A Characterisation of Smooth Maps into a Homogeneous Space

We generalize Cartan's logarithmic derivative of a smooth map from manifold into Lie group $G$ to maps homogeneous space $M=G/H$, and determine the global monodromy obstruction reconstructing such infinitesimal data. The embedding submanifold $\Sigma \subset M$ becomes an invariant $ under symmetries "Klein geometry" $M$ whose analysis is taken up in [SIGMA 14 (2018), 062, 36 pages, arXiv:1703.03851].