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ID 33175
FullText URL
Author
Otsuki, Tominosuke
Abstract

In the present paper, we shall investigate the conditions under which a given Riemannian space Vn can be imbedded, as a hypersurface, into a Riemannian space Vn+1 which has the following properties I) and II). I) The group of holonomy of the space with a normal projective .connexion corresponding to Vn+1 fixes a hyperquadric and Vn is its image in Vn+1 , that is, the locus of points lying on the parallel displaced hyperquadrics, regarded as points in the tangent projective spaces. If Vn+1 has the property above, there exist a scalar y such that the hypersurface is given by the relation y -= 0. II) The orthogonal trajectories of the family of the hypersurfaces on which y is constant are geodesics in Vn+1. If the group of holonomy of the space with a normal projective eonnexion corresponding to a Vn+1 fixes a hyperquadric, it is projectively equivalent to an Einstein space2 ). In the previous paper, the author have studied the problem of the same kind as this under the conditions I) and II') Vn+1 is an Einstein space. The imbedding problem of Vn into Vn+1 under the only condition I) is very complicated in structure. The purpose of the present paper is also to search for the methods dealing with the problem, as the previous one.

Published Date
1952-10
Publication Title
Mathematical Journal of Okayama University
Volume
volume2
Issue
issue1
Publisher
Department of Mathematics, Faculty of Science, Okayama University
ISSN
0030-1566
NCID
AA00723502
Content Type
Journal Article
language
英語
File Version
publisher
Refereed
True
Submission Path
mjou/vol2/iss1/4