Isometric immersions of SO(5) (Q1060433)

It was proved by the author and \textit{E. Kaneda} [Proc. Jap. Acad., Ser. A 58, 440-442 (1982; Zbl 0515.53041)] that a Riemannian symmetric space SO(n) cannot be isometrically immersed into the Euclidean space \({\mathbb{R}}^{1/4\cdot (3n^ 2-3n-2[n/2])-1}\) even locally. But in the case \(n\geq 5\), this estimate is not best possible. The main result of this paper is that SO(5) cannot be isometrically immersed into \({\mathbb{R}}^{15}\) even locally. Moreover, SO(5) cannot be globally isometrically immersed into \({\mathbb{R}}^{16}\). In the case \(n\geq 6\) the problem remains open. The author proved only that SO(6) cannot be locally isometrically immersed into \({\mathbb{R}}^{22}\).