If this is the first course in Differential geometry, you should not go further than Gauss--Bonnet for surfaces. I would not even consider anything with dimension >2. By the way here is our textbook on the subject. If they like Differential geometry, they could take another course.
If you cover more, then it is easy to produce lammers. If you skip these topics, then (most likely) your student will have no idea what is differential geometry at the end of the course.