By Izu Vaisman

ISBN-10: 1475719604

ISBN-13: 9781475719604

ISBN-10: 1475719620

ISBN-13: 9781475719628

The current paintings grew out of a research of the Maslov classification (e. g. (37]), that's a basic invariant in asymptotic research of partial differential equations of quantum physics. one of many many in terpretations of this category used to be given by way of F. Kamber and Ph. Tondeur (43], and it shows that the Maslov category is a secondary attribute category of a posh trivial vector package endowed with a true relief of its constitution staff. (In the fundamental paper of V. I. Arnold in regards to the Maslov type (2], it's also mentioned with out information that the Maslov category is attribute within the type of vector bundles pointed out pre viously. ) hence, we needed to check the complete diversity of secondary attribute periods concerned with this interpretation, and we gave a brief description of the implications in (83]. It grew to become out entire exposition of this thought was once relatively long, and, in addition, I felt that many strength readers must use loads of scattered references so as to locate the mandatory info from both symplectic geometry or the speculation of the secondary attribute sessions. at the otherhand, either those topics are of a miles greater curiosity in differential geome attempt to topology, and within the purposes to actual theories.

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**Extra info for Symplectic Geometry and Secondary Characteristic Classes**

**Example text**

L-t)y 0 Then J ' = J Y, is a dif- ferentiable family of positive, w-compatible complex structures of S such that J0 , J1 are the given structures. This ends the proof of Theorem 2,3,3. d. In the sequel, we always refer only to positive compatible co~ plex structures. Remark 2,3,4. The reason to oaU the foUOUJing one. 3,10) h(a,b) = g(a,b) - r-T w(a,b) (a,b E S) is a usuaZ hermitian metria of the oompZex veotor spaoe (S,J). d w as the imaginary part of h. 11) The mapping a E S 1-+..!... /"[ (a - r-T Ja) yieZds an isomorphism of the hermitian "linear spaces (S r 'gc) • (S,J,h) and g ap- 44 Proof.

3. ea: structures and hermitian structures. ex structures on S which starts with J 0 and ends with A very simple way to get J Proof. 4) The compatibility and positivity of J follow straightfor- wardly. A more invariant formulation of this construction is: the transversal pair of Lagrangian subspaces scalar product on JI L = b -1 Then J 0 take L, L', and a Euclidean L, with the induced isomorphism c : L ~ L* ; c, where b is w-duality, and is the requested complex structure; J/L' such that it sends L to J2 define = -Id.

D. 3 'A is isotropic. A subspace isotropic is caZZed a A c (S,w) subspace A correspond to as the symplectic isomorphism which sends ~ the first basis onto the second one. 9. ,~ which is both isotropic and co- L c (S,w) ~ is said to be coisotropic subspace. 9 introduce the most important types of subspaces, and we shall indicate some more properties of these subspaces. 3) that iff A c 'A, hence iff A= 'A. A is coisotropic iff 5 A c A, and A is Lagrangian On the other haad, A is symplectic iff k ~ 1A {o}.

### Symplectic Geometry and Secondary Characteristic Classes by Izu Vaisman

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