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Let $\varphi(t,\cdot,u)$ be the flow of a control system on a
Riemannian manifold $M$ of constant curvature. For a given initial
orthonormal frame $k$ in the tangent space $T_{x_{0}}M$ for some
$x_{0}\in M$, there exists a unique decomposition
$\varphi_{t}=\Theta_{t}\circ\rho_{t}$ where $\Theta_{t}$ is a
control flow in the group of isometries of $M$ and the remainder
component $\rho_{t}$ fixes $x_{0}$ with derivative
$D\rho_{t}(k)=k\cdot s_{t}$ where $s_{t}$ are upper triangular
matrices. Moreover, if $M$ is flat, an affine component can be
extracted from the remainder.