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

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


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, noncontractible 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 (FubiniStudy) 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 $2\pi$? 


There is also this long review paper by Yau from 2000: http://www.intlpress.com/AJM/p/2000/4_1/AJM41235278.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... 


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:



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


One of my favourite open problems is this: Are there any examples of irreducible, compact, simplyconnected, riemannian manifolds with vanishing Ricci curvature with generic (i.e., not special) holonomy? 


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


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. 


Just to make this answer selfcontained, I pasted the content from the wiki article here: http://en.wikipedia.org/wiki/ShingTung_Yau#Open_problems Yau has compiled an influential set of open problems in geometry.
One of Yau’s problems is about bounded harmonic functions, and harmonic functions on noncompact manifolds of polynomial growth. After proving nonexistence 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.
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 nonpositive 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 nonpositive curvature by Ballmann and BurnsSpatzier. This leaves rank 1 manifolds of nonpositive curvature as the focus of research. They behave more like manifolds of negative curvature, but remain poorly understood in many regards.
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.
He has collaborated with string theorists including Strominger, Vafa and Witten, and as postdoctorals 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. 


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). 


Here are two more to add to the listed above: Conjecture of LeBrun and Salamon : QuaternionicKahler metrics whose universal covers have only discrete isometry groups? Alekseevsky Conjecture: http://www2.math.ou.edu/~mjablonski/math/ 

