Regular Maps on Cartesian Products and Disjoint Unions of Manifolds

TL;DR

This paper investigates the minimal Euclidean space dimensions needed for 2-regular and k-regular maps from manifolds, especially Cartesian products and disjoint unions, providing exact bounds for spheres and projective spaces.

Contribution

It establishes precise lower bounds for the dimensions of ambient Euclidean spaces for k-regular maps on various manifolds, extending previous studies to new manifold classes.

Findings

Exact lower bounds for 2-regular maps on spheres and projective spaces.

Lower bounds for 3-regular maps on spheres and projective spaces.

Generalization to maps with non-degeneracy conditions from disjoint unions.

Abstract

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.