Oriented robot motion planning in Riemannian manifolds (Q1737981)

In [Discrete Comput. Geom. 29, No. 2, 211--221 (2003; Zbl 1038.68130)], \textit{M. Farber} has introduced the notion of topological complexity $\text{TC}$ of a space $X$ as the Schwarz genus of the map $p: PX\to X\times X$ where $PX$ denotes the path space of $X$, mapping a path $\alpha \in PX$ to the pair $(\alpha(0), \alpha(1))$. The author notes that a real robot has a shape, and hence one loses a lot of information by simply modeling the robot as a point in $X$. In this way, the author suggests to consider not points in $PX$ but frames on a Riemannian manifold. More accurately, we have the following motion problem (the author calls it the oriented motion problem): Given an oriented Riemannian manifold $(M,g)$ where $g$ denotes the metric, let $F(M,g)$ denote the family of all positive orthonormal frames of tangent spaces of $(M,g)$. For any two points $(x,y)\in M$ and positive orthonormal bases $B_x$ for $T_xM$ and $B_y$ for $T_yM$, find a path $\gamma \in P(F(M,g))$ that joins $\gamma(0) = B_x$ with $\gamma(1) = B_y$. The author considers the invariant $\text{TC}(F(M,g))$ (it depends on $M$ only and, in fact, does not depend on $g$), and notes that this invariant provides the above mentioned desired formalization that takes into account the \textit{shape} of a robot. This idea is the main contribution of the paper. Furthermore, the author makes some calculations of $\text{TC}(F(M,g))$, via a more or less standard technique.