You need differential calculus in R^n, but there's no need for the full force of differential geometry. Also, I don't think that measure gains much in terms of understanding, but it certainly is needed to do some proofs the proper way.

I agree that cutting corners is something I would be super skeptical of in this degree. It should really be an offshoot of a mathematics program, not a CS program with the bare minimal mathematics included. It's end goal is probably PR, money grab, and pumping out students that are really attractive for doing analytics grunt work.