Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks
type: Item
instance of: scholarly article
zbMATH Open document ID: 1276.37047
DOI: 10.1137/10081678X
author: Mario Micheli
author: Peter W. Michor
author: David Mumford
published in: SIAM Journal on Imaging Sciences
publication date: 2012-09-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.2637
Mathematics Subject Classification ID: 37K65
Mathematics Subject Classification ID: 53B21
Mathematics Subject Classification ID: 68U10
zbMATH DE Number: 6082565
zbMATH Keywords: shape spaces
zbMATH Keywords: landmark points
zbMATH Keywords: cometric
zbMATH Keywords: sectional curvatute
Wikidata QID: Q56624028
MaRDI profile type: Publication
arXiv ID: 1009.2637
title: Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks
review text: From the abstract: This paper deals with the computation of the sectional curvature for the manifolds of \(N\) landmarks in \(D\) dimensions with respect to the Riemannian metric induced by the diffeomorphism group. The inverse metric tensor in local coordinates for these manifolds has the property that each of its elements depends on at most \(2D\) points out of \(ND\) coordinates. This makes the matrices of partial derivatives of the cometric very sparse in nature, and so it solves the highly nontrivial problem of developing a formula, called the Mario's formula, that expresses the sectional curvature in terms of the cometric and its first and second partial derivatives. We apply such a formula to the manifolds of landmarks, explore the case of geodesics on which only two points have nonzero momenta, and compute the sectional curvature of \(2\)-planes spanned by the tangents to such geodesics.
reviewed by: Jin Hong Kim
description: scientific article; zbMATH DE number 6082565