TITLE: Learning modern differential geometry before curves and surfaces
QUESTION [13 upvotes]: What might one miss by learning modern differential geometry without first learning about curves and surfaces?
I'm currently reading this book on differential geometry which starts with manifolds and builds from there. I'm already deep inside it and it's a perfect fit for me. 
Still, I wonder what i might have missed by skipping on learning the "classical" differential geometry. The book has a chapter about hypersurfaces but i'm still a bit worried that i might miss something important.
What are some important notions from classical differential geometry i better know?

REPLY [4 votes]: I agree that you'll be fine if you go straight into manifolds before curves and surfaces. But it is simply NOT true (as stated in another answer) that surface is of little interest in modern different geometry, in particular in area related to PDE and analysis. 
In case of curves, that is indeed not so interesting as all 1 dimensional objects are locally isometric (So there are essentially no intrinsic properties to study). But there are still something extrinsic to talk about (for examples geodesic in Riemannian manifolds and closed orbit in symplectic manifolds). The studies are quite complete though (So really not so interesting from the research point of view).   
But that is not the same for dimensional two. The main reason is that the most natural operator, the Laplace operator, is conformal invariant when dimensional is two. This results in a large contrast between surface theory and general manifold theory. Indeed, there has been large progress in minimal surface theory, mean curvature flow of surfaces, when compared to that in general dimension. 
On the other hand, the use of surface theory is extremely essential in studying manifolds of positive curvatures, puesdo-holomorphic curves in symplectic manifolds etc, which are all active research directions in modern differential geometry. 
Similar special phenomenon occurs when you restrict to 3 and 4 dimensional manifolds. These "low dimensional geometry" are also very active research directions. 
However, I do agree that you can put aside the curve and surface theory for the moment. In a limit amount of time, it is reasonable to go into the general pictures first. I just feel like I have to say something for surface.