Abstract
We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup, and satisfies a number of nice properties of the Einstein addition. We also prove that a set of cogyrolines for the Einstein addition is the same as a set of gyrolines of another binary operation. This operation is found directly and it turns out to be commutative. The same results are obtained for the binary operation of the Beltrami–Poincare disk model, known as Möbius addition. We find a canonical representation of metric tensors of binary operations isomorphic to the Einstein addition, and a canonical representation of metric tensors defined by cogyrolines of these operations. Finally, we derive a formula for the Gaussian curvature of spaces with canonical metric tensors. We obtain necessary and sufficient conditions for the Gaussian curvature to be equal to zero.
Highlights
The theory of gyrogroups and gyrovector spaces has been intensively developed over recent years
We find a canonical representation of metric tensors of binary operations isomorphic to the Einstein addition, and a canonical representation of metric tensors defined by cogyrolines of these operations
(i) Einstein gyrovector spaces are based on the Einstein addition, and they provide the algebraic setting for the Klein ball model of hyperbolic geometry
Summary
The theory of gyrogroups and gyrovector spaces has been intensively developed over recent years. We developed in [21] a differential geometry approach to the theory of gyrogroups and gyrovector spaces based on local properties of underlying binary operations and, on properties of canonical metric tensors (see Definition 1) of corresponding Riemannian manifolds. For instance, the Einstein addition and Möbius addition in the ball are neither commutative nor associative They are both gyrocommutative and gyroassociative, giving rise to gyrogroups and gyrovector spaces [20]. The new results presented in this paper split up into three classes: Class 1: Einstein addition and Möbius addition are isomorphic to each other, giving rise to an isomorphism between corresponding gyrogroups and gyrovector spaces.
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