Oriented robot motion planning in Riemannian manifolds

TL;DR

This paper investigates the topological complexity of robot motion planning in oriented Riemannian manifolds by analyzing the oriented frame bundle, providing bounds that relate to the manifold's dimension.

Contribution

It introduces a topological approach to robot motion planning in Riemannian manifolds and derives bounds for the complexity based on topological invariants.

Findings

Topological complexity bounds are established for oriented frame bundles.

For spin manifolds, the complexity is at least the dimension of the base manifold.

Upper and lower bounds are derived from cup length computations.

Abstract

We consider the problem of robot motion planning in an oriented Riemannian manifold as a topological motion planning problem in its oriented frame bundle. For this purpose, we study the topological complexity of oriented frame bundles, derive an upper bound for this invariant and certain lower bounds from cup length computations. In particular, we show that for large classes of oriented manifolds, e.g. for spin manifolds, the topological complexity of the oriented frame bundle is bounded from below by the dimension of the base manifold.