Coordinate free differential geometry?

Question

Many texts about differential geometry, such as for example Frankel's book, rely more or less heavily on doing things in coordinates.

Are there any (introductory) resources to approaches that present ways to build up the concepts of differential geometry and their properties (including the proofs!) without relying on coordinates or explicitly defined ranks of the appearing structures?

Comments

Just a comment, I recommend to be fluent in both approaches. In particular, I know too many theoretical physicists who obsess over coordinate-free formulations (as compared to abstract-index formalisms) because of their "simplicity" and "elegance", but in practice they are absolutely unable to apply the formalism to concrete physical scenarios. As a rule of thumb, I find that for non-metric differential structure and related computations, and very simple field theories (Abelian gauge theory,...), coordinate-free formalisms can be more practical. However, once you have a metric, symplectic, and other structures and fields involved, the usefulness of coordinate-free formalisms starts to drop very quickly, at least if you are looking to do physics.

Answer 1

I have not read the book you mention but usually physics texts on differential geometry do indeed heavily use coordinate systems. In my math course on differential geometry we used Lee's "Introduction to Smooth Manifolds" , "Riemannian Manifolds", and Do Carmo's "Riemannian Geometry" all of which I found pretty nice.

Answer 2

Coordinates have their use, and coordinate-free formalism has its use.

This is long understood very deeply by algebraic geometers:

a) An affine scheme in the sense of a solution locus to polynomial equations is manifestly a coordinate-driven space. (The fact that we are talking about polynomials involves the choice of coordinates, namely of their free variables.)

b) By considering "functorial geometry" over the category of affine schemes, one obtaines a very abstract coordinate-free picture of geometry, via "schemes" and "algebraic spaces" and so forth.

When doing geometry, generally we don't want to  discard either of these.

There is a differential-geometric version on precisely this picture of algebraic geometry, this goes by the name synthetic differential geometry.

A development of perturbative quantum field theory in these terms is available here:

PhysicsForums Insights: Mathematical Quantum Field Theory

see:

•  chapter 1. "Geometry"
  (this is all coordinates on Cartesian spaces, analogous to affine schemes);
•  chapter 3. "Fields"
  (this introduces the abstract coordinate-free functorial geometry induced thereby)