Manifolds with positive sectional curvature almost everywhere

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Nonnegatively and Positively curved manifolds (2007, result): “Finally we should mention that prior to [Wilking, 2002], Petersen and Wilhelm [1999] constructed a slightly different metric on the unit tangent bundle of S4, the universal cover of PRTRP4, with positive curvature on an open dense set.”
Examples of Riemannian Manifolds with non-negative sectional curvature (2007, background): “But in [Wi1, ’99]) B.Wilking observed that the Aloff-Wallach space W1,1 is missing since it can be written as SU(3) SO(3)/U(2) where a biinvariant metric induces positive curvature. b)In [Wa,’72] Wallach also proved that if one adds to the assumptions in Proposition 4.3 that the fiber is a symmetric”
Positively curved manifolds with symmetry (2006, background): “In fact it was shown in [21] that the projective tangent bundle PCTCP of CP admits a cohomogeneity 2 metric with positive sectional curvature on an open dense set of points.”
An exotic sphere with positive sectional curvature (2008, methodology): “This changed around the year 2000 with the body of work [PetWilh], [Tapp1], [Wilh2], and [ Wilk ] that gave us many examples of almost positive curvature.”
Torus actions on manifolds of positive sectional curvature (2003, background): “In fact in [25] it was shown that the projective tangent bundles P R T R P ~, P c T C P ~ and P H T H P ~ of R P ~, C P ~ and H P ~ admit metrics with positive sectional curvature on open and dense sets of points.”