### 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.

pdf (394 kb)   tex (21 kb)

References

1. Blaom A.D., Lie algebroid invariants for subgeometry, SIGMA 14 (2018), 062, 36 pages, arXiv:1703.03851.
2. Burstall F.E., Calderbank D.M.J., Submanifold geometry in generalized flag manifolds, Rend. Circ. Mat. Palermo (2) Suppl. 72 (2004), 13-41.
3. Burstall F.E., Calderbank D.M.J., Conformal submanifold geometry I-III, arXiv:1006.5700.
4. Cannas da Silva A., Weinstein A., Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, Vol. 10, Amer. Math. Soc., Providence, RI, Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999.
5. Crainic M., Fernandes R.L., Integrability of Lie brackets, Ann. of Math. 157 (2003), 575-620, arXiv:math.DG/0105033.
6. Crainic M., Fernandes R.L., Lectures on integrability of Lie brackets, in Lectures on Poisson Geometry, Geom. Topol. Monogr., Vol. 17, Geom. Topol. Publ., Coventry, 2011, 1-107, arXiv:math.DG/0611259.
7. Dufour J.P., Zung N.T., Poisson structures and their normal forms, Progress in Mathematics, Vol. 242, Birkhäuser Verlag, Basel, 2005.
8. Ivey T.A., Landsberg J.M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, Vol. 61, Amer. Math. Soc., Providence, RI, 2003.
9. Mackenzie K.C.H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.
10. Sharpe R.W., Differential geometry. Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics, Vol. 166, Springer-Verlag, New York, 1997.