Preservation of immersed or injective properties by composing generic generalized distance-squared mappings

TL;DR

This paper demonstrates that composing a generic generalized distance-squared mapping preserves the non-singular and injective properties in equidimensional cases, despite singularities in such mappings.

Contribution

It shows that the non-singular and injective properties are preserved under composition with generic generalized distance-squared mappings in equidimensional cases.

Findings

Non-singular property preserved under composition

Injective property preserved under composition

Singularities are complex in higher dimensions

Abstract

Any generalized distance-squared mapping of equidimensional case has singularities, and their singularity types are wrapped into mystery in higher dimensional cases. Any generalized distance-squared mapping of equidimensional case is not injective. Nevertheless, in this paper, it is shown that the non-singular property or the injective property of a mapping is preserved by composing a generic generalized distance-squared mapping of equidimensional case.