18.965 | Fall 2004 | Graduate

Geometry of Manifolds


Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Course Overview

This course is meant to bring graduate students who will be using ideas from differential topology and differential geometry up to speed on these topics. Below is list of some of the highlights of the first semester.

  1. Banach Manifolds and the inverse and implicit function theorems.

  2. Sard’s theorem (for smooth maps). Parametric transversality and some applications.

  3. Whitney’s embedding theorems both easy and hard.

  4. Pontryagin-Thom construction. Fruedenthal suspension theorem. Computation of the low dimensional stable homotopy groups of spheres.

  5. Morse theory.

  6. Differential forms. de Rham’s theorem.

  7. Smales immersion theorem.


Analysis II (18.101) and Algebraic Topology (18.905)


100% of the grading is based on the assignments.


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.

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.

Hirsch, Morris W. “Differential Topology.” Graduate Texts in Mathematics. Vol. 33. Reprint edition. New York: Springer-Verlag, November 1, 1988. ISBN: 0387901485.

Course Info

As Taught In
Fall 2004
Learning Resource Types
Lecture Notes
Problem Sets