An experiment in Low Earth Orbit (LEO) is proposed to measure components of the Riemann curvature tensor using atom interferometry. We show that the difference in the quantum phase ∆φ of an atom that can travel along two intersecting geodesics is given bymR0i0j/~ times the spacetime volume contained within the geodesics. Our expression for ∆φ also holds for gravitational waves in the long wavel...

Let X be an n-dimensional compact complex manifold, L → X a positive holomorphic line bundle, and H the space of positively curved hermitian metrics on L. The purpose of this article is to prove that geodesics in the infinite-dimensional symmetric space H can be uniformly approximated by geodesics in the finite-dimensional symmetric spaces Hk = GL(Nk + 1,C)/U(Nk + 1,C), where Nk + 1 = dim(H(L))...

In this paper, we prove that on every Finsler n-sphere (S, F ) with reversibility λ satisfying F 2 < ( λ )g0 and l(S , F ) ≥ π(1 + 1 λ ), there always exist at least n prime closed geodesics without self-intersections, where g0 is the standard Riemannian metric on S n with constant curvature 1 and l(S, F ) is the length of a shortest geodesic loop on (S, F ). We also study the stability of thes...

The present note deals with the dynamics of metric connections with vectorial torsion, as already described by E. Cartan in 1925. We show that the geodesics of metric connections with vectorial torsion defined by gradient vector fields coincide with the Levi-Civita geodesics of a conformally equivalent metric. By pullback, this yields a systematic way of constructing invariants of motion for su...

We investigate the rudiments of Riemannian geometry on orbit spaces M/G for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space M/G and they can hit strata which are more singular only at the end points. This is phrased as convexity result. The geodesic spray, viewed as a (strata-preserving) vector field on TM/G,...

Hierarchical linear models (HLMs) are a standard approach for analyzing data where individuals are measured repeatedly over time. However, such models are only applicable to longitudinal studies of Euclidean data. In this paper, we propose a novel hierarchical geodesic model (HGM), which generalizes HLMs to the manifold setting. Our proposed model explains the longitudinal trends in shapes repr...