Types of the canonical isometric imbeddings of symmetric \(R\)-spaces (Q1207791)

Considering the space of infinitesimal isometric deformations of an isometric imbedding \(f\) of a Riemannian manifold \(M\) into a Euclidean space \(\mathbb{R}^ n\), \textit{N. Tanaka} [Nagoya Math. J. 51, 137-160 (1973; Zbl 0265.53050)] proved that there is a differential operator \(L\) associated with \(f\) so that the considered space is isomorphic to the solution space of the equation \(L\varphi = 0\). The isometric imbedding \(f\) is said to be of type \((E)\) (resp. type \((F)\)) if \(L\) is of elliptic type (resp. finite type). These types are known (N. Tanaka a.o.) for the canonical isometric imbeddings of the Hermitian symmetric spaces of compact type. Now the following results are proved for the canonical isometric imbeddings \(f\) of the symmetric \(R\)-spaces \(G/K\) associated with finite dimensional simple graded Lie algebras \(\mathbb{L}\) of the first kind over \(R\) such that \(\mathbb{L} ^ c\) is simple over \(\mathbb{C}\): (1) \(f\) is of type \((F)\) if \(G/K\) does not coincide with any of the real quadrics \(Q^{p,q}(\mathbb{R})\) \((p,q>1)\), the real projective spaces \(P^ n(\mathbb{R})\) \((n>1)\) nor the real Grassmann manifolds \(G^{2,q}(\mathbb{R})\) \((q>2)\); (2) \(f\) is of type \((E)\) but not of type \((F)\) if \(G/K = G^{2,q}(\mathbb{R})\) \((q>3)\); (3) \(f\) is of type \((E)\) if \(G/K\) does not coincide with any of the real quadrics \(Q^{ p,q}(\mathbb{R})\) \((p,q > 1)\) nor the real projective spaces \(P^ n(\mathbb{R})\) \((n>1)\).