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Math 433/533: Introduction to Differential Geometry, Spring 2022


Instructor: Prof. Boris Botvinnik : e-mail botvinn[at]uoregon.edu

Office: Fenton 304

Lectures: Monday, Wednesday, Friday at 11:00--11:50 am, 251 Straub Hall.

Office hours: Wednesday 3:00--3:50 pm, or by appointment.

Recommended online textbooks:

  • Richard Koch: Notes on Differential Geometry

  • Theodore Shifrin: Differential Geometry: A First Course in Curves and Surfaces.

  • Chuu-Lian Terng: Lecture notes on Curves and Surfaces, Part I

    Course Goals:

    Students will develop a deep understanding of the differential geometry of curves and surfaces,including the various relevant notions of curvature. This will require a synthesis of geometric visualization, symbolic and numerical calculation, and rigorous reasoning and communication. At the end of the course, students should be prepared for a graduate course in the differential geometry on manifolds.

    Prerequisites:

    Students are expected to have a solid grounding in multivariable calculus, differential equations and linear algebra, equiv-alent to the content offered in Math 341-342, Math 256 and Math 281-282. It would be great to take Math 307 or/and Math 316-317.

    Course Content:

  • Local theory of curves in the plane and in 3-space: Parametrized curves, arclength, curvature and torsion, Frenet frame, fundamental theorem.

  • Advanced topics in local theory: Higher dimensions, Bishop frame, approximations by curves of constant curvature and torsion.

  • Global differential geometry of plane curves: Hopf Umlaufsatz, Four Vertex Theorem, isoperimetric inequality.

  • Global differential geometry of curves in R3.

  • Regular surfaces and parametrizations: Definitions, examples, differentiable curves on surfaces, differentiable functions on surfaces.

  • Tangent planes and first fundamental form: Differential, tangent plane, intrinsic definitions, inner products, surface area.

  • Gauss map: Definition, second fundamental form, curvature of curves on surfaces,principal curvatures, Gaussian and mean curvatures.

  • Theorem Egregium: Christoffel symbols, Codazzi equations, Gauss equations, Gaussian curvature is an isometry invariant.

  • Parallel transport and geodesics: Vector fields, covariant derivative, parallelfields, geodesics and distance.

  • The Gauss-Bonnet theorem: Triangulations, local Gauss-Bonnet, Euler characteristic, global Gauss-Bonnet.

    Homework

    Homework is due online on Wednesdays by 11:59 pm, beginning April 6. Late homework will not be accepted. There will be 9 homework assignments, the last of which will not be graded. Your lowest homework score will be dropped. You may collaborate with other class members on your homework, although you must each write up your solutions independently and in your own words. To avoid falling behind, you should do the reading and homework as the material is presented in class, rather than leaving it all until the last minute.

    The homework should be uploaded to Canvas by the deadline. You can solve the homework by:

    1. Solving the problems on paper and then scanning your solutions. You can use a scanning app like Adobe Scan, Office Lens, Scanner Pro, Scanbot, or others, but your scans should be easy to read and contained in a single file. If you do not have a tablet, this is probably the easiest option.
    2. Writing solutions using a tablet (iPad, Surface, etc.) and exporting and uploading the result. If you have a tablet, this is probably the easiest option.
    3. Typing your solutions using LaTeX, Word, or Pages. You need to show your work, so typing your solutions is a substantial amount of effort. I would do this only if you're trying to learn to type mathematics in LaTeX.

    Graded homework will be returned electronically, using Canvas.

    Exams:

    There will be one home-take midterm exam and in-class final exam.

  • Midterm Exam: April 27, due to May 1, 11:59 pm.

  • Final Exam: 10:15 am -- 12:15 pm, Thursday, June 7, 2022

    Grading:
    Homework:  30%
    Midterm Exam:  30%
    Final Exam:  40%

    Last modified March 22, 2022 by Boris Botvinnik.