Regular maps on Cartesian products and disjoint unions of manifolds

A map from a manifold to a Euclidean space is said to be k-regular if the images of any distinct k points are linearly independent. For k-regular maps on manifolds, lower bounds on the dimension of the ambient Euclidean space have been extensively studied. In this paper, we study the lower bounds on the dimension of the ambient Euclidean space for 2-regular maps on Cartesian products of manifolds. As corollaries, we obtain the exact lower bounds on the dimension of the ambient Euclidean space for 2-regular maps and 3-regular maps on spheres as well as on some real projective spaces. Moreover, generalizing the notion of k-regular maps, we study the lower bounds on the dimension of the ambient Euclidean space for maps with certain non-degeneracy conditions from disjoint unions of manifolds into Euclidean spaces.