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 image of any distinct k points are linearly in- dependent. For k-regular maps on manifolds, lower bounds of the dimension of the ambient Euclidean space have been exten- sively studied. In this paper, we study the lower bounds of the dimension of the ambient Euclidean space for 2-regular maps on Cartesian products of manifolds. As corollaries, we obtain the exact lower bounds of 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 of the di- mension of the ambient Euclidean space for maps with certain non-degeneracy conditions from disjoint unions of manifolds into Euclidean spaces.