A lower bound for the curvature invariant $p(G/K)$ associated with a Riemannian symmetric space $G/K$

Abstract

We investigate the curvature invariant p(G/K) associated with a Riemannian symmetric space G/K, which was introduced in [3] in order to estimate the least dimension of the Euclidean space R N into which G/K can be locallyisometrically imbedded. We calculate, in a systematic method, a lower bound of p(G/K) for any compact irreducible Riemannian symmetric space G/K. Further, we calculate p(G/K) for compact rank one Riemannian symmetric spaces and establish a non-existence theorem of isometric imbeddings. It is conjectured that the lower bound obtained by our method coincides with p(G/K) for almost every compact irreducible Riemannian symmetric space G/K.