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

SIGMA 14 (2018), 062, 36 pages      arXiv:1703.03851      https://doi.org/10.3842/SIGMA.2018.062

Lie Algebroid Invariants for Subgeometry

Anthony D. Blaom
Waiheke Island, New Zealand

Received November 15, 2017, in final form June 13, 2018; Published online June 18, 2018

Abstract
We investigate the infinitesimal invariants of an immersed submanifold $\Sigma$ of a Klein geometry $M\cong G/H$, and in particular an invariant filtration of Lie algebroids over $\Sigma$. The invariants are derived from the logarithmic derivative of the immersion of $\Sigma$ into $M$, a complete invariant introduced in the companion article, A characterization of smooth maps into a homogeneous space. Applications of the Lie algebroid approach to subgeometry include a new interpretation of Cartan's method of moving frames and a novel proof of the fundamental theorem of hypersurfaces in Euclidean, elliptic and hyperbolic geometry.

Key words: subgeometry; Lie algebroids; Cartan geometry; Klein geometry; differential invariants.

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