Notion of ℍ-orientability for surfaces in the Heisenberg group ℍⁿ

2019/10/02 by Canarecci, Giovanni
#53A35 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1910.01158

Abstract

This paper aims to define and study a notion of orientability in the Heisenberg sense (ℍ-orientability) for the Heisenberg group ℍⁿ. In particular, we define such notion for ℍ-regular 1-codimensional surfaces. Analysing the behaviour of a Möbius Strip in ℍ¹, we find a 1-codimensional ℍ-regular, but not Euclidean-orientable, subsurface. Lastly we show that, for regular enough surfaces, ℍ-orientability implies Euclidean-orientability. As a consequence, we conclude that non-ℍ-orientable ℍ-regular surfaces exist in ℍ¹.

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