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This book is addressed to graduate students and researchers in the
fields of mathematics and physics who are interested in
mathematical and theoretical physics, differential geometry,
mechanics, quantization theories and quantum physics, quantum
groups etc., and who are familiar with differentiable and
symplectic manifolds. The aim of the book is to provide the reader
with a monograph that enables him to study systematically basic and
advanced material on the recently developed theory of Poisson
manifolds, and that also offers ready access to bibliographical
references for the continuation of his study. Until now, most of
this material was dispersed in research papers published in many
journals and languages. The main subjects treated are the
Schouten-Nijenhuis bracket; the generalized Frobenius theorem; the
basics of Poisson manifolds; Poisson calculus and cohomology;
quantization; Poisson morphisms and reduction; realizations of
Poisson manifolds by symplectic manifolds and by symplectic
groupoids and Poisson-Lie groups. The book unifies terminology and
notation. It also reports on some original developments stemming
from the author's work, including new results on Poisson cohomology
and geometric quantization, cofoliations and biinvariant Poisson
structures on Lie groups.
This book proposes a new approach which is designed to serve as an
introductory course in differential geometry for advanced
undergraduate students. It is based on lectures given by the author
at several universities, and discusses calculus, topology, and
linear algebra.
This volume discusses the classical subjects of Euclidean, affine
and projective geometry in two and three dimensions, including the
classification of conics and quadrics, and geometric
transformations. These subjects are important both for the
mathematical grounding of the student and for applications to
various other subjects. They may be studied in the first year or as
a second course in geometry. The material is presented in a
geometric way, and it aims to develop the geometric intuition and
thinking of the student, as well as his ability to understand and
give mathematical proofs. Linear algebra is not a prerequisite, and
is kept to a bare minimum. The book includes a few methodological
novelties, and a large number of exercises and problems with
solutions. It also has an appendix about the use of the computer
programme MAPLEV in solving problems of analytical and projective
geometry, with examples.
This volume discusses the classical subjects of Euclidean, affine
and projective geometry in two and three dimensions, including the
classification of conics and quadrics, and geometric
transformations. These subjects are important both for the
mathematical grounding of the student and for applications to
various other subjects. They may be studied in the first year or as
a second course in geometry.The material is presented in a
geometric way, and it aims to develop the geometric intuition and
thinking of the student, as well as his ability to understand and
give mathematical proofs. Linear algebra is not a prerequisite, and
is kept to a bare minimum.The book includes a few methodological
novelties, and a large number of exercises and problems with
solutions. It also has an appendix about the use of the computer
program MAPLEV in solving problems of analytical and projective
geometry, with examples.
This book proposes a new approach which is designed to serve as an
introductory course in differential geometry for advanced
undergraduate students. It is based on lectures given by the author
at several universities, and discusses calculus, topology, and
linear algebra.
This book presents to the reader a modern axiomatic construction of
three-dimensional Euclidean geometry in a rigorous and accessible
form. It is helpful for high school teachers who are interested in
the modernization of the teaching of geometry.
The present work grew out of a study of the Maslov class (e. g.
(37]), which is a fundamental invariant in asymptotic analysis of
partial differential equations of quantum physics. One of the many
in terpretations of this class was given by F. Kamber and Ph.
Tondeur (43], and it indicates that the Maslov class is a secondary
characteristic class of a complex trivial vector bundle endowed
with a real reduction of its structure group. (In the basic paper
of V. I. Arnold about the Maslov class (2], it is also pointed out
without details that the Maslov class is characteristic in the
category of vector bundles mentioned pre viously. ) Accordingly, we
wanted to study the whole range of secondary characteristic classes
involved in this interpretation, and we gave a short description of
the results in (83]. It turned out that a complete exposition of
this theory was rather lengthy, and, moreover, I felt that many
potential readers would have to use a lot of scattered references
in order to find the necessary information from either symplectic
geometry or the theory of the secondary characteristic classes. On
the otherhand, both these subjects are of a much larger interest in
differential geome try and topology, and in the applications to
physical theories."
Everybody having even the slightest interest in analytical
mechanics remembers having met there the Poisson bracket of two
functions of 2n variables (pi, qi) f g ~(8f8g 8 8 ) (0.1) {f,g} =
L...~[ji - [ji~ ,;=1 p, q q p, and the fundamental role it plays in
that field. In modern works, this bracket is derived from a
symplectic structure, and it appears as one of the main in-
gredients of symplectic manifolds. In fact, it can even be taken as
the defining clement of the structure (e.g., [TIl]). But, the study
of some mechanical sys- tems, particularly systems with symmetry
groups or constraints, may lead to more general Poisson brackets.
Therefore, it was natural to define a mathematical structure where
the notion of a Poisson bracket would be the primary notion of the
theory, and, from this viewpoint, such a theory has been developed
since the early 19708, by A. Lichnerowicz, A. Weinstein, and many
other authors (see the references at the end of the book). But, it
has been remarked by Weinstein [We3] that, in fact, the theory can
be traced back to S. Lie himself [Lie].
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