Split torus

Sheaf in fpqc topology 117 Smooth group scheme 88 Solvable group scheme 73 Spec A 41 Split torus 56 Strictly upper triangular 62 Subcomodule 23 Symplectic group 99... 

A droplet is initially flattened to an oblate, lenticular ellipsoid and then may be converted into a torus, depending on the magnitude of the internal forces causing the deformation. The torus subsequently becomes stretched and splits into smaller droplets. 

One may naively suppose that this phenomenon is caused by Rayleigh instability, that is, a single torus may split upon the accumulation of the charge. 

We can use the previous theorem to show that every torus is nearly made up of two extreme types. Call a torus split (deploye) if it is actually diagonalizable, or in other words the Galois action on the character group is trivial. At the other extreme, call it anisotropic if it has no nontrivial maps to Gm, or in other words zero is the only fixed element in the character group. 

Theorem. Every torus T has a largest split subtorus Td and a largest anisotropic subtorus Ta. The intersection Td n Ta is finite, and T equals Ta - Td in the sense that no proper closed subgroup contains them both. 

The drop looks like an oblate spheroid (Fig. 17.11, a). At further deformation the drop becomes a torus and then stretches and splits into fine droplets. 

A torus 7 splits into two tori as it passes through the centre of the trousers... 

When passing through the critical value a, the torus splits into two tori, Ti g and T2, . The separatrix diagram P. goes from the critical surface and, when descending, meets the torus along two circles 71 and 72 (see above). Lemmas... 

Denote by Ki and K2 both rings, into the union of which the circles 71 and 72 split the torus Construct a new surface Pi by adding to the separatrix... 

If a critical torus has the dimension n, then it is either the set of the local minimum or of the local maximum of the energy H. In this case, either two close nonsingular Liouville tori flow into one torus or the torus T splits into two tori r. Let P2 = P2 T ) and = P T ) be, respectively, in and out separatrix diagrams of the critical submanifold... 

Definition 3.4.6 A symplectic structure M w) will be called isotrivial if there exists a finite nonbranching covering x M — M, such that M splits into the direct product of an n-dimensional complex torus, completely isotropic with respect to the form... 

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