幾何學

The goal of this course is to introduce to undergraduate students the basic concepts in differential geometry such as the geometry of a surface embedded in the three-dimensional Euclidean space, as well as the more abstract notion of Riemannian manifolds. In this course, we will take a higher viewpoint which will ultimately allow us to handle differential geometric calculations in the setting of a surface embedded in the three space , as well as that of the more general abstract Riemannian manifolds. The main difficulty for one to learn the subject basically comes from the use of differential forms and tensors as tools in carrying out computations. One could say that, without the use of differential forms and tensors, one could not get a full understanding of the subject matter of differential geometry. So, our course will start with a full discussion about the essense of the use of differential forms and tensors. So, a student taking this course should always keep in mind that the routine use of tools from multi-variable calculus and linear algebra will arise from time to time.

A student taking this course should have a very solid working knowledge of undergraduate level analysis, as well as that of linear algebra.

The grading policy will be based on the following: Homework will contribute 20% towards the final grade. (The homework score will be based on the weekly lecture notes as taken and prepared by the student. Please note that handwritten lecture notes are expected, and each student has to provide handwritten lecture notes prepared by themselves for the determination of the total homework score. ) A take home final exam counts 10% towards the final grade. Attendance record counts 40% towards the final grade. A final written report on a selected topic and a presentation based on the written report will count 30% towards the final grade.

There is no required textbook for the course.