We may define a topological manifold to be a second-countable Hausdorff space such that every point has an open neighborhood homeomorphic to an open subset of $\mathbb{R}^n$. We can further define a smooth manifold to be a topological manifold equipped with a structure sheaf of rings of smooth functions by transport of structure from $\mathbb{R}^n$, since $\mathbb{R}^n$ has a canonical sheaf of differentiable functions $\mathbb{R}^n\to \mathbb{R}$, with a canonical restriction sheaf to any open subset. This gives a manifold as a locally ringed space. (Of course this definition generalizes to all sorts of other kinds of manifolds with minor adjustments). 


Then the questions:
If we totally ignore the definition using atlases, will we at some point hit a wall? Can we fully develop differential geometry without ever resorting to atlases? 


Regardless of the above answer, are there any books that develop differential geometry *primarily* from a "locally ringed space" viewpoint, dropping into the language of atlases only when necessary? I looked at Kashiwara & Schapira's "Sheaves on Manifolds", but that's much more focused on sheaves of abelian groups and (co)homology. 


Edit:


To clarify (Since Pete and Kevin misunderstood): It's easy to show that the approaches are equivalent, but proofs using charts don't *always* translate easily to proofs using sheaves. 



How much of differential geometry can be developed entirely without atlases?
There is the book by Ramanan "Global Calculus" which develops differential geometry relying heavily on sheaf theory (you should see his definition of connection algebra...).
He avoids the magic words "locally ringed space" by requiring the structure sheaf to be a subsheaf of the sheaf of continuous functions (hence maximal ideal of stalks = vanishing functions).


