Moduli of triangles in the Heisenberg group

作者：Cappadocia Chris; Nicas Andrew^*

来源：Geometriae Dedicata, 2012, 161(1): 189-219.

DOI：10.1007/s10711-012-9701-9

• Moduli of triangles
• Heisenberg group
• Carnot-Caratheodory metric
• Trigonometry
• Angle deficit formula

We study triangles in the three-dimensional Heisenberg group, , equipped with the Carnot-Carath,odory geometry. The set of ordered triangles in (excluding certain degenerate triangles), up to congruence is naturally identified via a parametrization map to a fine moduli space of parameters. We determine the homeomorphism type of this moduli space and also that of the coarse moduli space of unordered triangles. We describe a boundary for the fine moduli space and construct a compactification for it, up to similarity under the non-isotropic dilation of . Additionally, some trigonometric results for the Carnot-Carath,odory geometry of are given: an angle deficit formula and an analog of the Law of Sines in Euclidean geometry.

• 出版日期2012-12