Stationaere Kurven auf endlichdimensionalen Mannigfaltigkeiten

TL;DR

This paper explores the concept of weak geodesics on finite-dimensional Riemannian manifolds, providing foundational insights and discussing potential generalizations to infinite-dimensional spaces for applications like image matching.

Contribution

It offers a detailed, geometric understanding of weak geodesics on finite-dimensional manifolds and outlines how these ideas can be extended to infinite dimensions for practical algorithms.

Findings

Detailed description of weak geodesics on finite-dimensional manifolds

Framework for generalizing geodesics to infinite-dimensional spaces

Potential applications in image matching algorithms

Abstract

In this work we discuss the notion of stationary curves of the length functional, the so-called (weak) geodesics, on a Riemannian manifold. The motivation behind this work is to give a detailed description of many key concepts from differential geometry that one needs in order to understand the important notion of a (weak) geodesic. For this, we mainly focus on finite-dimensional smooth manifolds, so that we can develop an intuitive and geometric understanding of the concepts that we want to discuss. At the end of this work, we also provide a rough description of how one can generalise these ideas into infinite dimensions and how one can use (weak) geodesics in special algorithms for image matching (see [21]).