Manifolds of absolutely continuous curves and the square root velocity framework

2016/12/08 by Alexander Schmeding, Schmeding, Alexander
Computer Science · Engineering · Mathematics · #22E65 #58B10 #58B20 (secondary) #58D15 (primary) #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Morphological variations and asymmetry #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1612.02604

openalex publication_date 2016/12/08 · openalex created_date 2017/07/14 · openalex updated_date 2026/07/28

Abstract

A classical result in Riemannian geometry states that the absolutely continuous curves into a (finite-dimensional) Riemannian manifold form an infinite-dimensional manifold. In the present paper this construction and related results are generalised to absolutely continuous curves with values in a strong Riemannian manifolds. As an application we consider extensions of the square root velocity transform (SRVT) framework for shape analysis. Computations in this framework frequently lead to curves which leave the shape space (of smooth curves), and are only contained in a completion. In the vector valued case, this extends the SRVT to a space of absolutely continuous curves. We investigate the situation for shape spaces of manifold valued (absolutely continuous) curves.

Citations

• Lie groups over non-discrete topological fields
• Fundamentals of submersions and immersions between infinite-dimensional\n manifolds
• Measurable regularity properties of infinite-dimensional Lie groups

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