What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.
[Math] Open questions in Riemannian geometry
big-listdg.differential-geometryopen-problemsriemannian-geometrysoft-question
Related Solutions
To get a better feel of the Riemann curvature tensor and sectional curvature:
- Work through one of the definitions of the Riemann curvature tensor and sectional curvature with a $2$-dimensional sphere of radius $r$.
- Define the hyperbolic plane as the space-like "unit sphere" of $3$-dimensional Minkowski space, defined using an inner product with signature $(-,+,+)$. Work out the sectional and Riemann curvature of that
- Repeat #1 and #2 for the $n$-dimensional sphere and hyperbolic space, as well as flat space
Sectional curvature determines Riemann curvature:
That the sectional curvature uniquely determines the Riemann curvature is a consequence of the following:
- The Riemann curvature tensor is a quadratic form on the vector space of $\Lambda^2T_xM$
- The sectional curvature function corresponds to evaluating the Riemann curvature tensor (as a quadratic form) on decomposable elements of $\Lambda^2T_xM$
- There is a basis of $\Lambda^2T_xM$ consisting only of decomposable elements
Added in response to Anirbit's comment
Perhaps you shouldn't try to compute the curvature too soon. First, make sure you understand the Riemannian metric of the unit sphere and hyperbolic space inside out. There are many ways to do this. But the most concrete way I know is to use stereographic projection of the sphere onto a hyperplane orthogonal to the last co-ordinate axis. Either the hyperplane through the origin or the one through the south pole works fine. This gives you a very nice set of co-ordinates on the whole sphere minus one point. Work out the Riemannian metric and the Christoffel symbols. Also, work out formulas for an orthonormal frame of vector fields and the corresponding dual frame of 1-forms. Figure out the covariant derivatives of these vector fields and the corresponding dual connection 1-forms.
After you do this, do everything again with hyperbolic space, which is the hypersurface
$-x_0^2 + x_1^2 + \cdots + x_n^2 = -1$ with $x_0 > 0$
in Minkowski space with the Riemannian metric induced by the flat Minkowski metric. You can do stereographic projection just like for the sphere but onto the unit $n$-disk given by
$x_1^2 + \cdots + x_n^2 = 1$ and $x_0 = 0$,
where the formula for the hyperbolic metric looks just like the spherical metric in stereographic co-ordinates but with a sign change in appropriate places. This is the standard conformal model of hyperbolic space.
After you understand this inside out, you can use these pictures to figure out why the $n$-sphere and its metric is given by $O(n+1)/O(n)$ and hyperbolic space by $O(n,1)/O(n)$ and why the metrics you've computed above correspond to the natural invariant metric on these homogeneous spaces. You can then check that the formulas for invariant metrics on homogeneous spaces give you the same answers as above.
Use references only for the general formulas for the metric, connection (including Christoffel symbols), and curvature. I recommend that you try to work out these examples by hand yourself instead of trying to follow someone else's calculations. If possible, however, do it with another student who is also trying to learn this at the same time.
If, however, you want to peek at a reference for hints, I recommend the book by Gallot, Hulin, and Lafontaine. I suspect that the book by Thurston is good too (I studied his notes when I was a student). For invariant Riemannian metrics on a homogeneous space, I recommend the book by Cheeger and Ebin (available cheap from AMS! When I was a student, I had to pay a hundred dollars for this little book but it was well worth it).
But mostly when I was learning this stuff, I did and redid the same calculations many times on my own. I was never able to learn much more than a bare outline of the ideas from either books or lectures. Just try to get a rough idea of what's going on from the books, but do the details yourself.
These are some big problems I know about:
$e$-positivity of Stanley's chromatic-symmetric functions for incomparability graphs obtained from $3+1$-avoiding posets. Shareshian and Wachs have some recent results related to this that connects these polynomials to representation theory, and they refine this conjecture with a $q$-parameter. Note that Schur-positivity is already known, and that the $e_\lambda$ expands positively into Schurs. It all boils down to finding a partition-valued combinatorial statistic on certain acyclic orientations.
Give a combinatorial description of the Kronecker coefficients.
- Find a combinatorial interpretation of the Littlewood-Richardson coefficients for the Jack polynomials, $J_{\mu} J_{\nu} = \sum_\lambda c^\lambda_{\mu\nu}(\alpha) J_\lambda$. It is conjectured (but not proved) that $c^\lambda_{\mu\nu}(\alpha)$ is a polynomial in $\alpha$ with non-negative integer coefficients. (Here, one needs to be a bit careful with which normalization one chooses).
Find a combinatorial description of the multiplicative structure constants for the Schubert polynomials (analogue of the Littlewood-Richardson coefficients in the Schur polynomial case).
The different variants of the shuffle conjecture. UPDATE: There is a recent proof on arxiv. There are still some generalizations of this that remains unproved.
Prove that LLT polynomials have positive Schur expansion. This is related to the shuffle conjecture and the $qt$-Kostka polynomials, see N. Loehr's notes.
Give a combinatorial formula for the non-homogeneous symmetric Jack polynomials (similar to the Knop-Sahi formula for the ordinary Jack polynomials).
Give combinatorial descriptions of structure constants that appear in plethystic substitutions, $a^\nu_{\lambda\mu} = \langle s_\lambda[s_\mu], s_\nu \rangle$
Find a combinatorial description of the $qt$-Kostka polynomials.
Find a combinatorial description of the polynomials $c^\nu_{\lambda\mu}(t)$ in $$s_\lambda(t) \cdot P_\mu(x;t) = \sum_\nu c^\nu_{\lambda\mu}(t) P_\mu(x;t)$$ where $P_\lambda(x;t)$ is the Hall-Littlewood polynomial.
Prove (or disprive) that the map $k \mapsto K_{k\lambda,k\mu}$ is a polynomial with non-negative coefficients, where $K_{\lambda\mu}$ is the Kostka coefficient. Or in more general, same question for $k \mapsto c^{k\nu}_{k\lambda,k\mu}$ for the Littlewood-Richardson coefficients. This question goes back to King, Tollu and Toumazet. Note that a proof of the latter would imply the famous saturation conjecture proved by A. Knutson and T. Tao. The conjecture is stated in Rassart's paper, here, where the polynomiality property is proved.
I have looked on the special case concerning Koskta coefficients, and this lead me to ask this somewhat related question. The negative answer to this question hints that the positivity conjecture might possibly be false, but a minimal counter example lives in a very high dimension, out of reach by exhaustive computer search. This concerns Ehrhart polynomials for non-integral polytopes, and very little is know about this case. One can show that such Ehrhart (quasi)polynomials have properties that Ehrhart polynomials corresponding to integral polytopes lack.
Whenever one seeks a combinatorial description, it is understood that the polynomial or number in question is conjectured to be a polynomial with non-negative integer coefficients.
Best Answer
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 $2\pi$?