Isomorphism of Binary Operations in Differential Geometry

S S symmetry

Article Isomorphism of Binary Operations in Differential Geometry

Nikita E. Barabanov

Department of Mathematics, North Dakota State University, Fargo, ND 58108, USA; [email protected]

Received: 28 August 2020; Accepted: 30 September 2020; Published: 3 October 2020

Abstract: We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic.

Keywords: differential geometry; canonical metric tensor; binary operation; Einstein addition; Euclidean addition; isomorphism

1. Introduction The theory of the binary operation of the Beltrami–Klein ball model, known as Einstein addition, and the binary operation of the Beltrami–Poincare ball model of hyperbolic geometry, known as Möbius addition has been extensively developed for the last twenty years [1–36]. There appeared a theory that may be called gyrogeometry, based on nice algebraic properties of the aforementioned operations and the concepts of gyrogroups, gyrovector spaces, gyrotrigonometry, and gyrogeometric objects [21–30]. This theory has been extended to the space of matrices with indeﬁnite scalar products, which is closely related to the phenomenon of entanglement in quantum physics [32–34], and to harmonic analysis [5–7]. Recently there appeared a new approach in this theory based on the analysis of local properties of underlying operations, corresponding metric tensors and Riemannian manifolds. It turned out that both Einstein addition and Möbius addition may be recovered from corresponding metric tensors using standard operations of differential geometry: logarithmic mapping, parallel transport, and geodesics [37]. Basic properties of Einstein and Möbius gyrogroups and gyrovector spaces were derived using this approach [37]. Einstein and Möbius additions are special cases of more general binary operations, namely, operations invariant with respect to unitary transformations. A differential geometry approach to the theory of such operations has been developed in [37,38]. The central object in this approach is a metric tensor in a special form, which was called the canonical form. This form depends on two scalar functions, m0 and m1, which determine the value of the ﬁrst fundamental form at x in the directions orthogonal and parallel to x respectively.

Symmetry 2020, 12, 1634; doi:10.3390/sym12101634 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1634 2 of 30

We adressed the following problems in this paper.

1. Is it true that for every binary operation invariant with respect to unitary transformations the metric tensor has a canonical form (8) parametrized by a pair of functions (m0, m1)? 2. What are the necessary and sufﬁcient conditions on functions m0, m1 such that their canonical metric tensor is smooth and is generated by a binary operation? 3. Find necessary and sufﬁcient conditions under which two binary operations having canonical metric tensors are isomorphic. 4. Show that every algebraic isomorphism of binary operations generates an isomorphism of corresponding gyrogroups. 5. Show that an isomorphism of binary operations is transitive. 6. Find necessary and sufficient conditions for binary operations to be isomorphic to Euclidean addition. 7. Find necessary and sufficient conditions for binary operations to be isomorphic to Einstein addition. 8. Find a binary operation in Rn isomorphic to every binary operation isomorphic to Einstein addition in the unit ball.

9. Find necessary and sufﬁcient conditions for two operations having the same function m0 to be isomorphic.

10. Describe all binary operations isomorphic to Einstein addition and having the same function m0. 11. Find necessary and sufﬁcient conditions for two operations having the same function m1 to be isomorphic.

The organization of the paper is as follows. Following the introduction, in Section2, we prove that all the metric tensors of smooth binary operations invariant with respect to unitary transformations have the canonical form (8). In Section3, we find the necessary and sufficient conditions on functions m0, m1 to generate metric tensors of binary operations. We give the definition of isomorphism of binary operations and necessary and sufficient conditions for binary operations to be isomorphic in terms of the functions m0, m1 in Section4. We also show how to construct binary operations isomorphic to a given binary operation. In Section5, we show that every algebraic isomorphism between binary operations gives rise to an isomorphism between corresponding gyrogroups. The necessary and sufficient conditions for binary operations to be isomorphic to Euclidean addition are presented in Section6. The same problems for Einstein addition are solved in Section7. In this section, a set of binary operations in Rn parametrized by positive numbers and isomorphic to Einstein addition are presented. In Section8 we describe all the binary operations isomorphic to Einstein addition and having the same function m0 in the representation of their canonical metric tensors. The necessary and sufficient conditions for binary operations, having the same function m0 in the representation of their canonical metric tensors, to be isomorphic are given in Section9. In Section 10, we prove that every two binary operations, having the same function m1 in the representation of their canonical metric tensors, are isomorphic if and only if they are related by the standard scalar multiplication in a corresponding gyrovector space.

