# A Characterisation of Smooth Maps into a Homogeneous Space

## TL;DR

This paper extends Cartan's logarithmic derivative concept to smooth maps into homogeneous spaces, analyzing the global obstructions to reconstructing these maps from infinitesimal data and invariants of submanifolds.

## Contribution

It generalizes the logarithmic derivative to homogeneous spaces and identifies the global monodromy obstructions involved in reconstruction.

## Key findings

- Derived a generalized logarithmic derivative for maps into homogeneous spaces
- Identified the global monodromy obstruction to reconstruction
- Analyzed invariants of submanifolds under Klein geometry symmetries

## 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].