I see that someone has mentioned this in the comments already, but I think it deserves to be left as an answer.
Here at UGA we do have a regular undergraduate course fitting your approximate description. It is an undergraduate course in differential geometry. The prerequisites are multivariable calculus and linear algebra (it is hard to see how one could get away with any less than that!). Especially, real analysis is not a prerequisite for the course: in fact, to get an undergraduate math major at UGA one needs to take only one of: (a) real analysis (b) complex analysis (c) this differential geometry course. (To be honest, I am not thrilled that real analysis is not required, but I most certainly digress.)
This course is often taught by Ted Shifrin, the most distinguished and veteran teacher on our faculty and someone with more years of experience than he'd probably like me to quantify on the subject of differential geometry (his thesis advisor was Chern). In particular the course is now being taught by Ted, out of a preliminary draft of a book written by Ted. The course webpage is here, from which you can find links to the course text, the syllabus, the problem sets, and so forth.
