Super Hyperbolic Law of Cosines: same formula with different content

TL;DR

This paper extends classical trigonometric laws to the super hyperbolic plane using Minkowski supergeometry, revealing formulas with fermionic corrections and applications to supergeodesics and supernumber theory.

Contribution

It derives super hyperbolic Laws of Cosines and Sines with fermionic modifications, and explores their geometric and number-theoretic implications.

Findings

Identical formulas to classical laws with fermionic corrections

Existence of unique common orthogonal supergeodesic for non-ultraparallel supergeodesics

Insights into supernumber theory and potential Gauss product analogue

Abstract

We derive the Laws of Cosines and Sines in the super hyperbolic plane using Minkowski supergeometry and find the identical formulae to the classical case, but remarkably involving different expressions for cosines and sines of angles which include substantial fermionic corrections. In further analogy to the classical case, we apply these results to show that two parallel supergeodesics which are not ultraparallel admit a unique common orthogonal supergeodesic, and we briefly describe aspects of elementary supernumber theory, leading to a prospective analogue of the Gauss product of quadratic forms.