My personal opinion is that many problems in differential geometry could be avoided if one starts with the classical geometry of curves and surfaces, however, strictly sticks to differential forms and tensors. This has the following advantages:
- No prior knowledge of differential geometry is assumed.
- Not much background with something beyond calculus and basic linear algebra is assumed. Interestingly, there are books that do not even introduce topological spaces in detail to treat surfaces.
- Curves and Surfaces are very intuitive as to some extent one can still imagine all that. When one deals with Jacobi-fields on Riemannina manifolds, it is very ckever to try to imagine some surface in order to visualize this abstract concept.
Especially point three should clarify that it advisable to design an elemantary course which sticks to the easier/or: classical aspects of differential geometry. In Munich, it is possible to take such a course in semester number 4. But students motivated enough are already taking it in the first half of their second year. Sometimes with astonishing results...
Classical differential geometry has wide applications, simply think of Gauss who used his theory of surfaces to measure Hannover (the country Hannover in that days). I think after having understood the differential geometry of surfaces it will be more easy to follow a course on differential geometry which is designed to meet the mathematical prerequisites of for instance general relativity.
