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

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    $\begingroup$ You need at least 10 reputation points to make a post CW. Also see Open Problem Garden. $\endgroup$
    – Jason
    Commented Jan 3, 2011 at 22:11
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    $\begingroup$ I"m not sure how to react to the wording "what are some major open problems...?" What exactly are you looking for and why? $\endgroup$
    – Deane Yang
    Commented Jan 3, 2011 at 22:30
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    $\begingroup$ The OP's previous question was similarly broad. mathoverflow.net/questions/50687 Presumably he is not so interested in all the minor open problems... $\endgroup$
    – Yemon Choi
    Commented Jan 4, 2011 at 6:05
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    $\begingroup$ 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. $\endgroup$
    – Deane Yang
    Commented Jan 4, 2011 at 15:17
  • $\begingroup$ Retagged. "Tag-removed" is only supposed to be used when there are no other tags. $\endgroup$
    – Douglas Zare
    Commented Feb 10, 2011 at 21:29

11 Answers 11

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There are many surveys and books with open problems, but it would be nice to have a list of a dozen problems that are open and yet embarrasingly simple to state. A list that is "folklore" and that every graduate student in differential geometry should keep in his/her pocket.

Here are the ones I like best:

1. Does every Riemannian metric on the $3$-sphere have infinitely many prime closed geodesics? Does it have at least three (prime) closed geodesics?

2. If the volume of a Riemannian $3$-sphere is equal to 1, does it carry a closed geodesic whose length is less that $10^{24}$? Same question with $S^1 \times S^2$ if you like it better than the $3$-sphere.

3. Does $S^2 \times S^2$ admit a Riemannian metric with positive sectional curvature?

4. If a Riemannian metric on real projective space has the same volume as the canonical metric, does it carry a closed, non-contractible geodesics whose length is at most $\pi$ ?

5. What are the solutions of the isoperimetric problem in the complex projective plane provided with its canonical (Fubini-Study) metric?

6. Up to constant multiples, is the canonical metric on the complex projective plane the only Riemannian metric on this manifold for which all geodesics are closed?

7. Does every Riemannian metric on the $2$-sphere that is sufficiently close to the canonical metric and whose area is $4\pi$ carry a closed geodesic whose length is at most $2\pi$?

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

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There is also "Review of Geometry and Analysis" by S.-T. Yau (Asian Journal of Mathematics, vol. 4, no. 1, pp. 235-278, March 2000), 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...

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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.

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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.

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One of my favourite open problems is this:

Are there any examples of irreducible, compact, simply-connected, riemannian manifolds with vanishing Ricci curvature with generic (i.e., not special) holonomy?

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Just to make this answer self-contained, I pasted the content from the wiki article here: http://en.wikipedia.org/wiki/Shing-Tung_Yau#Open_problems

Yau has compiled an influential set of open problems in geometry.

  • Harmonic functions with controlled growth

One of Yau’s problems is about bounded harmonic functions, and harmonic functions on noncompact manifolds of polynomial growth. After proving non-existence of bounded harmonic functions on manifolds with positive curvatures, he proposed the Dirichlet problem at infinity for bounded harmonic functions on negatively curved manifolds, and then proceeded to harmonic functions of polynomial growth. Dennis Sullivan tells a story about Yau's geometric intuition, and how it led him to reject an analytical proof of Sullivan's. Michael Anderson independently found the same result about bounded harmonic function on simply connected negatively curved manifolds using a geometric convexity construction.

  • Rank rigidity of nonpositively curved manifolds

Again motivated by Mostow's strong rigidity theorem, Yau called for a notion of rank for general manifolds extending the one for locally symmetric spaces, and asked for rigidity properties for higher rank metrics. Advances in this direction have been made by Ballmann, Brin and Eberlein in their work on non-positive curved manifolds, Gromov's and Eberlein's metric rigidity theorems for higher rank locally symmetric spaces and the classification of closed higher rank manifolds of non-positive curvature by Ballmann and Burns-Spatzier. This leaves rank 1 manifolds of non-positive curvature as the focus of research. They behave more like manifolds of negative curvature, but remain poorly understood in many regards.

  • Kähler–Einstein metrics and stability of manifolds

It is known that if a complex manifold has a Kähler–Einstein metric, then its tangent bundle is stable. Yau realized early in 1980s that the existence of special metrics on Kähler manifolds is equivalent to the stability of the manifolds. Various people including Simon Donaldson have made progress to understand such a relation.

  • Mirror symmetry

He has collaborated with string theorists including Strominger, Vafa and Witten, and as post-doctorals from theoretical physics with B. Greene, E. Zaslow and A. Klemm . The Strominger–Yau–Zaslow program is to construct explicitly mirror manifolds. David Gieseker wrote of the seminal role of the Calabi conjecture in relating string theory with algebraic geometry, in particular for the developments of the SYZ program, mirror conjecture and Yau–Zaslow conjecture.

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Here are two more to add to the listed above:

Conjecture of LeBrun and Salamon : Quaternionic-Kahler metrics whose universal covers have only discrete isometry groups?

Alekseevsky Conjecture: http://www2.math.ou.edu/~mjablonski/math/

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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'.

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    $\begingroup$ This is a great list, but maybe a little outdated by the other suggestions? $\endgroup$
    – Deane Yang
    Commented Jan 4, 2011 at 22:56
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    $\begingroup$ 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". $\endgroup$
    – Spiro Karigiannis
    Commented Jan 4, 2011 at 23:38
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    $\begingroup$ 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. $\endgroup$
    – Daniel Pape
    Commented Jan 4, 2011 at 23:38
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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.

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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/

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