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.

**Read Online or Download Holomorphic Curves in Symplectic Geometry PDF**

**Best differential geometry books**

**Differential Geometry: Theory and Applications (Contemporary Applied Mathematics)**

This ebook supplies the elemental notions of differential geometry, akin to the metric tensor, the Riemann curvature tensor, the elemental types of a floor, covariant derivatives, and the basic theorem of floor conception in a self-contained and obtainable demeanour. even supposing the sector is frequently thought of a classical one, it has lately been rejuvenated, due to the manifold functions the place it performs a vital function.

**Compactifications of Symmetric and Locally Symmetric Spaces (Mathematics: Theory & Applications)**

Introduces uniform structures of many of the identified compactifications of symmetric and in the community symmetric areas, with emphasis on their geometric and topological constructions quite self-contained reference aimed toward graduate scholars and learn mathematicians attracted to the functions of Lie thought and illustration thought to research, quantity thought, algebraic geometry and algebraic topology

**An Introduction to Multivariable Analysis from Vector to Manifold**

Multivariable research is a crucial topic for mathematicians, either natural and utilized. except mathematicians, we think that physicists, mechanical engi neers, electric engineers, structures engineers, mathematical biologists, mathemati cal economists, and statisticians engaged in multivariate research will locate this e-book super helpful.

- Integral geometry, Radon transforms and complex analysis: Lectures
- Notes on differential geometry
- Differential Geometry in the Large: Seminar Lectures New York University 1946 and Stanford University 1956
- A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition
- The Language of shape: the role of curvature in condensed matter--physics, chemistry, and biology
- Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics

**Additional resources for Holomorphic Curves in Symplectic Geometry**

**Example text**

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 [11]). 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 [14], [15], [20] 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.