Quartic $L^p$-convergence of cubic Riemannian splines

TL;DR

This paper establishes that cubic Riemannian spline interpolation converges at a quartic rate as the grid size decreases, extending classical Euclidean results to curved spaces using intrinsic geometric methods.

Contribution

It proves quartic convergence of cubic Riemannian splines, adapting classical Euclidean spline theory to the Riemannian setting with intrinsic formulations.

Findings

Quartic convergence rate proven for Riemannian cubic splines.

Extension of classical Euclidean spline results to curved manifolds.

Use of intrinsic geometric methods to avoid charts.

Abstract

We prove quartic convergence of cubic spline interpolation for curves into Riemannian manifolds as the grid size of the interpolation grid tends to zero. In contrast to cubic spline interpolation in Euclidean space, where this result is classical, the interpolation operator is no longer linear. Still, concepts from the linear setting may be generalized to the Riemannian case, where we try to use intrinsic Riemannian formulations and avoid charts as much as possible.