We study geodesics of the form γ(t) = π(exp(tX) exp(tY )), X, Y ∈ g = Lie(G), in homogeneous spaces G/K, where π : G → G/K is the natural projection. These curves naturally generalise homogeneous geodesics, that is orbits of one-parameter subgroups of G (i.e. γ(t) = π(exp(tX)), X ∈ g). We obtain sufficient conditions on a homogeneous space implying the existence of such geodesics for X, Y ∈ m = T o (G/K). We use these conditions to obtain examples of Riemannian homogeneous spaces G/K so that all geodesics of G/K are of the above form. These include total spaces of homogeneous Riemannian submersions endowed with one parameter families of fiber bundle metrics, Lie groups endowed with special one parameter families of left-invariant metrics, generalised Wallach spaces, generalized flag manifolds, and k-symmetric spaces with k-even, equipped with certain one-parameter families of invariant metrics.Mathematics Subject Classification. Primary 53C25; Secondary 53C30.

Abstract. A geodesic orbit manifold (GO manifold) is a Riemannian manifold (M, g) with the property that any geodesic in M is an orbit of a one-parameter subgroup of a group G of isometries of (M, g). The metric g is then called a G-GO metric in M . For an arbitrary compact homogeneous manifold M = G/H, we simplify the general problem of determining the G-GO metrics in M . In particular, if the isotropy representation of H induces equivalent irreducible submodules in the tangent space of M , we obtain algebraic conditions, under which, any G-GO metric in M admits a reduced form. As an application we determine the U (n)-GO metrics in the complex Stiefel manifolds V k C n .

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Solutions to linear controlled differential equations can be expressed in terms of iterated path integrals of the driving path. This collection of iterated integrals encodes essentially all information about the driving path. While upper bounds for iterated path integrals are well known, lower bounds are much less understood, and it is known only relatively recently that some type of asymptotics for the n-th order iterated integral can be used to recover some intrinsic quantitative properties of the path, such as the length of C 1 paths.In the present paper, we investigate the simplest type of rough paths (the rough path analogue of line segments), and establish uniform upper and lower estimates for the tail asymptotics of iterated integrals in terms of the local variation of the path. Our methodology, which we believe is new for this problem, involves developing paths into complex semisimple Lie algebras and using the associated representation theory to study spectral properties of Lie polynomials under the Lie algebraic development.