Open Questions in Riemannian Geometry - MathOverflow

Open Questions in Riemannian Geometry

2011-01-03T22:04:54Z

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

Answer by boris_qd for Open Questions in Riemannian Geometry

2011-01-03T22:18:56Z

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/

Answer by Daniel Miller for Open Questions in Riemannian Geometry

2011-01-03T22:58:42Z

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

Answer by Joseph O'Rourke for Open Questions in Riemannian Geometry

2011-01-04T01:42:26Z

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.

Answer by Spiro Karigiannis for Open Questions in Riemannian Geometry

2011-01-04T13:37:54Z

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...

Answer by Thomas Riepe for Open Questions in Riemannian Geometry

2011-01-04T14:42:24Z

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

Answer by Daniel Pape for Open Questions in Riemannian Geometry

2011-01-04T22:38:48Z

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

Answer by Mauricio for Open Questions in Riemannian Geometry

2011-02-10T19:40:22Z

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.