2. Binary Operations and Metric Tensors Invariant with Respect to Unitary Transformations

n Let Bq be the open ball in the Euclidean space R with radius q ∈ (0, ∞]:

n Bq = {x ∈ R : kxk < q}. (1)

n The set B∞ is identiﬁed with R . We consider smooth binary operations f : Bq × Bq → Bq invariant with respect to unitary transformations. More precisely, we make the following assumption. Symmetry 2020, 12, 1634 3 of 30

Assumption 1. The function f is differentiable, and

f (Ua, Ub) = U f (a, b). (2) for all vectors a, b ∈ Bq and all n × n unitary matrices U.

For every vector x ∈ Bq we consider the matrix

∂ f (−x, y) g(x) = | = , (3) ∂y y x and deﬁne the Hermitian positive semi-deﬁnite matrix function

G(x) = g(x)>g(x). (4)

Lemma 1. Under Assumption1 the matrix function G satisﬁes the following condition: for any vector x ∈ Bq and a unitary n × n matrix U we have G(Ux) = UG(x)U>. (5)

Proof. We have

−1 ∂ f (−Ux, y) ∂ f (−x, U y) −1 > g(Ux) = | = = | = = g(x)U = g(x)U . (6) ∂y y Ux ∂y y Ux

Therefore G(Ux) = (g(Ux))>g(Ux) = Ug(x)>g(x)U> = UG(x)U>. (7)

We consider G as a metric tensor in Bq.

Deﬁnition 1. The metric tensor G in (4) is said to be associated with the binary operation f .

The quadratic form F(x, y) = y>G(x)y is called the ﬁrst fundamental form. For every curve x: [0, 1] → Bq the length of x in the space Bq with the metric tensor G is equal to Z 1 q l(x) = F(x(t), x˙(t))dt. 0 We are looking for a general form of symmetric matrix functions G satisfying (5).

Theorem 1. Let condition (5) be satisﬁed for all x ∈ Bq and all unitary matrices U. Then there exist functions 2 m0, m1: [0, q ) → [0, ∞) such that for all non zero x ∈ Bq we have " # xx> xx> G(x) = m (kxk2) I − + m (kxk2) . (8) 0 kxk2 1 kxk2

Proof. For an arbitrary number α ∈ (0, q) consider a vector e = α(1, 0, ... , 0)>. For an arbitrary unitary (n − 1) × (n − 1) matrix U˜ consider the unitary matrix ! 1 0 U = . 0 U˜ Symmetry 2020, 12, 1634 4 of 30

Notice that Ue = e, so that Equation (5) implies

G(e) = UG(e)U>.

We split up the matrix G(e) in a way similar to U, obtaining the block matrix ! a b> G(e) = , b C where a ∈ R, b ∈ Rn, and C is a symmetric (n − 1) × (n − 1) matrix. Then ! a b>U˜ > G(e) = . Ub˜ UC˜ U˜ >

Following (5) we have Ub˜ = b, UC˜ U˜ > = C for any unitary (n − 1) × (n − 1) matrix U˜ . This means that b = 0, and there exists a number d such that C = dI. Thus, ! a 0 G(e) = . 0 dI

For an arbitrary vector x such that kxk = α we consider a unitary matrix Ux such that Uxe = x. Then, this matrix can be split as follows:

x U = U˜ , x kxk x where U˜ x is an n × (n − 1) matrix. Since the matrix U is unitary, we have

xx> I = U U> = + U˜ U˜ >. x x kxk2 x x

Equation (5) determines the value of G(x): " # xx> xx> xx> G(x) = G(U e) = U G(e)U> = a + dU˜ U˜ > = a + d I − . x x x kxk2 x x kxk2 kxk2

If we set 2 2 m0(α ) = d, m1(α ) = a, then Equation (8) holds.

Deﬁnition 2. The representation (8) is called [38] the canonical representation of the metric tensor G. We say that the pair of functions (m0, m1) parametrizes G.

3. Binary Operations Invariant with Respect to Unitary Transformations In this section, we deﬁne binary operations with given smooth metric tensors G satisfying condition (5), and conditions on the function m1 necessary and sufﬁcient for the existence of such operations. In (8) the matrix function G(x) is a metric tensor. Therefore the functions m0, m1 should be non negative. We impose some smoothness conditions: the matrix function G is differentiable. In terms of the functions m0, m1 it means that the functions m0, m1 are differentiable, and m0(0) = m1(0). Symmetry 2020, 12, 1634 5 of 30

If we multiply G by a positive number, then all properties of the object remain the same. Therefore, we further impose the following assumption.

