Is classical differential geometry still useful?

[petergreat]

Is "classical" differential geometry still useful?

As a physics major I have seen in general relativity the power of modern differential geometry such as coordinate-free treatment of manifolds and Riemannian geometry. However, I've also encountered math textbooks devoted to "classical" differential geometry which is very tied up to curves and surfaces in 3D and does not consider higher-dimensional cases. It is my impression that many topics, especially geodesics, is treated in a much more systematic way in the modern, arbitrary dimensional formulation of differential geometry, and much of the techniques in the classical differential geometry is obsolete. In addition, classical differential geometry lacks the techniques that are widely applied in theoretical physics, such as differential forms.

So my question is, is classical differential geometry still worth studying?

 

[lavinia]

classical differential geometry has a rich store of techniques and examples and many subjects are still researched. There are modern books that develop the classical theories with modern calculus techniques and some that use connections on the tangent circle bundle of the surface.

The theory of Riemann surfaces deeply involves classical differential geometry.