Lie Algebroid Invariants for Subgeometry

Abstract

We investigate the infinitesimal invariants of an immersed submanifold Σ of a Klein geometry M ≅ G/H, and in particular an invariant filtration of Lie algebroids over Σ. The invariants are derived from the logarithmic derivative of the immersion of Σ 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.