Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 029, 15 pages      arXiv:1702.02717      https://doi.org/10.3842/SIGMA.2022.029

A Characterisation of Smooth Maps into a Homogeneous Space

Anthony D. Blaom
University of Auckland, New Zealand

Received June 25, 2021, in final form April 04, 2022; Published online April 10, 2022

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 $\Sigma \subset M$ becomes an invariant of $\Sigma $ under symmetries of the ''Klein geometry'' $M$ whose analysis is taken up in [SIGMA 14 (2018), 062, 36 pages, arXiv:1703.03851].

Key words: homogeneous space; subgeometry; Lie algebroids; Cartan geometry; Klein geometry; logarithmic derivative; Darboux derivative; differential invariants.

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