18.966 Homework 3 due Thursday April 19, 2007. 1. Let (M, ) be a symplectic manifold, J a compatible almost-complex structure, and g the corresponding Riemannian metric. Show that two-dimensional almost-complex sub manifolds of M are absolutely volume min
SYMPLECTIC GEOMETRY, LECTURE 17
Prof. Denis Auroux
The Hodge decomposition stated last time places strong constraints on H of Khler manifolds, e.g. dim H k a is even for k odd because C conjugation gives isomorphisms Hp,q Hq,p (note that this is false for
SYMPLECTIC GEOMETRY, LECTURE 12
Prof. Denis Auroux
1. Existence of Almost-Complex Structures Let (M, ) be a symplectic manifold. If J is a compatible almost-complex structure, we obtain invariants cj (T M , J ) H 2j (M, Z) of the deformation equivalence c
SYMPLECTIC GEOMETRY, LECTURE 8
Prof. Denis Auroux
1. Almost-complex Structures Recall compatible triples (, g , J ), wherein two of the three determine the third (g (u, v ) = (u, J v ), (u, v ) = g (Ju, v ), J (u) = g 1 ( (u) where g , are the induced iso
SYMPLECTIC GEOMETRY, LECTURE 7
Prof. Denis Auroux
1. Floer homology For a Hamiltonian dieomorphism f : (M, ) (M, ), f = 1 , Ht : M R 1-periodic in t, we want to look H for xed points of f , i.e. 1-periodic orbits of XH , x (t) = XHt (x(t). We consider the
SYMPLECTIC GEOMETRY, LECTURE 6
Prof. Denis Auroux
1. Applications (1) The work done last time gives us a new way to look at Tid Symp(M, ) (using C 1 -topology, wherein fi : X Y converges to f i fi f uniformly on compact sets and same for dfi : T X T Y . N
SYMPLECTIC GEOMETRY, LECTURE 5
Prof. Denis Auroux
Last time we proved: Theorem 1 (Moser). Let M be a compact manifold, (t ) symplectic forms, [t ] constant = (M, 0 ) = (M, 1 ). Theorem 2 (Darboux). Locally, any symplectic manifold is locally isomorphic to
SYMPLECTIC GEOMETRY, LECTURE 4
Prof. Denis Auroux
1. Hamiltonian Vector Fields Recall from last time that, for (M, ) a symplectic manifold, H : M R a C function, there exists a vector eld XH s.t. iXH = dH . Furthermore, the associated ow t of this vector
SYMPLECTIC GEOMETRY, LECTURE 2
Prof. Denis Auroux
1. Homology and Cohomology Recall from last time that, for M a smooth manifold, we produced a graded dierential algebra ( (M ), , d) giving us a cohomology H (M ) with cup product [] [ ] = [ ] (which is we
SYMPLECTIC GEOMETRY, LECTURE 1
Prof. Denis Auroux
1. Differential forms Given M a smooth manifold, one has two natural bundles: the tangent bundle T M = cfw_v = the cotangent bundle T M = cfw_ = i dxi . Under C maps, tangent vectors pushforward: (1) f
18.966 Homework 3 Solutions.
1. Given a point p C (a two-dimensional oriented submanifold), let (e, f ) be an oriented basis of Tp C , orthonormal with respect to the metric g induced by and J . Then (e, f ) = g (Je, f ) |Je| |f | = |e| |f | = 1. Meanwhi
18.966 Homework 2 Solutions.
1. Equip R7 = Im O = cfw_a + be, Re a = 0 with the cross-product x y = Im(xy ). By denition of the octonion product, if x, y Im O then Re(xy ) = x, y (the usual Euclidean scalar product on R7 ). Indeed, Re(a + be)(a + b e) =
18.966 Homework 1 Solutions.
1. Let E be a Lagrangian subspace of a symplectic vector space (V, ), and let e1 , . . . , en be a basis of E . We proceed by induction, assuming we have constructed f1 , . . . , fk1 V such that the family (e1 , . . . , en ,
SYMPLECTIC GEOMETRY, LECTURE 15
Prof. Denis Auroux
1. Hodge Theory Theorem 1 (Hodge). For M a compact Kahler manifold, the deRham and Dolbeault cohomologies are related p k by HdR (M, C) = p,q H ,q (M ), with H p,q H q,p . = Before we discuss this theorem