MIT OpenCourseWare

Browse materialsMIT 18.966 · Spring 2007

Geometry of Manifolds

Subjects: Algebra · Geometry · Linear Algebra · Mathematics

Art: Johannes Kepler, Konjukce Saturnu a Jupiteru Originál, 1596 · Public domain

Instructor: Prof. Denis Auroux
Term: Spring 2007
Level: Graduate
What's inside: 1 lecture note · 1 reading · 31 other files · 2 web pages
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About this course

Source: MIT OpenCourseWare 18.966 Geometry of Manifolds, Spring 2007, Prof. Denis Auroux — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

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Prerequisites are not listed here. Check MIT's own syllabus for this course.

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Lecture Notes

25 files

Assignments

MIT OpenCourseWare, Prof. Denis Auroux, 18.966 Geometry of Manifolds, Spring 2007.

License: CC BY-NC-SA 4.0 · Edition: Spring 2007 · Open the original course on OCW ↗

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Geometry of Manifolds

Before you start

What's here, and what isn't

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Materials by section

Lecture Notes

Assignments

Lecture 1: Review of differential forms, Lie derivative, and de Rham cohomology ( PDF )

Lecture 2: Cup-product and Poincaré duality in de Rham cohomology; symplectic vector spaces and linear algebra; symplectic manifolds, first examples; symplectomorphisms ( PDF )

Lecture 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 ( PDF )

Lecture 4: Symplectic vector fields, flux; isotopy and deformation equivalence; Moser’s theorem; Darboux’s theorem ( PDF )

Lecture 5: Tubular neighborhoods; local version of Moser’s theorem; Weinstein’s neighborhood theorem ( PDF )

Lecture 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 ( PDF )

Lecture 7: More Floer homology; almost-complex structures; compatibility with a symplectic structure; polar decomposition; compatible triples ( PDF )

Lecture 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 ( PDF )

Lecture 9: Horizontal distributions; metric connections; curvature of a connection: Intrinsic definition; expression in terms of connection 1-form ( PDF )

Lecture 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 ( PDF )

Lecture 11: Naturality properties of Chern classes and topological definition; equivalence between the two definitions; classification of complex line bundles ( PDF )

Lecture 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 ( PDF )

Lecture 13: Nijenhuis tensor; integrability; square of the dbar operator; Newlander-Nirenberg theorem; Kähler manifolds; complex projective space ( PDF )

Lecture 14: Kähler forms; strictly plurisubharmonic functions; Kähler potentials; examples; Fubini-Study Kähler form; complex projective manifolds; Hodge decomposition theorem ( PDF )

Lecture 15: Hodge * operator on a Riemannian manifold; d* operator; Laplacian, harmonic forms; Hodge decomposition theorem; differential operators; symbol, ellipticity; existence of parametrix ( PDF )

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

Lecture 17: Hodge diamond; hard Lefschetz theorem; holomorphic vector bundles; canonical connection and curvature ( PDF )

Lecture 18: Holomorphic sections and projective embeddings; ampleness; Donaldson’s proof of the Kodaira embedding theorem: local model; concentrated approximately holomorphic sections ( PDF )

Lecture 19: Donaldson’s proof of the Kodaira embedding theorem: Estimates; concentrated sections; approximation lemma ( PDF )

Lecture 20: Proof of the approximation lemma; examples of compact 4-manifolds without almost-complex structures, without symplectic structures, without complex structures; Kodaira-Thurston manifold ( PDF )

Lecture 21: Symplectic fibrations; Thurston’s construction of symplectic forms; symplectic Lefschetz fibrations, Gompf and Donaldson theorems ( PDF )

Lecture 22: Symplectic sum along codimension 2 symplectic submanifolds; Gompf’s construction of symplectic 4-manifolds with arbitrary pi_1 ( PDF )

Lecture 23: Symplectic branched covers of symplectic 4-manifolds ( PDF )

Lecture 24: Homeomorphism classification of simply connected 4-manifolds; intersection pairings; spin^c structures; spin^c connections; Dirac operator ( PDF )

Lecture 25: Seiberg-Witten equations; gauge group; moduli space; linearized equations; compactness of moduli space ( PDF )

Lecture 26: Seiberg-Witten invariant; properties; vanishing for manifolds of positive scalar curvature; vanishing for connected sums; Taubes non-vanishing for symplectic manifolds; examples of non-symplectic 4-manifolds, of non-diffeomorphic homeomorphic manifolds