We study a well-known scalar quantity in Riemannian geometry, the Ricci scalar, in the context of diffusion tensor imaging (DTI), which is an emerging non-invasive medical imaging modality. We derive a physical interpretation for the Ricci scalar and explore experimentally its significance in DTI. We also extend the definition of the Ricci scalar to the case of high angular resolution diffusion...

Integrability, one of the classic issues in galactic dynamics and in general in celestial mechanics, is here revisited in a Riemannian geometric framework, where newtonian motions are seen as geodesics of suitable “mechanical” manifolds. The existence of constants of motion that entail integrability is associated with the existence of Killing tensor fields on the mechanical manifolds. Such tens...

Two Riemannian manifolds are said to have C-conjugate geodesic flows if there exist an C diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step nilmanifolds the geodesic flow is C rigid. For special classes of 2-step nilmanifolds, we show t...

Abstract Optimisation algorithms such as the Newton method were first generalised to manifolds by generalising components of algorithm directly: gradients replaced Riemannian gradients, straight lines geodesics, and so forth. This meant having endow manifold with a metric. Traditionally then, attention focused on geometry underlying manifold. However, we argue is not right focus because it does...

Abstract. The problem of the local summability of the sub-Riemannian mean curvature H of a hypersurface M in the Heisenberg group, or in more general Carnot groups, near the characteristic set of M arises naturally in several questions in geometric measure theory. We construct an example which shows that the sub-Riemannian mean curvature H of a C2 surface M in the Heisenberg group H1 in general...

We give a new characterisation of the unparametrised geodesics, or distinguished curves, for affine, pseudo-Riemannian, conformal, and projective geometry. This is type moving incidence relation. The used to provide very general theory construction quantities that are necessarily conserved along curves. formalism immediately yields explicit formulae these curve first integrals. usual role Killi...