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18.966

Geometry of Manifolds II

3-0-9 units · Graduate

Continuation of 18.965 , focusing more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology.

Spring 2007 · Geometry of Manifolds

This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.

Readings

  • Cannas da Silva, A. Lectures on Symplectic Geometry (Lecture Notes in Mathematics) . New York City, NY: Springer, 2001. ISBN: 9783540421955.
    Listed reading
  • Wells, R. O. Differential Analysis on Complex Manifolds . New York City, NY: Springer, 1980. ISBN: 9780387904191.
    Listed reading
  • McDuff, D., and D. Salamon. Introduction to Symplectic Topology . New York City, NY: Oxford University Press, 1999. ISBN: 9780198504511.
    Listed reading
  • 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.
    Listed reading
  • LEC #: 2 · TOPICS: Cup-product and Poincaré duality in de Rham cohomology; symplectic vector spaces and linear algebra; symplectic manifolds, first examples; symplectomorphisms · READINGS: Cannas. pp. 3-7.
    Reading assignment
  • LEC #: 3 · TOPICS: 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 · READINGS: Cannas. pp. 9-19, 35-37, and 105-107.
    Reading assignment
  • LEC #: 4 · TOPICS: Symplectic vector fields, flux; isotopy and deformation equivalence; Moser’s theorem; Darboux’s theorem · READINGS: Cannas. pp. 106 and 42-46.
    Reading assignment
  • LEC #: 5 · TOPICS: Tubular neighborhoods; local version of Moser’s theorem; Weinstein’s neighborhood theorem · READINGS: Cannas. pp. 37-40 and 45-52.
    Reading assignment
  • LEC #: 6 · TOPICS: 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 · READINGS: Cannas. pp. 53-56.
    Reading assignment
  • LEC #: 7 · TOPICS: More Floer homology; almost-complex structures; compatibility with a symplectic structure; polar decomposition; compatible triples · READINGS: Cannas. pp. 67-70.
    Reading assignment
  • LEC #: 9 · TOPICS: Horizontal distributions; metric connections; curvature of a connection: Intrinsic definition; expression in terms of connection 1-form · READINGS: Wells. pp. 70-74.
    Reading assignment
  • LEC #: 10 · TOPICS: 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 · READINGS: Wells. pp. 73-77 and 84-91.
    Reading assignment
  • LEC #: 11 · TOPICS: Naturality properties of Chern classes and topological definition; equivalence between the two definitions; classification of complex line bundles · READINGS: Wells. pp. 91-96.
    Reading assignment
  • LEC #: 12 · TOPICS: 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 · READINGS: Cannas. pp. 78-81 and 83-87.
    Reading assignment
  • LEC #: 13 · TOPICS: Nijenhuis tensor; integrability; square of the dbar operator; Newlander-Nirenberg theorem; Kähler manifolds; complex projective space · READINGS: Cannas. pp. 82 and 88-89.
    Reading assignment
  • LEC #: 14 · TOPICS: Kähler forms; strictly plurisubharmonic functions; Kähler potentials; examples; Fubini-Study Kähler form; complex projective manifolds; Hodge decomposition theorem · READINGS: Cannas. pp. 90-97.
    Reading assignment
  • LEC #: 17 · TOPICS: Hodge diamond; hard Lefschetz theorem; holomorphic vector bundles; canonical connection and curvature · READINGS: Wells. pp. 77-80.
    Reading assignment
  • LEC #: 20 · TOPICS: Proof of the approximation lemma; examples of compact 4-manifolds without almost-complex structures, without symplectic structures, without complex structures; Kodaira-Thurston manifold · READINGS: Cannas. pp. 101-103.
    Reading assignment
  • LEC #: 21 · TOPICS: Symplectic fibrations; Thurston’s construction of symplectic forms; symplectic Lefschetz fibrations, Gompf and Donaldson theorems · READINGS: McDuff-Salamon. pp. 197-203.
    Reading assignment
  • LEC #: 22 · TOPICS: Symplectic sum along codimension 2 symplectic submanifolds; Gompf’s construction of symplectic 4-manifolds with arbitrary pi_1 · READINGS: McDuff-Salamon. pp. 253-256.
    Reading assignment
  • LEC #: 24 · TOPICS: Homeomorphism classification of simply connected 4-manifolds; intersection pairings; spin^c structures; spin^c connections; Dirac operator · READINGS: Morgan.
    Reading assignment

Prerequisites & course information

Prereq: 18.965 G (Spring) 3-0-9 units

Sources