Foliated tori

Next considering the case of reactive motion at the same energy E, we realize that the situation at hand is not very different. The phase space of qi is still elliptical. The phase space of is not elliptical, but it is a simple closed curve, and it still therefore has the same topology as a one-dimensional sphere (every point on a closed curve can be uniquely mapped onto a sphere). Thus, the phase space of reactive motion consists of foliated tori that span both sides of the potential barrier. These reactive tori will be skinny when sliced along the ( 2 Pz) compared to the trapped tori, because they have less energy in the vibrational coordinate and more in the reaction coordinate. In Figure 8 these are labeled Qab j... 

We can construct the representation of the sphere with this parameterization. One must, however, be careful when 4 = 0 (resp. ly = 0) since then the corresponding angle 9 (resp. 0, ) is not defined. The two angles 0 < Qx,y < specify a 2-torus. To fully foliate the sphere, two distances 4,/y, with a linear relation, may be specified. A one-parameter family of tori of varying radii foliates the 3-sphere. Each trajectory is fully specified by a point on these tori. 

Since 4> does not appear explicitly in the Hamiltonian, we go one step further, exploiting the other constant of the motion, A (rotational invariance of the Hamiltonian). Let us define a torus T2 C S3 in the following way. Since A is a conserved quantity, the A = Aq surfaces foliate the S3 (Pi ) sphere in a... 

Figure 7 (Top left) Torus of constant-energy vibrational motion in two uncoupled degrees of freedom, projected upon the three-dimensional space (9, q, pz). (Below right) Three nested tori, sliced to reveal their foliated structure.

On an even-dimensional complex torus there are many symplectic structures. All of them are uniquely determined by their values at a certain point. It is therefore reasonable first to investigate separately the question of the existence of foliation into tori of smaller dimensions and then to find out with respect to which symplectic structure this foliation is isotropic. It turns out that on a general complex torus there are no foliations into tori of smaller dimensions, and if they do exist, they are locally trivial. 

If m = 1, then is a manifold. It is homeomorphic to a torus if m 1 and to a Klein bottle if m = — 1. As already mentioned, this manifold may be smooth or non-smooth, depending on whether intersects the strong-stable foliation transversely everywhere or not. When x and (p are... 

---