A Characterisation of Smooth Maps into a Homogeneous Space

Abstract

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