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 almo
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 gi
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
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 (J
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-per
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 u
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). Lo
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
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 cohom
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 d
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
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
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 ,
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