Lie Algebroid Invariants for Subgeometry

We investigate the infinitesimal invariants of an immersed submanifold $\mathrm{\Sigma }$ of a Klein geometry $M\cong G\mathrm{/}H$, and in particular an invariant filtration of Lie algebroids over $\mathrm{\Sigma }$. The invariants are derived from the logarithmic derivative of the immersion of $\mathrm{\Sigma }$ 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.