(Summary) There will be a reading course using Differential Geometry: Bundles, Connections, Metrics and Curvature by C.Taubes. What is the prerequisite for this book? Is its contents something that every mathematician should know? Do you recommend that I take this course?
(Details) I am currently an undergraduate student, and there will be a reading course on differential geometry that I can take. I am wavering in my decision in taking that course, and I would like some advice.
The text used for the reading course is the book by C.Taubes mentioned above. I am not sure if I should take this reading course because I feel like the contents in this book is too advanced for me. I glanced over its table of contents, but I have almost no idea what this book actually does. But the professor chose this book, so it must be something very important and what everyone should know.
I asked the professor if I have the prerequisite for reading this book, but he gave me an unsatisfactory answer, which was that if I understand the topics covered in the course on smooth manifolds, which is described below, then I am ready for this book. (To be fair, he has not met me before, so I believe he couldn't say anything more than that.)
Maybe, as the professor told me, I might be ready. I would love it if I can take this course. However, I know that I am quite dumb (in the sense that a lot of the books recommended in my past courses did not quite work well with me, and I often ended up working thorough one of the friendliest introductions to the subjects taught in the courses), so if there is a slightest concern, I would like to avoid taking this course. (I cannot opt out once I decide to take the course, so I am very cautious...)
I do not want to miss this opportunity for the reading course is offered only for students with good grades, and I worked hard to earn the "good grades". But I am unsure if I am ready to take this course. Do you recommend taking this course?
My background: I have completed a semester-long course on smooth manifolds, which covered essentially all the topics up to Chapter 16 of Introduction to Smooth Manifolds by J.Lee (a wonderful book!). I also worked through all the exercises and problems of this book up to Chapter 16, and I am, at least, not uncomfortable working with smooth manifolds. I've also read the its prequel Introduction to Topological Manifolds except for chapter 13, which covers homology.