Singularities of Frontals

TL;DR

This survey introduces frontals, a class of generalized submanifolds with singularities and well-defined tangent spaces, reviewing their theory and applications in singularity theory with detailed proofs.

Contribution

It provides a comprehensive review of frontals, including detailed proofs and insights that were previously omitted, advancing understanding in singularity theory.

Findings

Frontals generalize submanifolds with singularities.

Detailed proofs of key results on frontals are provided.

Frontals have applications in various geometric problems.

Abstract

In this survey article we introduce the notion of frontals, which provides a class of generalised submanifolds with singularities but with well-defined tangent spaces. We present a review of basic theory and known studies on frontals in several geometric problems from singularity theory viewpoints. In particular, in this paper, we try to give some of detailed proofs and related ideas, which were omitted in the original papers, to the basic and important results related to frontals.