Hyperspherical Trigonometry and Corresponding Elliptic Functions

TL;DR

This paper develops hyperspherical trigonometry in higher dimensions, linking it to elliptic functions and providing new addition formulae, with applications to multidimensional Euler tops and Double Elliptic models.

Contribution

It introduces hyperspherical trigonometry in multidimensional space and connects it to elliptic functions with two moduli, extending classical results to higher dimensions.

Findings

Derived basic formulae for hyperspherical trigonometry in multidimensional space.

Established addition formulae for elliptic functions related to algebraic curves.

Applied the formulae to multidimensional Euler tops and the Double Elliptic model.

Abstract

We develop the basic formulae of hyperspherical trigonometry in multidimensional Euclidean space, using multidimensional vector products, and their conversion to identities for elliptic functions. We show that the basic addition formulae for functions on the 3-sphere embedded in 4-dimensional space lead to addition formulae for elliptic functions, associated with algebraic curves, which have two distinct moduli. We give an application of these formulae to the cases of a multidimensional Euler top, using them to provide a link to the Double Elliptic model.