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Everyone:

I am a little frustrated with the progress in my bottom-up learning process, and I think I might get better results by doing some more top-down, i.e., by reading and trying to make sense of some papers in Differential Geometry (with an overlap with Riemannian), Analysis and in Topology. Still, since many papers seem so heavy to read, with unexplained notation, or poor presentation/writing, I was hoping to get some recommendations on papers that are accessible, nicely-written, and not overly-specialized. For context on my background, I took--and passed graduate-level classes , and my qual. exams ( but unfortunately had to drop out.)

Thanks in Advance,

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Any specific branches of analysis? (Real, complex, PDEs, foundations and logic, functional, probability and measure theory, harmonic... just to name a few.) –  Willie Wong Sep 15 '11 at 0:28
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This is totally vague. It sounds like you want nice accessible papers in basically 2/3 of pure mathematics (everything but algebra). Being more specific will get you much better answers. –  Adam Smith Sep 15 '11 at 0:54
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I recommend anything by Milnor. –  Grumpy Parsnip Sep 15 '11 at 1:25
    
Well, am a bit out-of-the-loop. I have gone over a good chunk of Royden, Lee's Smooth Manifolds, and Hatcher but my knowledge of PDE's is shaky. I would take anything that posters consider to be nicely-written and broad-enough, i.e., not hyperspecialized. Sorry not to be able to be more specific. –  MAK Sep 15 '11 at 3:15
    
For miscellaneous collection of analysis stuff, you can just read Terry Tao's blog. :-) –  Willie Wong Sep 15 '11 at 13:02
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Differential / Riemannian Geometry

Older papers

Here I list some papers that I personally enjoyed reading, has historical significance, and which are about stuff that are sometimes omitted from introductory textbooks.

  • Bochner (1946), "Vector fields and Ricci curvature", Bulletin of AMS
  • Sard (1942), "The measure of the critical values of differentiable maps", Bulletin of AMS
  • Yano (1952), "On harmonic and Killing vector fields", Annals of Math.
  • Kobayashi (1957), "Theory of Connections", Ann. Mat. Pur. Appl.
  • Sasaki (1958), "On the differential geometry of tangent bundles of Riemannian manifolds", Tohoku Mathematical Journal
  • Milnor (1964) "Microbundles. I", Topology
  • Smale (1965), "An infinite dimensional version of Sard's theorem", American J. Math.
  • Singer and Thorpe (1969), "The curvature of 4-dimensional Einstein spaces", in Global Analysis: Papers in Honor of K. Kodaira

More modern papers

I must admit that in the choice below there are quite some bit of personal bias. Modern research papers in differential geometry almost certainly assume more knowledge of "general background material" than their counterparts in the 50s through 70s. So in terms of accessibility sometimes you will need to invest some effort looking at references (but I will try to post only "accessible" papers in the sense that references are given to where the background material can be learned). I will put more emphasis on "nicely written" (a subjective value judgement), as "not overly specialised" is something I find hard to define.

To get better recommendations, you probably need to specify which aspects of differential/Riemannian geometry you are more interested in at the moment.

  • Galloway (1989) "The Lorentzian splitting theorem without completeness assumption", J. Diff. Geom
    • Requires some background in causal geometry which arises through general relativity; but the concepts are easily picked up.
  • Schoen (1989) "Variational theory for the total scalar curvature functional for Riemannian metrics and related topics" in Topic in Calculus of Variations
    • A very nice survey article; in particular includes a proof of the $n$-dimensional positive mass theorem.
  • Kühnel and Rademacher (1995) "Conformal diffeomorphisms preserving the Ricci tensor", Proc. AMS
    • Short and very accessible.
  • Galloway and Schoen (2006) "A Generalization of Hawking's Black Hole Topology Theorem to Higher Dimensions", Comm. Math. Physics
    • Don't be scared off by the mention of physics in the introduction. In fact, feel free to skip section 1. All the geometric concepts and terminologies needed are defined in section 2, so you can just start reading there.
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Thanks a lot, Willie! –  MAK Sep 15 '11 at 3:15
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