Isometric imbeddings of compact symmetric spaces

The purpose of this paper is to construct an equivariant isometric imbedding of each of the following symmetric spaces into a Euclidean space in a unified manner:(For (i), (ii), (iii) and (iv) our constructions are explicit and are given in § §2-5.The general principle of our imbeddings is given in §6.The general principle can be applied to the exceptional case (v).But no explicit constructions will be given for (v).For the hermitian symmetric spaces, our imbeddings coincide with those obtained in [3].For the real, complex and quaternionic projective spaces and for the Cayley projective plane F 4 /Sρin(9), our imbeddings coincide with those constructed in [5].For the classical groups, our imbeddings are nothing but the natural ones into the spaces of matrices of the same size.For the compact hermitian symmetric spaces, our imbeddings are minimal (and substantial) in the sense of total curvature, [1], The same is true for the real, complex and quaternionic projective spaces and the Cayley projective plane, [5].We conjecture that this is the case for all spaces given above 0 . % Grassmann πtanifolds.Let F denote the field R of real number's* the field C of complex numbers or the field Q of quaternions.Let U(n F) be the group of unitary matrices of degree n over F> i.e., U(n;F)={A; t AA=I n ]so that U(n R) = O(ή), U(n C) = U(n) and U(n Q) = Sp(n).Let H(n F) *) Supported partially by NSF Grant GP-5798. 1) ADDED IN PROOF: This conjecture has bean verified by Takeuchi and the author.These imbeddings give also minimal submanifolds of spheres.See M. Takeuchi & S. Kobayashi, Minimal imbeddings of -R-spaces, to appear in J. Diff.Geometry.