Abstract
Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n ∈ N , and discover the properties that qualify these operations to the title addition despite the fact that, in general, these binary operations are neither commutative nor associative. The binary operation of the Beltrami-Klein ball model of hyperbolic geometry, known as Einstein addition, and the binary operation of the Beltrami-Poincaré ball model of hyperbolic geometry, known as Möbius addition, determine corresponding metric tensors in the unit ball. For a variety of metric tensors, including these two, we show how binary operations can be recovered from metric tensors. We define corresponding scalar multiplications, which give rise to gyrovector spaces, and to norms in these spaces. We introduce a large set of binary operations that are algebraically equivalent to Einstein addition and satisfy a number of nice properties of this addition. For such operations we define sets of gyrolines and co-gyrolines. The sets of co-gyrolines are sets of geodesics of Riemannian manifolds with zero Gaussian curvatures. We also obtain a special binary operation in the ball, which is isomorphic to the Euclidean addition in the Euclidean n-space.
Highlights
Let B be the unit, open ball in the Euclidean n-space Rn,B = { x ∈ Rn : k x k < 1 }, (1)n ∈ N, where k · k is the Euclidean norm.Einstein addition is a binary operation, ⊕E, in the ball B ⊂ Rn that stems from his velocity composition law in the ball B ⊂ R3 of relativistically admissible velocities
Einstein addition turns out to be both gyrocommutative and gyroassociative, giving rise to the rich algebraic structures that became known as a gyrogroup and a gyrovector space, the definitions of which are presented in Definitions 1–3, Section 2
One may expect that the rich algebraic structure of Einstein addition can find home in differential geometry, giving rise to a novel branch called Binary Operations in the Ball
Summary
N ∈ N, where k · k is the Euclidean norm. Einstein addition is a binary operation, ⊕E , in the ball B ⊂ Rn that stems from his velocity composition law in the ball B ⊂ R3 of relativistically admissible velocities. One may expect that the rich algebraic structure of Einstein addition can find home in differential geometry, giving rise to a novel branch called Binary Operations in the Ball. For each metric tensor considered in this paper we define an operation t ⊗ a of scalar multiplication, which leads to corresponding gyrovector spaces. The lack of the commutative and associative laws is compensated by the gyrocommutative and gyroassociative laws that these binary operations obey As such, these binary operations give rise to the algebraic objects known as gyrogroups and gyrovector spaces. The special interest of our study of both Einstein addition and Möbius addition within the framework of differential geometry stems from the result that they are gyrocommutative gyrogroup operations.
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