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**ABSTRACT:**In this paper, we classify compact simply connected cohomogeneity one manifolds up to equivariant diffeomorphism whose isotropy representation by the connected component of the principal isotropy subgroup has three or less irreducible summands. The manifold is either a bundle over a homogeneous space or an irreducible symmetric space. As a corollary such manifolds admit an invariant metric with non-negative sectional curvature. Comment: 64 pages, 2 appendices and 19 tablesGeometriae Dedicata 06/2010; · 0.47 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We prove that S^2 x S^2 satisfies an intermediate condition between having metrics with positive Ricci and positive sectional curvature. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the 2-Grassmannian, is strictly positive; and this can be done with an arbitrarily small lower bound on the distance between the planes considered. Although they have positive Ricci curvature, these metrics do not have nonnegative sectional curvature. Such metrics also have positive biorthogonal curvature, meaning that the average of sectional curvatures of any two orthogonal planes is positive.09/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**This is a survey of recent results on manifolds with positive curvature from a series of lecture given in Guanajuato, Mexico in 2010. It also contains some hitsorical comments.10/2012;

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