This is a continuing course of Geometry (201 25300, MATH3301). It assumes that students have taken that course. No other backgrounds are required. The purpose of the course is two folds. On one aspect, from previous experiences we are ready to introduce the notion of differential manifolds and associate structure of vector fields, differential forms and tensors. At the same time, we can extend the discussion of intrinsic Geometry of surfaces last semester to Riemannian metrics and curvatures. On the other aspect, we also like to explore some global Differential Geometry properties.
We will first discuss some materials omitted last semester including the proof of fundamental Theorem of the local theory of surfaces, and part of §3.4, §3.5 in [1]. Then we will move to an introduction of differential manifolds and Riemannian structures. It will cover §4.6 to §5.6 in [1], but in the setting of Riemannian manifold. For this part, it will follow the instructor’s own notes and [2] will be a good reference. The students can also compare the contents of §4.6-§5.6 in [1] with this general setting. Additional subjects on curvature, space of constant curvature, and 2nd fundamental form will also be discussed. We will also try to cover some theorems in §5.7-§5.11 of [1] if time allowed. The meeting time of the course is possibly rescheduled to the students’ need. The interested students please let me know your schedule in advance. The focus of course can also be adjusted along the classes. There will be some overlapping on the basic material with “Differential Geometry I”. But here the pace will be slower and emphasize on the foundations. More precise and detailed plan for the course will be provided later.