MATH 641-2

Differential Geometry I-II

News

General Information

• Meets: MWF 2:10-3:00pm in Ritter 316.
• Instructor: Dr. Bryan Clair
• Syllabus: syllabus-642.pdf, (Old 641 syllabus syllabus-641.pdf).
• Textbook: Jeffrey M. Lee, Manifolds and Differential Geometry, AMS Graduate Studies in Mathematics, 2009. Here is a link to the author's web page, including the online supplement.
• Policies:

Math Stuff

• A short handout on the deRham cohomology of Sⁿ: deRham-on-Sn.pdf.
• A Note on Differential Calculus in Rⁿ, by James Hebda.
• Animations of a rotation of the three sphere in stereographic projection: cross section, cross section (with orbits), cutaway.
• Hopf fibration: hopf-fibration.jpg, animation.

Exams and Additional Problems

• Past SLU differential geometry qualifying exams.
• 642 Final Exam: problems, solutions.
• 642 Final Exam Review: problems, solutions.
• 642 Midterm Exam: problems, solutions.
• 642 Midterm Exam Review: problems, solutions.
• 641 Final Exam: problems, solutions.
• 641 Final Exam Review: problems, solutions.
• 641 Midterm Exam: problems, solutions.
• 641 Midterm Exam Review: problems, solutions.

Homework Assignments - Math 642

• week1.pdf (Flows for vector fields). Solutions.
• week2.pdf (1-forms). Solutions.
• week3.pdf (Line integrals). Solutions.
• week4.pdf (Bilinear forms and metrics). Solutions.
• week5-projects.pdf (Proofs involving Riemannian metrics).
• week6.pdf (Problems related to Week 5 projects). Solutions.
• week9.pdf (Alternating tensors, differential forms, orientation). Solutions.
• week10.pdf (Volume forms, exterior derivative). Solutions.
• week11.pdf (Cartan's formula, manifolds with boundary). Solutions.
• week12.pdf (Stokes' theorem, divergence). Solutions.
• week13.pdf (Rⁿ, Laplacian). Solutions.
• week14.pdf (Cohomology). Solutions.

Homework Assignments - Math 641

• week1.pdf (Stereographic projection, differentiable maps)
• week2.pdf (Derivatives, inverse function theorem)
• week3.pdf (Manifolds, diffeomorphisms), solution to problem 5
• week4.pdf (Projective space, partition of unity, group actions), solution to problem 4
• week5.pdf (Group actions)
• week6.pdf (Tangent vectors)
• week7.pdf (Tangent maps, tangent bundles)
• week8-9.pdf (Vector fields, vector bundles), solution to problem 1, problems 2 and 3
• week10.pdf (Regular submanifolds, critical values, submersions). Most of the solutions
• week11.pdf (Immersions, embeddings, Sard's Theorem). Solutions.
• week12.pdf (Embeddings, Transversality) Solutions.
• week13.pdf (Lie Bracket, Curves in R³)
• week14.pdf (Orientability, Shape operator) Solutions.