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

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 Möbius 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 ${\mathbb{H}}^1$.

19 pages