Invariant higher-order variational problems II

TL;DR

This paper studies Riemannian cubics on object manifolds influenced by Lie group actions, characterizing their properties and reductions, with applications in computational anatomy.

Contribution

It introduces a second-order variational framework for Riemannian cubics on object manifolds and analyzes their horizontal lifts and projections.

Findings

Characterization of cubics that lift horizontally to the transformation group.

Conditions under which non-horizontal geodesics project to cubics.

Reduced equations via second-order Lagrange–Poincaré reduction.

Abstract

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesics on the group of transformations project to cubics. Finally, we apply second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.