ﻻ يوجد ملخص باللغة العربية
We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full $H^2$-metric without zero order terms. We find isometries (called $R$-transforms) from some of these spaces into function spaces with simpler weak Riemannian metrics, and we use this to give explicit formulas for geodesics, geodesic distances, and sectional curvatures. We also show how to utilise the isometries to compute geodesics numerically.
We study completeness properties of reparametrization invariant Sobolev metrics of order $nge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions a
We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependenc
We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient
This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curves modulo shape-preserving transformations. We summarize the general constr
We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group. Th