By I. L Walker

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**Additional resources for Crystallography: An outline of the geometrical problems of crystals**

**Example text**

Fix a point (x0 , p0 ) ∈ T ∗ M and let ψ be a closed 1-form on M such that ψ(x0 ) = (x0 , p0 ). Since f is symplectic, the form µ = f ◦ ψ is also closed, and thus the map h = fµ−1 ◦ f ◦ fψ is a symplectomorphism of T ∗ M which preserves fibers and fixes the zero section ZM ⊂ T ∗ M . 1 that Dh is the identity at all points of ZM . Consequently, the fiber-derivative of f at the arbitrary point (x0 , p0 ) equals the identity, so f is a translation on each fiber. Defining β(x) = f (x, 0), we have f = fβ .

22 (Moser [45]) If {ωt } is a family of symplectic structures on a compact manifold P and ωt1 − ωt2 is exact for all t1 , t2 , then there is a diffeomorphism f : P → P with f1∗ ω1 = ω0 .

E. only for an exact lagrangian 36 submanifold. Since we are specifically trying to go beyond the exact case, this interpretation is not acceptable either. Instead, we will make the following compromise. 1 A projectable lagrangian submanifold (L, ι) ⊂ T ∗ M is quantizable if its Liouville class λL,ι is -integral for some ∈ R+ . The values of for which this condition holds will be called admissible for (L, ι). If L is quantizable but not exact, the set of all admissible forms a sequence converging to zero, consisting of the numbers 0 /k, where 0 is the largest such number, and k runs over the positive integers.