# 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