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What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.

edited Feb 10 2011 at 21:26

Douglas Zare
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asked Jan 3 2011 at 22:04

James G
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You need at least 10 reputation points to make a post CW. Also see [Open Problem Garden](garden.irmacs.sfu.ca). – Jason Jan 3 2011 at 22:11

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I"m not sure how to react to the wording "what are some major open problems...?" What exactly are you looking for and why? – Deane Yang Jan 3 2011 at 22:30

The OP's previous question was similarly broad. mathoverflow.net/questions/50687 Presumably he is not so interested in all the minor open problems... – Yemon Choi Jan 4 2011 at 6:05

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I have to say that the answers below are great. You can't do any better than learning what Berger, Donaldson, Gromov, and Yau think are the important open problems, and each definitely has his own unique perspective on the subject. – Deane Yang Jan 4 2011 at 15:17

Retagged. "Tag-removed" is only supposed to be used when there are no other tags. – Douglas Zare Feb 10 2011 at 21:29

Mauricio · Answer 1 · 2011-02-10 19:40:22Z UTC

You can try one of these: http://www.aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf. All of them concern with nonnegatively curved Riemannian manifolds and Alexandrov geometry. In the same context I know a couple of surveys: http://arxiv.org/abs/0707.3091 and http://arxiv.org/abs/math/0701389. It has been conjectured (you can check in those papers) that any nonnegatively curved manifold is rationally elliptic. This is an important open problem in Riemannian geometry.

Daniel Pape · Answer 2 · 2011-01-04 22:38:48Z UTC

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You can find some open problems in the last section, called 'Problem section', of Shing-Tung Yau's book 'Seminar on differential geometry'.

answered Jan 4 2011 at 22:38

Daniel Pape
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This is a great list, but maybe a little outdated by the other suggestions? – Deane Yang Jan 4 2011 at 22:56

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Yes, in fact Yau wrote the "Review of Geometry and Analysis" in 2000 when I was his student precisely as an updated, corrected, and massively expanded sequel to that "Problem Section". – Spiro Karigiannis Jan 4 2011 at 23:38

Surely there may be some redundancies with the other lists and I even guess that some problems from Yau's list are solved by now, but since it also contains some classic problems (like Hopf's conjecture) I thought might be worth mentioning it. – Daniel Pape Jan 4 2011 at 23:38

Thomas Riepe · Answer 3 · 2011-01-04 14:42:24Z UTC

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Gromov's "Spaces and Questions" sketches some big themes and associated questions in Geometry. Atiyah's lectures discuss themes inspired by physics.

answered Jan 4 2011 at 14:42

Thomas Riepe
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Spiro Karigiannis · Answer 4 · 2011-01-04 13:37:54Z UTC

There is also this long review paper by Yau from 2000:

http://www.intlpress.com/AJM/p/2000/4_1/AJM-4-1-235-278.pdf

where he discusses many big open problems in Riemannian geometry, symplectic geometry, algebraic geometry, and geometric analysis. This can keep you occupied for a long long time...

Joseph O'Rourke · Answer 5 · 2011-01-04 01:42:26Z UTC

Here are two possibly relevant references, a decade apart (1998 and 2008), neither of which I can knowledgeably assess:

(1) Thierry Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, 1998.

(2) Simon Donaldson, "Some problems in differential geometry and topology," Nonlinearity 21 T157, 2008.

Here is one sentence from Donaldson's paper:

The outstanding problem then, in 4-manifold topology, is to find if there is something which could play the role of Thurston’s geometrization conjecture, for the case of 3-manifolds, and which might guide further research.

Daniel Miller · Answer 6 · 2011-01-03 22:58:42Z UTC

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The book "A Panoramic View of Riemannian Geometry" by Marcel Berger includes a number of open problems.

answered Jan 3 2011 at 22:58

Daniel Miller
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boris_qd · Answer 7 · 2011-01-03 22:18:56Z UTC

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AIM maintains a list of open problems from workshops that it hosts. You could try looking there (but they may be too specific for your needs).

http://www.aimath.org/pastworkshops/

answered Jan 3 2011 at 22:18

boris_qd
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Open Questions in Riemannian Geometry

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