By Izu Vaisman

The current paintings grew out of a learn of the Maslov category (e. g. (37]), that is a primary invariant in asymptotic research of partial differential equations of quantum physics. one of many many in terpretations of this category was once given by means of F. Kamber and Ph. Tondeur (43], and it exhibits that the Maslov category is a secondary attribute type of a posh trivial vector package deal endowed with a true relief of its constitution workforce. (In the elemental paper of V. I. Arnold concerning the Maslov category (2], it's also mentioned with no info that the Maslov classification is attribute within the type of vector bundles pointed out pre viously. ) for that reason, we would have liked to check the complete variety of secondary attribute sessions enthusiastic about this interpretation, and we gave a brief description of the consequences in (83]. It grew to become out whole exposition of this idea used to be relatively long, and, in addition, I felt that many capability readers must use loads of scattered references with the intention to locate the mandatory info from both symplectic geometry or the speculation of the secondary attribute sessions. at the otherhand, either those matters are of a far better curiosity in differential geome try to topology, and within the purposes to actual theories.

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11. 12) F = {v o} E v;vw E v, ~(v,w) called the annihitator of "'oo, then dim F duces a symplectic form oo vector space. (S,oo) on S = V/F, = k. and Then oo obviously in(S,oo) is said to be obtained from is a symplectic (V,w) by reduction. In the remaining part of this Section we shall discuss about the so-called symptectic group. 6. (51 ,oo1 ) ~: two symplectic vector spaces. 13) Since a symplectic form is nondegenerate, a symplectic mapping is necessarily injective, and it is an isomorphism if dim 51 =dim 52 , A symplectic isomorphism is also called a sympleatomorphism.

Furthermore, always applying the definitions fiberwise, we shall distinguish isotropic, coisotropic, Lagrangian, and symplectic subbundles. 11 holds if, every- where in its formulation, vector spaces are replaced by vector bundles over M. 4 holds good. definition of the subbundle Let denote by (x EM). 5. 1) with the natural projection U £(E X ) +M, xEM TI', is again a locally trivial bundle, and we call it the Lagrange Grassmannian bundZe of E. , (80]). 10), and it acts at the right. 55 A Lagrangian subbundle cross-section L : M ~£(E).

E is equivalent with a global (T52 ,w) its usual volume form. Then ~ [80]. This does be the unit sphere, and w be is a symplectic vector bundle over 5 2, with 2-dimensional fiber, and any subbundle of rank 1 is a Lagrangian But it is well known that such a subbundle does not exist subbundle. since 52 has a nonzero Euler-Poincare characteristic. 4 we will be able to construct a transversaZ Lagrangian subbundZe E fiberwise) such that ture group of =L~ L' L'. 1. 12)). 2. morphism The above decomposition 1) E "' L ~ L* E = L ~ L' yieZds an iso- simiZar to the one encountered for vector spaces.