Abstract
On a Lie group , the restrictions of a homomorphism to intervals are geodesic according to the left- or right-invariant connection. If this affine connection is torsion free, is a Riemannian manifold. On Riemannian manifolds in general, it's difficult to compute geodesics (i.e. locally shortest lines) between two points, which are far away from each other. But any curve connecting the two points can be shortened by using a ruler, which allows to construct short geodesics:In normed vector spaces, we look at the midpoint method and the reduced method, which shorten the curve iterative. Both let converge a curve to the geodesic: the straight line between starting point and end point of the curve. On Riemannian manifolds, we get a weaker result: The reduced method can be generalized to complete Riemannian manifolds and on some of them – as for example the hyperbolic spaces – the reduced method lets converge each curve to a geodesic between its starting point and end point.
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