Symplectic Geometry

An Introduction based on the Seminar in Bern, 1992

B. Aebischer, M. Borer, M. Kälin, C. Leuenberger, Hans Martin Bach (Autoren)

Buch | Softcover

XII, 244 Seiten

2012
Springer Basel (Verlag)
978-3-0348-7514-1 (ISBN)

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The seminar Symplectic Geometry at the University of Berne in summer 1992 showed that the topic of this book is a very active field, where many different branches of mathematics come tog9ther: differential geometry, topology, partial differential equations, variational calculus, and complex analysis. As usual in such a situation, it may be tedious to collect all the necessary ingredients. The present book is intended to give the nonspecialist a solid introduction to the recent developments in symplectic and contact geometry. Chapter 1 gives a review of the symplectic group Sp(n,R), sympkctic manifolds, and Hamiltonian systems (last but not least to fix the notations). The 1/Iaslov index for closed curves as well as arcs in Sp(n, R) is discussed. This index will be used in chapters 5 and 8. Chapter 2 contains a more detailed account of symplectic manifolds start ing with a proof of the Darboux theorem saying that there are no local in variants in symplectic geometry. The most important examples of symplectic manifolds will be introduced: cotangent spaces and Kahler manifolds. Finally we discuss the theory of coadjoint orbits and the Kostant-Souriau theorem, which are concerned with the question of which homogeneous spaces carry a symplectic structure.

1 Introduction.- 2 Darboux' Theorem and Examples of Symplectic Manifolds.- 3 Generating Functions.- 4 Symplectic Capacities.- 5 Floer Homology.- 6 Pseudoholomorphic Curves.- 7 Gromov's Compactness Theorem from a Geometrical Point of View.- 8 Contact structures.- A Generalities on Homology and Cohomology.- A.1 Axioms for homology.- A.2 Axioms for cohomology.- A.3 Homomorphisms of (co)homology sequences.- A.4 The (co)homology sequence of a triple.- A.5 Homotopy equivalence and contractibility.- A.6 Direct sums.- A.7 Triads.- A.8 Mayer-Vietoris sequence of a triad.- References.

Erscheint lt. Verlag	5.11.2012
Reihe/Serie	Progress in Mathematics
Zusatzinfo	XII, 244 p.
Verlagsort	Basel
Sprache	englisch
Gewicht	299 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Contact Geometry • Differential Geometry • manifold • Symplectic Geometry • Topology
ISBN-10	3-0348-7514-2 / 3034875142
ISBN-13	978-3-0348-7514-1 / 9783034875141
Zustand	Neuware