# Lie Algebroid Invariants for Subgeometry

## TL;DR

This paper introduces Lie algebroid invariants for immersed submanifolds in Klein geometries, providing new insights into subgeometry and classical theorems through a Lie algebroid framework.

## Contribution

It develops a Lie algebroid-based approach to subgeometry, offering a new interpretation of Cartan's moving frames and a novel proof of the fundamental theorem of hypersurfaces.

## Key findings

- Invariants derived from the logarithmic derivative of immersions.
- A new interpretation of Cartan's method of moving frames.
- A novel proof of the fundamental theorem of hypersurfaces.

## 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.