Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 036, 21 pages      arXiv:1304.7430      https://doi.org/10.3842/SIGMA.2013.036
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces

Jeongoo Cheh
Department of Mathematics & Statistics, The University of Toledo, Toledo, OH 43606, USA

Received May 14, 2012, in final form April 19, 2013; Published online April 28, 2013

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 kth 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 R3 subject to rotations.

Key words: congruence; nonhomogeneous space; equivariant moving frame; constant-structure invariant coframe field.

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