See http://en.wikipedia.org/wiki/Affine_manifold for a number of other such --- "Geometry of affine manifolds is essentially a network of longstanding conjectures; most of them proven in low dimension and some other special cases." WFL On 3/11/11, Dan Asimov <dasimov@earthlink.net> wrote:

Speaking of affine, there's a famous conjecture in differential geometry (by Shiing-Shen Chern):

Define an *affine manifold* to be one having an atlas all of whose transition functions are affine.

For example, R^n / x ~ 2x is an affine manifold that's topologically the product of spheres S^1 x S^(n-1).

CONJECTURE: The Euler characteristic of any compact affine manifold is equal to 0.

This has remained unresolved for over 50 years.

--Dan

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