18.966 | Spring 2007 | Graduate

Geometry of Manifolds


The readings in this section are not required.

The recommended texts are:

Cannas da Silva, A. Lectures on Symplectic Geometry (Lecture Notes in Mathematics). New York City, NY: Springer, 2001. ISBN: 9783540421955.

Wells, R. O. Differential Analysis on Complex Manifolds. New York City, NY: Springer, 1980. ISBN: 9780387904191.

McDuff, D., and D. Salamon. Introduction to Symplectic Topology. New York City, NY: Oxford University Press, 1999. ISBN: 9780198504511.

Morgan, J. W. The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (Mathematical Notes 44). Princeton, NJ: Princeton University Press, 1995, ISBN: 9780691025971.

2 Cup-product and Poincaré duality in de Rham cohomology; symplectic vector spaces and linear algebra; symplectic manifolds, first examples; symplectomorphisms Cannas. pp. 3-7.
3 Symplectic form on the cotangent bundle; symplectic and Lagrangian submanifolds; conormal bundles; graphs of symplectomorphisms as Lagrangian submanifolds in products; isotopies and vector fields; Hamiltonian vector fields; classical mechanics Cannas. pp. 9-19, 35-37, and 105-107.
4 Symplectic vector fields, flux; isotopy and deformation equivalence; Moser’s theorem; Darboux’s theorem Cannas. pp. 106 and 42-46.
5 Tubular neighborhoods; local version of Moser’s theorem; Weinstein’s neighborhood theorem Cannas. pp. 37-40 and 45-52.
6 Tangent space to the group of symplectomorphisms; fixed points of symplectomorphisms; Arnold’s conjecture; Morse theory: Gradient trajectories, Morse complex, homology; action functional on the loop space, and the basic idea of Floer homology Cannas. pp. 53-56.
7 More Floer homology; almost-complex structures; compatibility with a symplectic structure; polar decomposition; compatible triples Cannas. pp. 67-70.
8 Almost-complex structures: Existence and contractibility; almost-complex submanifolds vs. symplectic submanifolds; Sp(2n), O(2n), GL(n,C), and U(n); connections: definition, connection 1-form

Cannas. pp. 71-76

Wells. pp. 65-70.

9 Horizontal distributions; metric connections; curvature of a connection: Intrinsic definition; expression in terms of connection 1-form Wells. pp. 70-74.
10 Twisted de Rham operator; Levi-Civita connection on (TM,g); Chern classes of complex vector bundles (via curvature and Chern-Weil); Euler class and top Chern class Wells. pp. 73-77 and 84-91.
11 Naturality properties of Chern classes and topological definition; equivalence between the two definitions; classification of complex line bundles Wells. pp. 91-96.
12 Chern classes of the tangent bundle; cohomological criterion for existence of almost-complex structures on a 4-manifold, examples; splitting of tangent and cotangent bundles of (M,J), types; complex manifolds, Dolbeault cohomology Cannas. pp. 78-81 and 83-87.
13 Nijenhuis tensor; integrability; square of the dbar operator; Newlander-Nirenberg theorem; Kähler manifolds; complex projective space Cannas. pp. 82 and 88-89.
14 Kähler forms; strictly plurisubharmonic functions; Kähler potentials; examples; Fubini-Study Kähler form; complex projective manifolds; Hodge decomposition theorem Cannas. pp. 90-97.
15 Hodge * operator on a Riemannian manifold; d* operator; Laplacian, harmonic forms; Hodge decomposition theorem; differential operators; symbol, ellipticity; existence of parametrix

Cannas. pp. 98-99.

Wells. pp. 114-116 and 136.

16 Elliptic regularity, Green’s operator; Hodge * operator and complex Hodge theory on a Kähler manifold; relation between real and complex Laplacians

Cannas. pp. 99-100

Wells. pp. 136-141, 154-163, and 191-199.

17 Hodge diamond; hard Lefschetz theorem; holomorphic vector bundles; canonical connection and curvature Wells. pp. 77-80.
20 Proof of the approximation lemma; examples of compact 4-manifolds without almost-complex structures, without symplectic structures, without complex structures; Kodaira-Thurston manifold Cannas. pp. 101-103.
21 Symplectic fibrations; Thurston’s construction of symplectic forms; symplectic Lefschetz fibrations, Gompf and Donaldson theorems McDuff-Salamon. pp. 197-203.
22 Symplectic sum along codimension 2 symplectic submanifolds; Gompf’s construction of symplectic 4-manifolds with arbitrary pi_1 McDuff-Salamon. pp. 253-256.
24 Homeomorphism classification of simply connected 4-manifolds; intersection pairings; spin^c structures; spin^c connections; Dirac operator Morgan.

Course Info

As Taught In
Spring 2007
Learning Resource Types
Problem Sets with Solutions
Lecture Notes