A characterization of oriented Grassmann manifolds
type: Item
instance of: scholarly article
title: A characterization of oriented Grassmann manifolds
zbMATH Open document ID: 0547.53026
DOI: 10.1216/RMJ-1985-15-2-573
published in: Rocky Mountain Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: The geodesic spheres in two-point homogeneous spaces are hypersurfaces whose second fundamental forms have some nice properties. T. J. Willmore and the reviewer have used these properties to characterize, at least locally, the two-point homogeneous spaces. Jacobi vector fields are used extensively in their work. It is natural to ask if the other symmetric spaces can be characterized locally in a similar way. This paper is a first step in answering this problem. The authors consider a special class of symmetric spaces, namely the real Grassmann manifolds and their compact duals. A characterization is given in terms of a special parallel tensor field of type (1,3) satisfying several additional algebraic properties and with a special relation with the second fundamental form of the geodesic spheres. These results include some of the results of T. J. Willmore and the reviewer as a special case. This paper has been extended by the second author [Indian J. Pure Appl. Math. 15, 99-112 (1984; Zbl 0531.53036)] to the complex Grassmann manifolds and a unified theory for all the Grassmannians will be published in one of his forthcoming papers.
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 3872204
zbMATH Keywords: symmetric spaces
zbMATH Keywords: Grassmann manifolds
zbMATH Keywords: parallel tensor field
zbMATH Keywords: geodesic spheres
author: David E. Blair
author: Arthur J. Ledger
reviewed by: Lieven Vanhecke
MaRDI profile type: Publication
Wikidata QID: Q124798863
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description: scientific article; zbMATH DE number 3872204