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18.965

Geometry of Manifolds I

3-0-9 units · Graduate

Differential forms, introduction to Lie groups, the DeRham theorem, Riemannian manifolds, curvature, the Hodge theory. 18.966 is a continuation of 18.965 and focuses 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. Prior exposure to calculus on manifolds, as in 18.952 , recommended.

Fall 2004 · Geometry of Manifolds

Topics

  • Manifolds: Definitions and Examples
  • Smooth Maps and the Notion of Equivalence
  • The Derivative of a Map between Vector Spaces
  • Inverse and Implicit Function Theorems
  • More Examples
  • Vector Bundles and the Differential: New Vector Bundles from Old
  • Vector Bundles and the Differential: The Tangent Bundle
  • Connections
  • The Embedding Manifolds in R N
  • Sard’s Theorem
  • Stratified Spaces
  • Fiber Bundles
  • Whitney’s Embedding Theorem, Medium Version
  • A Brief Introduction to Linear Analysis: Basic Definitions
  • A Brief Introduction to Linear Analysis: Fredholm Operators
  • Smale’s Sard Theorem
  • Parametric Transversality
  • The Strong Whitney Embedding Theorem
  • Morse Theory
  • Canonical Forms: The Lie Derivative
  • Canonical Forms: The Frobenious Integrability Theorem
  • Differential Forms and de Rham’s Theorem: The Exterior Algebra
  • Differential Forms and de Rham’s Theorem: The Poincaré Lemma and Homotopy Invariance of the de Rham Cohomology
  • Refinement
  • The Poincaré Lemma Implies the Equality of Cech Cohomology and de Rham Cohomology
  • The Immersion Theorem of Smale

Readings

  • Bott, Raoul, R. Bott, and Loring W. Tu. Differential Forms in Algebraic Topology (Graduate Texts in Mathematics; 82) . Reprint edition. New York: Springer-Verlag, June 1, 1995. ISBN: 0387906134.
    Listed reading
  • Abraham, Ralph, Jerrold E. Marsden, and Tudor Ratiu. “Manifolds, Tensor Analysis, and Applications.” Applied Mathematical Sciences . Vol. 75. New York: Springer Verlag, May 1, 1998. ISBN: 3540967907.
    Listed reading
  • Hirsch, Morris W. “Differential Topology.” Graduate Texts in Mathematics . Vol. 33. Reprint edition. New York: Springer-Verlag, November 1, 1988. ISBN: 0387901485.
    Listed reading

Prerequisites & course information

Prereq: 18.101 , 18.950 , or 18.952 G (Spring) 3-0-9 units

Sources