Showing papers in "Tohoku Mathematical Journal in 1975"

On some types of isoparametric hypersurfaces in spheres i

Abstract: Introduction. This paper is a continuation of Part I [13]. In the first half of the present paper, we study the homogeneous isoparametric hyper surf aces in spheres. Every homogeneous hyper surf ace in a sphere is represented as an orbit of a linear isotropy group of a Riemannian symmetric space of rank 2, due to Hsiang-Lawson [8]. In §1, we study the linear isotropy representations of Riemannian symmetric spaces and their orbits in general. §2 and §3 are devoted to a study of the homogeneous isoparametric hyper surf aces, their classification and invariant polynomials. In § 4 and § 5, we construct explicitly the defining polynomial F for each homogeneous isoparametric hypersurface in a sphere, which was done by Cartan [3] in case g — 3. In the second half, we prove that every closed isoparametric hypersurface in a sphere in case g = 4 and m^ or w2 = 2 is homogeneous. Cartan [4] indicated, without proof, that in case g = 4, every closed isoparametric hypersurface in a sphere with the same multiplicities is homogeneous. In case m^ = m2 = 2, we give a brief outline of its proof in §9. In §6, we exhibit explicit forms of {pa, qa] for some of the homogeneous examples. We see that, for a homogeneous isoparametric hypersurface with g = 4, m^ — 4 and m2 = 3, its defining polynomial — F does not satisfy the condition (B) given in § 6 of Part I. Thus one can conclude that our example constructed in Theorem 2 of Part I for F = H and r = 1 is not homogeneous. Consequently, there are at least two types of isoparametric hypersurfaces in S with the same multiplicities; one is homogeneous, and the other is not. It seems to be an interesting problem to seek a local geometric quantity in order to distinguish them.

Conformally flat Riemannian manifolds admitting a transitive group of isometries, II

Abstract: 1.Introduction. In the present paper, we shall classify conformally flat Riemannian manifolds admitting a transitive group of isometries. This class of manifolds contains the homogeneous Riemannian manifolds of constant curvature classified by J. A. Wolf ([4], [5]). Theorem A in Section 2 imposes a restriction on tie local Riemannian structure of the manifold in consideration. Using Theorem A, we get Theorem B in Section 3 and Theorem C in Section 4. They give the classification together with Theorem D in Section 4. The author wishes to express his sincere thanks to Prof. S. Tanno and Prof. T. sakai who gave him many valuable suggestions and guidantes.