By Solomon Lefschetz

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**Example text**

1), some twodimensional crystal lattices are realized. Another interesting class of orbits in infinite-dimensional spaces arises in quantum mechanics when constructing systems of coherent states. 3 Coherent States I begin with the classical definition of coherent states for the simplest physical system, a harmonic oscillator with one degree of freedom. Let Q be the Hermitian coordinate operator and P the momentum operator satisfying the commutation relation [Q, P] = iii, where li = hl21r and h is Planck's constant.

The fiber bundle whose fiber is S 2 n- 1 and the structure group the unitary group SU(n) is called a (2n - I)-dimensional sphere bundle. (c) G = O(n), H = O(n - I), K = O(n - k). In the real case, the situation is similar. We have the fiber bundle Vn,k = O(n)IO(n- k) O(n-1)/0((n-k)- v. 1k -. 33) where Vn, k is a real Stiefel manifold. 2), we can consider the sequence of fiber bundles Vn,n = O(n)-+ Vn,n-1-+ ... -+ Vn,1 = sn- 1. The corresponding fiber bundle Vn,n-k-+ Vn-k-l is called the (n- I)dimensional sphere bundle.

Besides principal bundles, a special class of associated bundles of vector bundles ~ = (E, F, B) with the fiber Rn is essential. Including the tangent bundle TM already encountered above, they admit a number of algebraic operations enabling us to construct fiber bundles. Direct product. Let ~1 and b be two vector fiber bundles with projections p;: E;-+ B;, i = 1, 2. The fiber bundle with the total space E1 x E2, base space Preliminaries in Mathematical Setting 55 B1 x Bz, and the projection p = P1 x ]J2:E1 x Ez-+ B1 x Bz, where each fiber (p1 X P2) - 1(b1, bz) = Fb, X Fb 2 , b1 E B1, bz E Bz, is also endowed with the struc- ture of a vector space, is called the direct product 6 x ~2.