Definition of a differentiable manifold and papers in Riemannian geometry

Question

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?