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0. Introduction. The Lagrange singularity theory connects the Lagrange classification of Lagrange immersions with the classification of families of functions, that is, generating families. (The theory of Hormander, Arnol’d, Zakalyukin, . . ., [AGV]). Though there exist detailed studies on singular Lagrange varieties (e.g. [A1], [A2], [DP], [G1], [G2], [J], [Z]), the nice connection between Lagrange singularities and generating families seems to break, in particular, when we try to classify generic Lagrange non-immersions or generic isotropic mappings. Our attempt is then to study singularities of isotropic mappings in the framework of singularity theory of differential mappings (Thom-Mather theory: e.g. [T], [M2]), and to classify their generic singularities under the Lagrange equivalence. In the singularity theory, the transversality theorem is a powerful tool to grasp generic conditions for differentiable mappings, with respect to partial derivatives (see, for instance, [M1], [M2]). In general, if we fix a space of mappings, a generic condition on a mapping should be described by the transversality of the jet section of the mapping to a stratification naturally defined in a jet space of sufficiently higher order. Then, “the transversality theorem” claims the density of the subspace of mappings satisfying the transversality condition, relatively to an appropriate topology on the space of mappings. As a rule, the validity of “the theorem”, however, depends on the space of mappings. Then, the purpose of this paper is to formulate explicitly and to prove the transversality theorem for the space of isotropic mappings of corank at most one, endowed with

Transversalities for Lagrange singularities of isotropic mappings of corank one