By Michele Audin, Jacques Lafontaine
This booklet is dedicated to pseudo-holomorphic curve equipment in symplectic geometry. It comprises an creation to symplectic geometry and correct suggestions of Riemannian geometry, proofs of Gromov's compactness theorem, an research of neighborhood homes of holomorphic curves, together with positivity of intersections, and functions to Lagrangian embeddings difficulties. The chapters are in keeping with a chain of lectures given formerly via the authors M. Audin, A. Banyaga, P. Gauduchon, F. Labourie, J. Lafontaine, F. Lalonde, Gang Liu, D. McDuff, M.-P. Muller, P. Pansu, L. Polterovich, J.C. Sikorav. In an try to make this ebook obtainable additionally to graduate scholars, the authors give you the worthwhile examples and strategies had to comprehend the purposes of the idea. The exposition is largely self-contained and comprises a variety of workouts.
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Additional resources for Holomorphic Curves in Symplectic Geometry
The homotopy exact sequence of the fibration An ---+ Sl gives 0= 1f1(SU(n)jSO(n)) ---+ 1f1(An) ---+ 1f1(Sl) ---+ 1fo(SU(n)jSO(n)) = O. Therefore 1f1(An) ~ 1f1(Sl) ~ Z. 3. Exercise. n(n + l)j2. Deduce that An is a compact manifold of dimension The tangent space to the Lagrangian Grassmann manifold (see ). Fix a Lagrangian subspace>. E An. A n with the space S(>') of all symmetric bilinear forms on >.. A can be represented as 1tA(t)>'lt=o, where A(t) is a path of linear symplectic transformations of en with A(O) = id.
Exercise. - (A - Id) 1, that is, that IIAx - xl1 2 < Check that IIAx + xl1 2 - IIAx - xll Now let S be a matrix such that and that the endomorphism J 2 = 4w(x, Jx) > O. IISII < 1, which implies that Id -S is invertible = Jo 0 (Id +S) 0 (Id -SrI is well defined. 9. Exercise. Check that J is an almost complex structure (1-1 = -J) if and only if JoS + SJo = 0 and that it is tamed by w. Obviously the map S 1-+ J is the inverse of the one we are considering, so that we have proved the proposition for tamed structures.
J is an almost complex structure) and that n(JX, JY) = n(X, Y). Moreover, letting g(X, Y) = go(RX, Y), we have g(JX, JY) = g(X, Y) and g(JX, Y) = n(X, Y). For more details, we refer to the basic texts , ,  and to chapter II of the present book. 2. Symplectic Manifolds If an almost symplectic form n is closed, then we say that n is a symplectic form. The couple (M, n) of a smooth manifold M and a symplectic form n on it is called a symplectic manifold. 1. For x E R 2n, ns(X)(~, 1]) = as(~(x), 1](x)), 2 2 where ~,1]: R n ----+ R n are vector fields on R2n.