Definition of a differentiable manifold and papers in Riemannian geometry

There are at least two ways of introducing a definition of differentiable manifolds. I read John Lee's excellent book "Introduction to smooth manifolds" before, but there is too much bundles there for me. Do Carmo's way of introducing definition of a differentiable manifold is quite interesting. Who first introduced this way of definition of differentible manifold and which paper do you recommend me to read immidiately after reading Do Carmo's book on Riemannian geometry? Who follows this definition in papers?

• Not everyone has Do Carmo's book at hand. Please include the definitions you refer to. – Michael Albanese Jan 8 '14 at 9:18
• Definition-wise, I think his way is different from other definitions I have seen in one respect: the local coordinate functions are defined as an inverse of what I normally encountered; they map subsets of $R^n$ into neighborhoods in the manifold, and not the other way around. Perhaps this reduces the number of $f^{-1}$s throughout the further constructions :). Other than this, in my opinion, he doesn't introduce all the short hand terms for various objects, which could have turned a lot of long repetitions in setting the ground at beginnings of theorems and proofs. – user76568 Jan 8 '14 at 10:01
• His definition is more on generalizing a definition of regular surface on $\mathbb{R}^{n}$. It is one step lower in difficulty, because after that there is an easier calculation to do. I see this definition much beautiful than the ordinary, because it gives you better insight what manifold should be. – Alem Jan 8 '14 at 10:18