Computing the Riemannian Logarithm on the Stiefel Manifold: Metrics, Methods, and Performance

A Matrix-Algebraic Algorithm for the Riemannian Logarithm on the Stiefel Manifold under the Canonical Metric

Sobolev Metrics on the Riemannian Manifold of All Riemannian Metrics

On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L-metric as decribed first by [10]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition...

ON THE LIFTS OF SEMI-RIEMANNIAN METRICS

In this paper, we extend Sasaki metric for tangent bundle of a Riemannian manifold and Sasaki-Mok metric for the frame bundle of a Riemannian manifold [I] to the case of a semi-Riemannian vector bundle over a semi- Riemannian manifold. In fact, if E is a semi-Riemannian vector bundle over a semi-Riemannian manifold M, then by using an arbitrary (linear) connection on E, we can make E, as a...

Sobolev Metrics on the Manifold of All Riemannian Metrics

On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L-metric as described first by [11]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a conditio...

On the Implementation of Lie Group Methods on the Stiefel Manifold on the Implementation of Lie Group Methods on the Stiefel Manifold *

There are several applications in which one needs to integrate a system of ODEs whose solution is an n × p matrix with orthonormal columns. In recent papers algorithms of arithmetic complexity order n×p have been proposed. The class of Lie group integrators may seem like a worth while alternative for this class of problems, but it has not been clear how to implement such methods with O(np) comp...