Differential Geometry

For honours courses please contact the School Office.

Description

This course is part of the course offerings for Honours Pure Mathematics (Level IV). Assumed knowledge: Multivariable Calculus. A basic understanding of topology as would be obtained from Analysis & Topology is helpful but not mandatory. A basic understanding of abstract linear algebra is also helpful, but the necessary material will developed during the course.

Objective

This course aims to introduce students to the basic ideas and techniques of modern differential geometry, as is applied in contemporary mathematical analysis and mathematical physics. By the end of the course, students should understand the concepts involved in differential forms on manifolds, including Stokes' theorem and de Rham cohomology. Students should also understand the concepts involved in connections on vector bundles over manifolds, including curvature and Chern classes, as well as understand the statement and proof of the Gauss-Bonnet theorem.

Content

1. Review of multivariable calculus; linear algebra. 2. Differential forms in Euclidean space: exterior derivative, pull-back, integral, Poincar'e Lemma. 3. Manifolds: Tangent spaces, differentiable functions, the derivative, differential forms, Stokes' theorem. 4. de Rham and Cech cohomology. 5. Vector bundles and connections: Vector bundles, connections, curvature, Chern classes. 6. The Gauss-Bonnet theorem: The Euler characteristic of a surface, the Gauss-Bonnet theorem.

Nicholas Buchdahl Lecturer for this course

Delivery

2 one-hour lectures per week.

Assessment

3-hour final exam (75%), best 5 of 6 homework assignments (25%), optional mini-project worth up to 15%.

Graduate attributes

• Applying mathematical knowledge (1)
• Interpreting data and drawing conclusions (2)
• Formulating mathematical problems (3)
• Problem solving skills (4)
• Flexibility in working environment (5)
• Self-directed study (8)
• Career development (9)

Linkage past

No past linkages have been noted.

Linkage present

No present linkages have been noted.

Linkage future

This course is not recorded as prequisite for other courses.

Restrictions

None.

Recommended text

References: 1. Bott & Tu: Differential forms in algebraic topology. 2. Chern: Differentiable manifolds. 514.76 C521d 3. Choquet-Bruhat, DeWitt-Morette, Dillard-Bleick: Analysis, manifolds, and physics (Rev.\ ed.). 517 C5495a 4. Guillemin & Pollack: Differential Topology. 513.83 G958d 5. Lee: Introduction to smooth manifolds. 514.76 L478i