This paper, the second of a series, deals with the function space of all smooth Kähler metrics in any given closed complex manifold M in a fixed cohomology class. This function space is equipped with a pre-Hilbert manifold structure introduced by T. Mabuchi [10], where he also showed formally it has non-positive curvature. The previous result of the second author [4] showed that the space is a ...

Cyclic evolutions of quantum states are accompanied by geometric phases. This should remain true not only for pure but also for mixed states, then resulting in noncommutative phases. After a short introduction expressions for the parallel transport of these phases along geodesic polygons are derived. They become rather explicit for two-level systems, predicting definite deviations from the pure...

The d-convex sets in a metric space are those subsets which include the metric interval between any two of its elements. Weak modularity is a certain interval property for triples of points. The d-convexity of a discrete weakly modular space X coincides with the geodesic convexity of the graph formed by the two-point intervals in X. The Helly number of such a space X turns out to be the same as...

Using the shallow water equations, a numerical framework on a spherical geodesic grid that conserves domainintegrated mass, potential vorticity, potential enstrophy, and total energy is developed. The numerical scheme is equally applicable to hexagonal grids on a plane and to spherical geodesic grids. This new numerical scheme is compared to its predecessor and it is shown that the new scheme d...

Isometric surfaces share the same geometric structure also known as the ‘first fundamental form’. For example, all possible bending of a given surface, that include all length preserving deformations without tearing or stretching the surface, are considered to be isometric. We present a method to construct a bending invariant canonical form for such surfaces. This invariant representation is an...

In this paper we answer to a question raised by Ambrosio and Rigot [2] proving that any interior point of a Wasserstein geodesic in the Heisenberg group is absolutely continuous if one of the end-points is. Since our proof relies on the validity of the so-called Measure Contraction Property and on the fact that the optimal transport map exists and the Wasserstein geodesic is unique, the absolut...

This paper addresses the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. Current methods like Principal Geodesic Analysis (PGA) and Geodesic PCA (GPCA) minimize the distance to a ”Geodesic subspace”. This allows to build sequences of nested subspaces which are consistent with a forward component analysis approach. However, these methods cannot be adapted to a backw...

Surface-constrained motion, i.e., motion constraint by a rigid surface, is commonly found in daily activities. The current work investigates the choice of hand paths constrained to a concave hemispherical surface. To gain insight regarding paths and their relationship with task dynamics, we simulated various control policies. The simulations demonstrated that following a geodesic path (the shor...