An example of a compact non-C-analytic real subvariety of ${\mathbb R}^3$

TL;DR

This paper constructs a compact, irreducible real-analytic surface in three-dimensional space exhibiting pathological properties, illustrating complex behaviors of real-analytic subvarieties with elementary proofs and visualizations.

Contribution

It provides a novel example of a real-analytic subvariety with unusual properties, highlighting complex behaviors and pathologies in real-analytic geometry.

Findings

The subvariety has only the zero function as a real-analytic function.

Its singular set is not a subvariety and not contained in any one-dimensional subvariety.

It contains a proper subvariety of the same dimension.

Abstract

The purpose of this short expository note is to provide an example exhibiting some of the pathological properties of real-analytic subvarieties, where the pathology can be visualized, and the proofs use only elementary properties of analytic functions. We construct a compact irreducible real-analytic subvariety $S$ of ${\mathbb R}^3$ of pure dimension two such that 1) the only a real-analytic function is defined in a neighbourhood of $S$ and vanishing on $S$ is the zero function, 2) the singular set of $S$ is not a subvariety of $S$, nor is it contained in any one-dimensional subvariety of $S$, 3) the variety $S$ contains a proper subvariety of dimension two. The example shows how a badly behaved part of a subvariety can be hidden via a second well-behaved component to create a subvariety of a larger set. The pathology is visualized using several figures. Examples of these phenomena are known since the time of Cartan, but hard to find in the English language literature.