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We show that the unit tangent bundle of S 4 and a real cohomology CP 3 admit Riemannian metrics with positive sectional curvature almost everywhere. These are the only examples so far with positive curvature almost everywhere that are not also known to admit positive curvature. AMS Classification numbers (i) The connected component of the identity of the isometry group is isomorphic to SO(4) and contains a free S 3 -subaction. (ii) The set of points where there are 0 sectional curvatures contains totally geodesic flat 2-tori and is the union of two copies of SRemark In the course of our proof we will also obtain a precise description of the set of 0-curvature planes in the Grassmannian. This set is not extremely complicated, but the authors have not thought of a description that is succinct enough to include in the introduction. RemarkIn the sequel to this paper, [25], the second author shows that the metric on the Gromoll-Meyer sphere can be perturbed to one that has positive sectional curvature almost everywhere. In contrast to [25] the metric we construct here is a perturbation of a metric that has zero curvatures at every point. Peter Petersen and Frederick WilhelmGeometry and Topology, Volume 3 (1999) 332By taking a circle subgroup of the free S 3 -action in Theorem A(i) we get the following. Before outlining the construction of our metric we recall that the S 3 -bundles over S 4 are classified by Z ⊕ Z as follows ([14], [21]). The bundle that corresponds to (n, m) ∈ Z ⊕ Z is obtained by gluing two copies of R 4 × S 3 together via the diffeomorphism g n,m : ( Corollary B There is a manifoldwhere we have identified R 4 with H and S 3 with {v ∈ H | |v| = 1}. We will call the bundle obtained from g m,n "the bundle of type (m, n)", and we will denote it by E m,n .Translating Theorem 9.5 on page 99 of [15] into our classification scheme (0.1) shows that the unit tangent bundle is of type (1, 1). We will show it is also the quotient of the S 3 -action on Sp(2) given byManifolds with positive curvature almost everywhere Geometry and Topology, Volume 3 (1999) 333(It was shown in [20] that this quotient is also the total space of the bundle of type (2, 0), so we will call it E 2,0 .)The quotient of the biinvariant metric via A 2,0 is a normal homogeneous space with nonnegative, but not positive sectional curvature. To get the metric of Theorem A we use the method described in [8] to perturb the biinvariant of Sp(2) using the commuting S 3 -actionsWe call the new metric on, and will observe in Proposition 1.14 that A 2,0 is by isometries with respect to g ν 1 ,ν 2 ,l uOur metric on the unit tangent bundle is the one induced by the Riemannian submersion (Sp(2), g ν 1 ,ν 2 ,l uIn section 1 we review some generalities of Cheeger's method. In section 2 we study the symmetries of E 2,0 . In section 3 we analyze the infinitesimal geometry of the Riemannian submersion Sp(2) p 2,1 −→ S 7 , given by projection onto the first column. This will allow us to compute the curvature tensor of the metric, g ν 1 ,ν 2 , obtain...

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Abstract. We show how to lift positive Ricci and almost non-negative curvatures from an orbit space M/G to the corresponding G-manifold, M. We apply the results to get new examples of Riemannian manfiolds that satisfy both curvature conditions simultaneously.

We prove a new comparison lemma for Jacobi fields that exploits Wilking's transverse Jacobi equation. In contrast to standard Riccati and Jacobi comparison theorems, there are situations when our technique can be applied after the first conjugate point. Using it, we show that the focal radius of any submanifold N of positive dimension in a manifold M with sectional curvature greater than or equal to 1 does not exceed π2. In the case of equality, we show that N is totally geodesic in M and the universal cover of M is isometric to a sphere or a projective space with their standard metrics, provided that N is closed. Our results also hold for kth intermediate Ricci curvature, provided that the submanifold has dimension ⩾k. Thus, in a manifold with Ricci curvature ⩾n−1, all hypersurfaces have focal radius ⩽π2, and space forms are the only such manifolds where equality can occur, if the submanifold is closed. Example  and Remark  show that our results cannot be proven using standard Riccati or Jacobi comparison techniques.