A lower bound for the curvature invariant \(p(G/K)\) associated with a Riemannian symmetric space \(G/K\) (Q1876323)

Let \(M\) be a Riemannian manifold and \(M\to \mathbb R^N\) an isometric embedding. In [Hiroshima Math. J. 24, 77--110 (1994; Zbl 0808.53050)] the authors intoduced a curvature invariant \(p(M)\) and showed that \(N \geq 2 \dim M - p(M)\). The present paper deals with the problem of determining \(p(M)\) for a symmetric space \(M = G/K\). The invariant \(M\) is reformulated in Lie algebraic terms and lower bounds for \(p(G/K)\) are obtained. For compact rank one symmetric spaces (except for the complex projective plane) these lower bounds on \(p(G/K)\) are sharp and the authors therefore obtain a non-existence theorem for isometric embeddings. This theorem, however, is in general not optimal: The inequality \(N \geq 2 \dim M - p(M)\) can be improved for high dimensional complex projective spaces and the quaternionic and octonionic projective planes as a previous paper [J. Math. Kyoto Univ. 27, 501--505 (1987; Zbl 0633.53080)] by the first named author and the sequels [Hokkaido Math. J. 33, 399--412 (2004; Zbl 1094.53055)], [Hokkaido Math. J. 34, 331--353 (2005; Zbl 1129.53038)] to the present paper show.