First of all, it is not a duplicate of those questions a la "best textbooks for differential geometry". What I want to ask is which papers would you recommend reading while taking (self learning) differential geometry courses? I could list (and find advises on) such papers in classical statistics, or Bayesian statistics for example. But I cannot find any info about which papers would someone recommend for differential geometry. The scope I assume would be some sufficiently general topic, not too narrow, so that material would be usefull for those eager to learn differential geometry more in depth.
From time to time, I teach a graduate seminar at the University of Washington called "Research in Geometric Analysis." It's aimed at second-year and third-year math PhD students who have already taken at least a year's worth of graduate courses in differential geometry. I give them this list, which is my subjective list of "greatest hits" in differential geometry over the past 50 years or so (current as of 2005, the last time I taught the course).
You'll be surprised how difficult it is to read mathematics research papers. My students -- even the best prepared ones -- typically tell me that seminar is the hardest course they've ever taken. As you probably already know, mathematics is more cumulative than any other subject, so to understand any research paper demands an understanding and appreciation of lots of background and context. And most math papers aren't very easy to follow even for experts, since they're written when the ideas are still new and haven't had time to be cleaned up and made user-friendly.
If you're really serious about familiarizing yourself with the research literature, my suggestion would be to pick one of the "clusters" of three or four related papers on this list that interests you, and start skimming them. You'll quickly run into stuff you don't understand, so you'll have to look at the references in the bibliography and go read them. Then the same thing will happen. Keep working backwards until you start getting a sense of the mathematical context in which the paper was written, and then start reading the original paper again. Through a series of "successive approximations" like this, you'll eventually start understanding what's going on.
I don't mean to be discouraging, but this process is very hard, even for students who are enrolled in graduate programs and get lots of mentoring and support. So if you can do it successfully by self-study, my hat's off to you!