Immersions of manifolds

TL;DR

This paper introduces geometric methods related to manifold immersions, focusing on the Whitney-Graustein theorem for plane curves, aiming to make complex concepts accessible to a broader audience.

Contribution

It provides an accessible exposition of the Whitney-Graustein theorem as a step towards understanding the Smale-Hirsch theorem, emphasizing geometric intuition.

Findings

Simplified explanation of the Whitney-Graustein theorem

Connection to the broader context of the Smale-Hirsch theorem

Accessible presentation for non-specialists

Abstract

This text is based on my talk at the popular science conference ``Dark geometry fest'' which was related to geometric methods and their applications, July 17, 2022. We will move towards the Smale-Hirsch theorem. To this end we will deal with the simplest partial case of this theorem -- the Whitney-Graustein theorem on regular curves in the plane. The text is written in an accessible and sometimes informal way, it is intended primarily for people who are interested in mathematics, but have not yet studied it deeply enough.