The generalized Cartan-Ambrose-Hicks theorem
type: Item
instance of: scholarly article
title: The generalized Cartan-Ambrose-Hicks theorem
zbMATH Open document ID: 0674.53040
DOI: 10.1007/BF00182117
author: Robert A. Blumenthal
author: James Hebda
published in: Geometriae Dedicata
publication date: 1989-01-01T00:00:00Z
review text: The purpose of the present paper is to generalize the well-known Ambrose theorem and the Hicks theorem [\textit{N. Hicks}, Ill. J. Math. 3, 242-254 (1959; Zbl 0088.381), ibid. 10, 661-679 (1966; Zbl 0184.252)]. Though similar conditions are given which imply the local or global existence of connection-preserving mappings between manifolds endowed with Cartan connections, the new proof by the authors differs significantly from the original ones, since the authors construct the mapping between the frame bundles over the manifolds rather than between the manifolds themselves. Instead of broken geodesics considered in the Ambrose theorem, \(\omega\)- curves are considered where \(\omega\) is the 1-form of the Cartan connection and an \(\omega\)-curve is a curve generated by a vector field X such that \(\omega\) (X) is constant.
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 4103908
zbMATH Keywords: Cartan map
zbMATH Keywords: Ambrose theorem
zbMATH Keywords: Hicks theorem
zbMATH Keywords: connection-preserving mappings
zbMATH Keywords: Cartan connections
zbMATH Keywords: frame bundles
reviewed by: Yosio Mutō
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00182117
OpenAlex ID: W2026496382
description: scientific article; zbMATH DE number 4103908