# Recommendation on studying Smooth Manifold & Differential Geometry and related subjects.

My major was physics, but i'm changing my major to mathematics this year.

I took 2 years off to study mathematics myself and now i'm going back in this year.

Here's the list of books i have studied for last 2 years:

1. Elementary Set Theory - Jech (I have studied one more set theory text, but i don't remember its name)

2. Linear Algebra - Friedberg / Linear Algebra - Hoffman & Kunze

3. Principle of Mathematical Analysis - Rudin (Only First 8 Chapters: I have not studied multivariable calculus)

4. Topology - Munkres (Only Point-set-topology, NOT algebraic topology)

5. Real & Complex Analysis - Rudin (First 3 chapters)

I'm sure that i have a concrete understanding of books listed above.

I really like abstract approach, so i didn't study multivariable calculus since i thought it could be done in differential manifold. (BUT i'm not sure if this is possible to study differential manifold WITHOUT studying multivariable calculus). For example, I really feel comfortable with Measure theory than basic calculus.

Here's the course descriptions in my school:

Differential Geometry - "In this course, we understand the theory of curves and surfaces, then we investigate differential and integral calculus on surfaces"

Differential Manifold - "Linear Algebra, calculus and 1 year analysis are prerequisites to this course"

My questions: is it possible to study differential manifold without studying differential geometry, complex analysis and multivariable calculus?

This is a very important question to me.. When i studied RCA-rudin, it's written in the index that "only first 8 chapters of PMA are enough to start this book", but frankly it was not. It required concrete understanding of linear algebra and topology.

• It is definitely not possible to study differentiable manifolds without studying multivariable calculus first. In some sense, the language of differentiable manifolds just transforms calculus 'on manifolds' to calculus 'on $\mathbb{R}^n$'. Complex analysis is required if you want to study complex differentiable manifolds (ones that are locally $\mathbb{C}^n$). – Chaitanya Tappu Nov 25 '17 at 13:27

## 2 Answers

You should be fine in a "curves and surfaces" course in my humble opinion. The stereotypical "curves and surfaces" undergraduate course usually does not get into the nitty-gritty theory of differentiable manifolds, and the first 8 chapters of baby Rudin are probably more than sufficient as prerequisites. There's quite a few books covering this kind of material, a lot of people recommend this book. I used this one during my bachelor's and really liked it.

To understand differentiable manifolds, I really think it's useful to have seen some algebra before. Many of the real interesting examples arise as algebraic structures (i.e. Lie groups), and it would be difficult to study those if you didn't know what a group was. Algebraic topology and complex analysis are probably also useful to have, though differential geometry is likely a useful prerequisite to those classes as well.

Questions like this are probably best directed at the professor teaching the class, and I'm sure he or she can give a much better answer than anyone else can.

Read this, it is short and should suffice for most topics in differential geometry

http://books.google.com/books?id=POIJJJcCyUkC