On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces

TL;DR

This paper develops a comprehensive method to determine local congruence of immersions in various G-spaces, providing necessary and sufficient conditions based on jet bundle invariants, with applications to classical differential geometry.

Contribution

It introduces a general approach to solve the local congruence problem for immersions in G-spaces, extending classical invariants to nonhomogeneous spaces.

Findings

Derived conditions for local congruence in G-spaces

Bound on the differential order of invariants

Reconstructed classical invariants like curvature and Schwarzian derivative

Abstract

We show how to find a complete set of necessary and sufficient conditions that solve the fixed-parameter local congruence problem of immersions in $G$-spaces, whether homogeneous or not, provided that a certain $k^{\rm th}$ order jet bundle over the $G$-space admits a $G$-invariant local coframe field of constant structure. As a corollary, we note that the differential order of a minimal complete set of congruence invariants is bounded by $k+1$. We demonstrate the method by rediscovering the speed and curvature invariants of Euclidean planar curves, the Schwarzian derivative of holomorphic immersions in the complex projective line, and equivalents of the first and second fundamental forms of surfaces in ${\mathbb R}^3$ subject to rotations