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?

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 $2\pi$?

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?

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?