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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.
Mathematical Journal of Okayama University
Department of Mathematics, Faculty of Science, Okayama University