Math 8365: Manifold and Topology

Course Details

Time:
MWF 10:10 AM – 11:00 AM

Location:
Vincent Hall 20

Instructor Office:
Vincent Hall 365

Zoom Meeting ID:
973 584 3950

Office Hours

• To Be Determined (TBD)
• Or by appointment.

Textbooks

This course will primarily follow:

• Differential Geometry: Bundles, Connections, Metrics and Curvature by Clifford Taubes.

Additionally, the following reference is recommended:

• Differential Geometry: Connections, Curvature, and Characteristic Classes by L. W. Tu.

Course Topics

The course will cover:

• Riemannian metric, tensors, and vector bundles.
• Connections and curvature on a vector bundle, including the Levi-Civita connection.
• Positivity of curvature and the topology of a manifold.
• Geodesics, Lie derivatives, the exponential map, and completeness.
• Characteristic classes and the Chern-Weil theory.
• Bianchi identities and the Gauss-Bonnet theorem.
• An introduction to Lie groups.

Detailed course content is available here.

Prerequisites

A solid understanding of the following topics is essential:

• Basic point-set topology.
• Multivariate calculus.

Homework

Homework assignments will be posted weekly on Canvas. Collaboration on homework is encouraged.

Exams

There will be one midterm exam and one final exam, both of which are take-home exams.

• Midterm Exam:
  Exam Time: TBD

• Final Exam:
  Exam Time: TBD

Important:

• Collaboration is not allowed on exams.
• You may use books or notes, but internet use is strictly prohibited.

Grading Policy

Grades will be based on the following components:

• Homework: 60%
• Exams: 40%