Topics in Geometry I

Instructor

Prof. Paul Gunnells, LGRT 1115L, 545-6009, gunnells at math dot umass dot edu.

Office Hours

Overview

Topics in Geometry I develops the basic topology and geometry of differentiable manifolds. Topics include a review of differential maps between Euclidean spaces. Inverse and Implicit Function Theorems. Differentiable manifolds, definition and examples. Regular and critical values, Sard's Theorem, Submanifolds, immersions and embeddings, Vector bundles, tangent and cotangent bundles. Vector fields, ODE's on manifolds, Lie bracket, integrable distributions, Frobenius Theorem. Differential forms, Exterior differential. Additional topics may include group actions on manifolds, symmetric spaces, Lie groups, and Lie algebras.

Textbook

The required text is Lee, "Introduction to Smooth Manifolds." We are using the second edition. Others that you might find valuable are

• Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry"
• Warner, "Foundations of Differentiable Manifolds and Lie Groups"
• Spivak, "Differential Geometry, vol. I"
• Guillemin and Pollack, "Differential Topology."
• Gallot, Hulin, and Lafontaine, "Riemannian Geometry"
• Milnor, "Topology from the differentiable viewpoint"

Grading

There will be problem sets assigned during the term. Some problems will be graded, and will count for 25% of your grade. There will also be a midterm and a final exam, which will be worth 30% each. The remaining 15% will be based on course participation.

Problem Sets

Please make sure you are using the problems from the second edition of the textbook. The first edition has different problems.

• 1
• 2
• 3
• 4
• 5
• 6
• 7 (Extra time has been given for this one.)
• 8

Midterm exam

The evening exam will be the week of Election day, either W or Th from 6-8. The exam will cover through Ch.6 of the text (Sard's theorem). Here is a copy of the exam, with answers.

Final exam

The date and time of the final will be set later in the term.