The first author has called a Riemannian manifold (resp. a submanifold in a Euclidean space) strongly k-parabolic, if its nullity index (resp. extrinsic nullity index) is not less than k. The problem whether any such Riemannian manifold can be immersed as such a submanifold (with the same k), seems still to be open. Now it is claimed that the answer is negative. Namely, it is proved by means of Cartan’s method of moving frames, exterior differential calculus and a compatibility criterion for Pfaff systems that the analytic strongly 1-parabolic. 3-dimensional Riemannian manifolds depend on three functions of two variables, but such 1-parabolic 3-dimensional submanifolds depend on two variables
[Reviewer’s remark. The well known results of A. N. Kolmogorov, V. I. Arnold a.o. on the possibility to represent functions as the compositions of functions with smaller numbers of variables and problems connected with them indicate that the authors’ argument may not lead to a solution of the considered problem with absolute confidence.