Geodesic complexity of homogeneous Riemannian manifolds
Stephan Mescher, Maximilian Stegemeyer · Algebraic & Geometric Topology · 2023
X X such that, on each of the open sets, there exists a continuous motion planner.Here, a continuous motion planner is a map associating with each pair of points a continuous path from the first point to the second point which varies continuously with the endpoints.Such maps are interpreted as algorithms telling an autonomous robot in the workspace X how it is supposed to move from its position to a desired endpoint.Unfortunately, the topological complexity of a space does not tell us anything about the feasibility or efficiency of the paths taken by motion planners having TC.X / domains of continuity; see the discussion of Z Błaszczyk and J Carrasquel-Vera [3, Introduction].For example, the explicitly constructed motion planners for configuration spaces of Euclidean spaces by H Mas-Ku and E Torres-Giese [29] and Farber [16, Section 8] require few domains of continuity, but have paths among their values which are far from being length-minimizing.Considering a general metric space, paths taken by the motion planners might become arbitrarily long and thus be unsuited for practical motion planning problems.Recently, D Recio-Mitter [34] has introduced the notion of geodesic complexity of metric spaces.There, the paths taken by motion planners are additionally required to be lengthminimizing between their endpoints.Intuitively, this is seen as the complexity of efficient motion planning in metric spaces.Recio-Mitter's seminal article has already triggered research in geodesic complexity, especially computations of geodesic complexity for interesting classes of examples; see Davis, 9;10].In this article we study the geodesic complexity of complete Riemannian manifolds and derive new lower and upper bounds for their geodesic complexities by methods from Riemannian geometry.Before continuing, we recall the definition of geodesic complexity of geodesic spaces from [34, Definition 1.7] for the special case of a complete Riemannian manifold.Let .M; g/ be a complete connected Riemannian manifold and let PM WD C 0 .Œ0; 1; M / be equipped with the compact-open topology.We recall that a geodesic segment W Œ0; 1 !M is called minimal if it minimizes the length compared to all rectifiable paths from .0/ to .1/.For simplicity, we shall call a minimal geodesic segment simply a minimal geodesic.Consider GM WD f 2 PM j is a minimal geodesic in .M; g/g as a subspace of PM and let W GM !M M;. / D ..0/; .