We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive curvature operator and positive sectional curvature, and stems from the work of Thorpe [28]. We prove that this condition is preserved under Riemannian submersions and Cheeger deformations, and that most compact homogeneous spaces with positive sectional curvature satisfy it.Here, α = A * A is a positive-semidefinite operator obtained from the tensor A of the submersion, and b(α) is the component of α orthogonal to tensors that satisfy the Bianchi identity. It turns out that such component is precisely the projection of α onto the subspace of operators induced by 4-forms.This technique is at the core of many of our results. For instance, it allows us to prove that strongly positive curvature is also preserved under Cheeger deformations:Theorem B. Suppose that (M, g) has strongly positive curvature and an isometric action of a compact Lie group G. Then, the corresponding Cheeger deformation (M, g t ) also has strongly positive curvature for all t > 0.The second part of our paper uses the above tools to study which manifolds with sec > 0 also admit strongly positive curvature. It follows from Theorem A that all manifolds admitting a Riemannian submersion from a round sphere S n → M ,

Abstract. We prove a Slice Theorem around closed leaves in a singular Riemannian foliation, and we use it to study the C ∞ -algebra of smooth basic functions, generalizing to the inhomogeneous setting a number of results by G. Schwarz. In particular, in the infinitesimal case we show that this algebra is generated by a finite number of polynomials.

Abstract. We obtain a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds W 6 , W 12 and W 24 , which are respectively the manifolds of complete flags in C 3 , H 3 and Ca 3 . Together with our earlier work [4], this concludes the classification of simply-connected homogeneous spaces that admit a homogeneous metric with strongly positive curvature.

We present a new link between the Invariant Theory of infinitesimal singular Riemannian foliations and Jordan algebras. This, together with an inhomogeneous version of Weyl's First Fundamental Theorems, provides a characterization of the recently discovered Clifford foliations in terms of basic polynomials. This link also yields new structural results about infinitesimal foliations, such as the existence of non-trivial symmetries.

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