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It is well-known that a convex polytope is regular if and only if the action of its symmetry group on the set of complete flags is transitive. We use this criterion to extend the notion of regularity from convex polytopes to convex bodies in general.To achieve this extension, we introduce the notion of maximality for flags, which reduces to completeness for convex polytopes. A convex body whose symmetry group acts transitively on the set of maximal flags (and which satisfies a further condition which is automatic for convex polytopes) will be called a regular solid.It is easy to show that if B is a regular solid, then so is its polar B*. Moreover, all regular solids are perfect in the sense of [7]. Our main structure theorem concerns a projection n: ¥ R -> 0> R from the set Sf R of all regular solids onto the set 2P R of all regular polytopes. Conversely, a construction of Kostant [5] associates either one or two regular nonpolytopes to certain regular polytopes. The relation between n and Kostant's construction is not yet fully elucidated, but does at least suggest conjectures on the classification of regular solids, and of perfect solids in general.We also explore the less stringent notions of /-transitivity and complete transitivity, finding conditions under which these properties are preserved under product and coproduct operations.The work is organised in three sections. Section 1 examines convex bodies in general, introducing some ideas and constructions that are needed later. Section 2 deals with the notion of regularity, and Section 3 with perfection and transitivity.

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Let F be a totally geodesic foliation of dimension n and codimension p on a Riemannian manifold (M, g). Suppose that g is a bundle-like metric for F and M has at least one point at which none of its mixed sectional curvatures vanishes. Under these conditions we prove that n ≤ p − 1. We show that this inequality is optimal, and none of the above conditions can be removed. (2000): 53C12, 53C20. Mathematics Subject Classification

We introduce the notion of generalised 3-Sasakian structure on a manifold and show that a totally umbilical, but not totally geodesic, proper QR-submanifold of a quaternion Kaehlerian manifold is an extrinsic sphere and inherits such a structure.

Abstract. We define the Liouville distribution on the tangent bundle of a pseudo-Finsler manifold and prove that it is integrable. Also, we find geometric properties of both leaves of Liouville distribution and the vertical distribution.