A metric on the manifold of immersions and its Riemannian curvature
type: Item
instance of: scholarly article
title: A metric on the manifold of immersions and its Riemannian curvature
zbMATH Open document ID: 0547.58016
DOI: 10.1007/BF01507749
author: Gerd Kainz
published in: Monatshefte für Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178201
review text: \textit{E. Binz} [ibid. 89, 275-288 (1980; Zbl 0428.58006)] considered two canonical Riemannian metrics on the space of embeddings of a closed (n-1) dimensional manifold into \({\mathbb{R}}^ n\), and computed the geodesic sprays. Here we consider the space of immersions Imm(M,N) where M is without boundary, and we compute the covariant derivative (in the form of its connector) and the Riemannian curvature of one of these metrics, the nontrivial one. The setting is close to that used by \textit{P. Michor} [Manifolds of differentiable mappings (1980; Zbl 0433.58001)] and we refer the reader to this paper for notation.
Mathematics Subject Classification ID: 58D10
Mathematics Subject Classification ID: 58D15
Mathematics Subject Classification ID: 58D20
zbMATH DE Number: 3872325
zbMATH Keywords: Riemannian metrics
zbMATH Keywords: embeddings
zbMATH Keywords: immersions
zbMATH Keywords: covariant derivative
zbMATH Keywords: Riemannian curvature
Wikidata QID: Q115393390
MaRDI profile type: Publication
cites work: Two natural metrics and their covariant derivative on a manifold of embeddings
description: scientific article; zbMATH DE number 3872325