Curvature characterization and classification of rank-one symmetric spaces

Abstract

We characterize and classify rank-one symmetric spaces by two axioms on R( , v)v , the Jacobi part of the curvature tensor.1. Introduction.In his book [3], Chavel gave a beautiful account of the rank-one symmetric spaces from a geometric point of view up to the classification of them, which he left for the reader to pursue as a matter in Lie group theory.The purpose of this paper is to extend ChaveΓs approach to fill in this last step by classifying these spaces, on the Lie algebra level, based on geometric considerations.To be more precise, for each unit vector v , define the Jacobi operator K v = R( , υ)υ, where R(X, Y)Z denotes the curvature tensor.Then for a compact rank-one symmetric space one notes that (1) K v have two distinct constant eigenvalues (1 & 1/4) for all v if the space is not of constant curvature, and ( 2) E\(υ) 9 the linear space spanned by v and the eigenspace of K v with eigenvalue 1 is the tangent space of a totally geodesic sphere of curvature 1 (a projective line in fact) through the base point of υ , and consequently E\(w) = E\(υ) whenever w is in E\(v).These two properties will be adapted in the next section as two axioms, and we will prove then that they turn out to characterize locally rank-one symmetric spaces.Indeed, motivated by [5] and [8], we prove that the curvature tensor, under the two axioms, induces a certain Clifford module, from which the curvature components and the dimension of the space can be read off.It then follows that the space must be locally rank-one symmetric, and the list of such spaces falls out in a natural way.We would like to mention that there is another interesting geometric classification of the compact symmetric spaces by Karcher [11].Karcher's construction of the Cayley plane rests on some intriguing properties of isoparametric submanifolds and he has to assume the space is symmetric of positive curvature for the classification, whereas all our results follow from the the two axioms and the technique requires essentially no rίiore than linear algebra.Our analysis reveals by

Locations

• Pacific Journal of Mathematics - View - PDF
• Project Euclid (Cornell University) - View - PDF