1) Are there any obstructions to go from non-negative sectional curvature to positive sectional curvature on compact simply connected manifolds ?

2) (Hopf conjecture) If M×N admits a product metric with non-negative sectional curvature, e.g. S²×S², does it admit a metric with positive sectional curvature? Same for simply connected compact symmetric spaces with rank > 1.

3) (Hopf conjecture) Is the Euler characteristic c(M) > 0 for an even dimensional positively curved manifold?

4) Are there positively curved metrics on exotic spheres?

5) Are there any positively curved manifolds Mⁿ with n > 24 besides Sⁿ,CPⁿ, HPⁿ?

6) Are there any "towers" of positively curved manifolds besides Sⁿ,CPⁿ, HPⁿ. A tower is M₁ Ì M₂ Ì ¼ Ì M_i with all M_i positively curved and M_i totally geodesic in M_i+1.

7) Are there any manifolds with pinching d, i.e. 0 < d £ K £ 1, with 1/37 < d < 1/4 ?

8) A Riemannian submersion with totally geodesic fibers is called fat (Weinstein) if all vertizontal curvatures are positive. Show that for a fat principle G bundle, the structure group can not be reduced to a subgroup of G.

9) Are there any fat principle G bundles with dimG > 3? For all known examples G=S¹, SO(3) or SU(2).

10) Show that all Eschenburg spaces

diag(zk1, zk2 , zk3)\SU(3)/diag(1,1,zk1+k2+k3)

have a positively curved metric such that the projection to the weighted projective space

CP2[k1+k2,k1+k3,k2+k3]

is a Riemannian submersion with totally geodesic fibers. For the positively curved Eschenburg metric the fibers are not totally geodesic. So far this is known only for certain values of k_i (O.Dearricott).

11) For a metric on the total space of principle circle bundle to have a positively curved metric, such that the projection is a Riemannian submersion with totally geodesic fibers, reduces to the following "hyperfat" condition for the curvature w on the base manifold B (which must be symplectic):

(Ñxw)(x,y)2 < |ixw|2 á RB(x,y)y,xñ

For example, all fibrations S¹® W_p,q ® SU(3)/T² have a hyperfat principal connection.

Show that all fibrations

S1®diag(zk,zl,zk+l) \SU(3) /diag(1,1,z2k+2l)® SU(3)//T2

have a hyperfat connection. This would produce new examples of positive curvature, since the Eschenburg metrics requires that kl > 0.

12) In 1995 Hitchin produced self dual Einstein orbifold metrics g_k on S⁴, one for each positive integer k, which are invariant under the cohomogeneity one action by SO(3).

These metrics give rise to an SO(3) orbifold principle bundle P_k® S⁴ for each k such that P_k carries a 3-Sasakian metric. As was observed by Grove-Wilking-Ziller P_k is actually a manifold as opposed to just an orbifold.

Show that P_k carry a metric with positive sectional curvature, e.g. a principal bundle metric which is hyperfat.