***This is a course guideline.  Students should contact instructor for the updated information on current course syllabus, textbooks, and course content***

Prerequisites: MATH 2415 and six additional hours of 3000-4000 level Mathematics.

Course Description: Differential forms in R^n (particularly R^2 and integration, the intrinsic theory of surfaces through differential forms, the Gauss-Bonnet theorem, Stokes’ theorem, manifolds, Riemannian metric and curvature. Other topics at discretion of instructor.

Textbook: Instructor's Notes. Reference book: Differential Geometry: A first course in curves and surfaces, April, 2021 by Prof. Theodore Shifrin.

Topics Covered: *syllabus from Spring 2024 - Wu*

This course will cover fundamentals of differential geometry, holonomy and the Gauss-Bonnet theorem, introduction to hyperbolic geometry, surface theory with differential forms, calculus of variations and surfaces of constant mean curvature, abstract surfaces (2D Riemannian manifolds).

This course will introduce the theory of the geometry of curves and surfaces in three-dimensional space using calculus techniques, exhibiting the interplay between local and global quantities:

• Part 1: The Geometry of curves (this part contains the main topics of the curves: The curvature and torsion (and their geometric meanings), the Frenet frame, the Frenet formula).

• Part 2: The tangent spaces, the first and second fundamental forms, and the shape operator for surfaces. The first fundamental form gives the measurement (i.e., the length of the curve and surface area). The second fundamental form and the shape operator will be used in the next chapter to define various concepts of curvatures.

• Part 3: We introduce the concepts of various curvatures, including the normal, principal, Gauss and mean curvatures. We also study the curves on the surfaces. For curves, we have the concepts of the normal and geodesic curvatures.

• Part 4: We introduce the intrinsic geometry of surfaces, covariant derivatives, and the geodesics are also discussed.

• Part 5: Parallel translation, and geodesics. Holonomy and the Gauss-Bonnet theorem. An introduction to hyperbolic geometry.

• Part 6: Surface theory with differential forms. Calculus of variations and surfaces of constant mean curvature.

Grading & Make-up Policy/Assignment & Exam Details: Please consult your instructor's syllabus regarding any and all grading guidelines.

Justin Dart Jr. Center Accommodations:

Academic Adjustments/Auxiliary Aids: The University of Houston System complies with Section 504 of the Rehabilitation Act of 1973 and the Americans with Disabilities Act of 1990, pertaining to the provision of reasonable academic adjustments/auxiliary aids for students who have a disability. In accordance with Section 504 and ADA guidelines, University of Houston strives to provide reasonable academic adjustments/auxiliary aids to students who request and require them. If you believe that you have a disability requiring an academic adjustments/auxiliary aid, please visit Justin Dart Jr. Student Accessibility Center website at https://www.uh.edu/accessibility/ for more information.

UH CAPS

Counseling and Psychological Services (CAPS) can help students who are having difficulties managing stress, adjusting to college, or feeling sad and hopeless. You can reach (CAPS) by calling 713-743-5454 during and after business hours for routine appointments or if you or someone you know is in crisis. No appointment is necessary for the "Let's Talk" program, a drop-in consultation service at convenient locations and hours around campus.