Notion of $\mathbb{H}$-orientability for surfaces in the Heisenberg group $\mathbb{H}^n$

TL;DR

This paper introduces a new concept of orientability tailored to the Heisenberg group, exploring its implications for surfaces and demonstrating that non-Euclidean orientability can differ from classical notions.

Contribution

It defines $ ext{H}$-orientability for surfaces in the Heisenberg group and analyzes its relationship with Euclidean orientability, including examples and conditions for equivalence.

Findings

Existence of $ ext{H}$-regular, non-Euclidean-orientable surfaces in $ ext{H}^1$

$ ext{H}$-orientability implies Euclidean-orientability for regular surfaces

A M"obius strip in $ ext{H}^1$ is $ ext{H}$-regular but not Euclidean-orientable

Abstract

This paper aims to define and study a notion of orientability in the Heisenberg sense ($\mathbb{H}$-orientability) for the Heisenberg group $\mathbb{H}^n$. In particular, we define such notion for $\mathbb{H}$-regular $1$-codimensional surfaces. Analysing the behaviour of a M\"obius Strip in $\mathbb{H}^1$, we find a $1$-codimensional $\mathbb{H}$-regular, but not Euclidean-orientable, subsurface. Lastly we show that, for regular enough surfaces, $\mathbb{H}$-orientability implies Euclidean-orientability. As a consequence, we conclude that non-$\mathbb{H}$-orientable $\mathbb{H}$-regular surfaces exist in $\mathbb{H}^1$.