Spatio-temporal continuous wavelets for the analysis of motion on manifolds

Abstract

This paper presents kinematical Lie algebras and Lie groups that describe motion on differentiable spatiotemporal manifolds. Motion is assumed to be translational and rotational with position, velocity and acceleration. The general kinematical groups that are derived have action that depends up the local chart and the local curvature. Squared integrable representations of these Lie groups define continuous wavelet transforms. General conditions of admissibility are presented with an example on space of constant curvature. The wavelet construction matches perfectly to differential geometry "a la Cartan" and mechanics on manifolds. Applications concern optimum control, general relativity and quantum mechanics.