$-axis. Locally, we can choose a magnetic vector potential $$\tag{2} A_x ~=~ \partial_x\Lambda, \qquad A_y ~=~ Bx +\partial_y\Lambda, $$ where $\Lambda(x,y)$ is ...


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$U(1)$ Chern-Simons theory with (physical) space a 2-torus is such an example. Its phase space is the gauge equivalence classes of flat connections on the 2-torus. These are specified by the holonomies around two 1-cycles forming a basis of $H_1(T^2)$. This is of course a 2-torus $U(1) \times U(1)$. Because of the form of the Chern-Simons action, these ...


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In solid state physics, the bulk of a crystal is usually given periodic boundary conditions to avoid the sticky problem of what to do at the termination of the crystal. So the crystal is all bulk, no surface. This turns out to be a very good approximation to the bulk of a real crystal. It also gives the solid the topology of a 3-torus.


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I wondered about this one myself a while back. I'm not absolutely positive about this but it is definitely in the ballpark. Here's what I know for the background: I believe the first paper on exotic spheres in physics was by Witten [Commun. Math. Phys., 100, 197–229 (1985)] and centered around the idea that exotic spheres can be interpreted as ...


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Other answers have made clear the 'flat' only implies infinite given additional assumptions around the topology. In short: A universe which is the same everywhere but not simply connected can be finite. It's worth mentioning that whilst the main working model assumes that the universe is simply connected, the actual topology is an open and serious ...


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The phase space is a symplectic manifold, so any manifold $\mathcal{M}$ that admits a closed nondegenerate 2-form is a possible phase space. Now, what is necessary (or sufficent) for admitting such a form? First, as you mention, $\mathcal{M}$ must be even-dimensional. Second, $\mathcal{M}$ must be orientable. Why? Because orientability is equivalent to ...


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The bible for the mathematical formulation of classical Mechanics, namely Foundations of Mechanics by Abraham and Marsden, defines a hamiltonian system as a triple $(M, \omega, X_H)$ where $(M, \omega)$ is a symplectic manifold, and $X_H$ is the Hamiltonian vector field corresponding to a hamiltonian function $H:M\to\mathbb R$. Now, are there typically any ...


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This claim is simply wrong. The flat hyperplane is of course infinite, but non trivial topologies can be flat and still finite. The simplest example is the 3-torus, but there are even the Klein bottle and the Hantzsche-Wendt manifold. See for example page 27 of Janna Levin - Topology and the Cosmic Microwave Background, which show you ten different closed ...


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We need to be precise about the phrase the size of the universe. Specifically I'm going to take it to mean the maximum possible separation between any two points. In an infinite universe two points can be separated by an arbitrarily large distance, so if the maximum distance between two points is finite this means the universe must not be infinite. The ...


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I think that it is important to note that (almost) everyone doing cosmology works within the framework of the FLRW universe. This implies that we assume that the universe is spatially homogeneous and isotropic, i.e. 'every place is the same (at least on large scales)'. Now, think of a flat, finite universe: Is it possible to maintain that all places are ...


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I couldn't resist putting the sledgehammers to work that Robin Ekman alluded to, and I will try to present their general argumentation in a way one can understand without knowing all the details, but I don't intend to presume that this is in any sense a better answer, it is simply one that shows which things one must know to rigorously understand the weird ...


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From $$\operatorname{curl} A = 0 \tag{1} $$ in a region $U$, you can in general not conclude that $$A = \nabla f \tag{2}$$ for some function $f$ defined on all of $U$. Indeed this is related to the singularity, which removes a line through the origin. The degree to which (1) fails to imply (2) depends on the topology of $U$, more specifically it's de Rham ...


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This is simply a short addendum to Robin Ekman's answer and response to your comment So I could phrase it this way: Lorentz transformations are homeomorphisms, so even though they open sets not invariant, all topological notions are still preserved? Homeomorphism is indeed the key concept here, and I wish to add a very slight nitpick with Robin's ...



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