On the holonomy of Lie algebroids

What the math gives to ML

The paper constructs a canonical quotient of time-dependent Lie-algebroid paths by identifying paths whose induced local automorphism germ is identical. Its most transferable ingredient is the flow-product law: a noncommutative composition of time-dependent controls whose induced flows compose exactly, together with holonomy as the resulting path invariant. This suggests neural modules that represent transformations by composable algebra-valued controls while training on the induced holonomy rather than on arbitrary control parameterizations. The clearest initial target is a symmetry-aware sequence or geometric encoder, where path segments should compose consistently and equivalent transformation trajectories should produce identical latent actions.