Solving the so-called geodesic endpoint problem, i.e., finding a that connects two given points on manifold, is at basis of virtually all data processing operations, including averaging, clustering, interpolation and optimization. On Stiefel manifold orthonormal frames, this problem computationally involved. A remedy to use quasi-geodesics as replacement for Riemannian geodesics. Quasi-geodesic...

In a recent paper, we verifed a conjecture of Mej́ıa and Pommerenke that the extremal value for the Schwarzian derivative of a hyperbolically convex function is realized by a symmetric hyperbolic “strip” mapping. There were three major steps in the verification: first, a variational argument was given to reduce the problem to hyperbolic polygons bounded by at most two hyperbolic geodesics; secon...

The use of time–like geodesics to measure temporal distances is better justified than the use of space–like geodesics for a measurement of spatial distances. We give examples where a ”spatial distance” cannot be appropriately determined by the length of a space–like geodesic.

Our original results refer to dualistic structure on primaldual interior-point methods for symmetric cone programs with linear constraints. It is shown that scalings by the Nesterov-Todd direction are generated by middle points of geodesics joining with primal interior points and dual interior points. Finally we relate power classes of search directions with geodesics and weighted geometric mea...

In this paper the following phenomena of geodesics in an innnite-dimensional Teichm uller space are founded: a geodesic (locally shortest arc) need not be a straight line (an isometric embedding of a segment of R into the Teichm uller space), no sphere is convex with respect to straight lines, and some geodesics can intersect themselves.

Hedlund 18] constructed Riemannian metrics on n-tori, n 3 for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics 4] are optimal for nilpotent fundamental groups.

In this paper we study pointwise projectively related Einstein metrics (having the same geodesics as point sets). We show that pointwise projectively related Einstein metrics satisfy a simple equation along geodesics. In particular, we show that if two pointwise projectively related Einstein metrics are complete with negative Einstein constants , then one is a multiple of another.

In this paper we study homogeneous geodesics in a three-dimensional connected Lie group G equipped with a left invariant Randers metric and investigates the set of all homogeneous geodesics. We show that there is a three-dimensional unimodular Lie group with a left invariant non-Berwaldian Randers metric which admits exactly one homogeneous geodesic through the identity element. Mathematics Sub...

We develop a study of the relationship between geometry of geodesics and Markoff’s spectrum for Q(i). There exists a particular immersed totally geodesic twice punctured torus in the Borromean rings complement, which is a double cover of the once punctured torus having Fricke coordinates (2 √ 2, 2 √ 2, 4). The set of the simple closed geodesics on this once punctured torus is decomposed into tw...

We exhibit a canonical geometric pairing of the simple closed curves of the degree three cover of the modular surface, Γ\H , with the proper single self-intersecting geodesics of Crisp and Moran. This leads to a pairing of fundamental domains for Γ with Markoff triples. The routes of the simple closed geodesics are directly related to the above. We give two parametrizations of these. Combining ...