Was Einstein's version of differential geometry close to the modern version? The title pretty much gets the whole question across. In particular, did Einstein work with the modern definition of smooth manifolds, or was that formalized later? Did he know about/use the coordinate-free approach, or did he only consider vectors/tensors to be "things that transform as <rule>"?
Just curious, since I'm learning the modern version, and find it elegant but also quite abstract. So I'm partly interested in the history of differential geometry, and partly in the history of Einstein's formulation of GR.
 A: I would say it was close in spirit, but still different when it comes to the precise mathematical terminology, which we use today. For example, the "birth" of topology is usually dated back to Henri Poincare's paper "Analysis Situs" from 1895. However, the object what is called "topological space" nowadays was only defined in 1914 (in the formulation with neighbourhoods) by Felix Hausdorff in the special case which is nowadays called "Hausdorff space" and in 1922 by Kazimierz Kuratowski (in the formulation with closed sets) in full generality, i.e. without the Hausdorff assumption. The modern definition of manifolds dates back to Herman Weyl's work from 1912/13 about (Riemannian) surfaces. However, the modern definition with atlases was introduced by Hassler Whitney in the 1930s. So all of these modern definitions regarding topology and manifolds were established after Einstein's publications of General relativity around 1915.
Furthermore, also "tensor fields" were discussed in a different fashion. The modern (global) definition requires vector bundles and these differential topological objects were pioneered by Herbert Seifert and Hassler Whitney in the 1930s. To the time Einstein published his general relativity, tensor were usually discussed as geometric objects with indices, dating back to the work of Gregorio Ricci-Curbastro and Tullio Levi-Civita from the end of the 19th century (which in turn is based on previous works by Elwin Christoffel and Bernhard Riemann). So, this is basically still the notation which is nowadays often used in physics and goes under the name "abstract index notation".
However, when you look at the original papers about general relativity by Einstein, they are already very close in the formulation which is used today in the physics literature. If I remember correctly, Einstein's field equations have exactly the same notation (in its local form, i.e. with indices) as the one used in modern physics texts. The original paper from Einstein (1915) has the title "Die Feldgleichungen der Gravitation" and you can find it on google. It is in German, but you will see that the equations look quite similar to what we use today.
Let me stress that I am not an historian, so not all what I have said above might be 100% accurate, but it is often hard nowadys to understand the history of such topics, since it was often much more nonlinear then one would expect.