Assumption 2. The functions m0, m1 in (5) are differentiable, m0(s) > 0, m1(s) > 0 for all s ∈ [0, q), and m0(0) = m1(0) = 1.

A binary operation ⊕ in Bq is deﬁned as follows [37]. Consider a manifold with metric tensor G. Denote by expa(v) the exponential mapping from a point a at vector v, denote by log the logarithmic mapping: log(v) = w such that exp0(w) = v, and P0→a(v) is the parallel transport of a vector v from zero to a. Then a ⊕ b = expa(P0→a(log(b))). In [37] it is proved that if the metric tensor G is given by (3) and (4), then the operations ⊕ and f coincide, f (a, b) = a ⊕ b (9) for all a, b ∈ Bq. Further we need a procedure for calculating a ⊕ b for various functions m0 [37]. If b = 0, then a ⊕ b = a. If a = 0, the a ⊕ b = b. If a, b ∈ Bq\{0}, then we obtain a ⊕ b by the following three steps [37]:

1. Calculate a vector Z kbk q b 2 X0 = m1(s )ds. kbk 0 2. Calculate a vector " # 1 aa> 1 aa> X = I − X0 + X0. 1 p 2 2 p 2 2 m0(kak ) kak m1(kak ) kak

3. Solve the following initial value problem:

m0 (u) m0 (u) 2m0 (u) (x>x˙)2 + 0 | ( > ) + 1 − 0 | x¨ u=kxk2 2 x x˙ x˙ u=kxk2 2 (10) m0(u) m1(u) m0(u) kxk ! m (u) − m (u) − um0 (u) (x>x˙)2 + 1 0 0 | k k2 − = u=kxk2 x˙ 2 x 0, um1(u) kxk

x(0) = a, x˙(0) = X1.

The value a ⊕ b is equal to x(1), a ⊕ b = x(1).

The value of a ⊕ b is well-deﬁned if and only if the vector x(1) belongs to the ball Bq, that is, a solution of the initial value problem (10) exists on [0, 1], and kx(t)k < q for all t ∈ [0, 1].

Theorem 2. The value of a ⊕ b is well-deﬁned for all a, b ∈ Bq if and only if

Z q q 2 m1(s )ds = ∞. (11) 0

Proof. Assume the right hand side of (11) is ﬁnite. Denote it by A. Find a number r ∈ (0, q) such that

Z r q 2 A m1(s )ds > . (12) 0 2 Symmetry 2020, 12, 1634 6 of 30

Pick a vector a such that kak = r. Let us try to calculate a ⊕ a. The ﬁrst two steps of the procedure give Z kak q a 2 1 X0 = m (s )ds, X = X0. 1 1 p 2 kak 0 m1(kak ) In step 3, the values of x(0) and x˙(0) are parallel. Therefore, a solution x(·) has the form x(t) = aϕ(t), where ϕ is a scalar function. In this case

(x>x˙)2 = kak2(ϕ˙ )2, kxk2 and Equation (10) take the form

m0 ϕ¨ + 1 kak2 ϕ˙ 2 ϕ = 0, m1 ϕ(0) = 1, (13) 1 Z kak q ϕ˙ (0) = m (s2)ds. p 2 1 kak m1(kak ) 0

The differential equation in (13) is separable. Integrating of (13) yields

C 1 Z kak q ϕ˙ = 1 , C = m (s2)ds. p 2 2 1 1 m1(kak ϕ ) kak 0

Therefore Z 1 q 2 2 m1(kak s )ds = C1, 0 and Z kakϕ(1) q Z kak q 2 2 m1(s )ds = m1(s )ds. kakϕ(0) 0 Since kx(1)k = kakϕ(1) and ϕ(0) = 1, we have

Z kx(1)k q Z kak q 2 2 m1(s )ds = m1(s )ds, kak 0 and Z kx(1)k q Z kak q 2 2 m1(s )ds = 2 m1(s )ds > A. 0 0 Inequality (12) implies that a solution x(·) of the initial value problem (10) on the interval [0, 1] such that kx(t)k < q for all t ∈ [0, 1] does not exist. Hence a ⊕ a is not deﬁned. n Conversely, assume that equality (11) holds. Consider arbitrary vectors a ∈ Bq, X1 ∈ R and a solution x of the initial value problem (10). We need to show that x(t) ∈ Bq for all t ∈ [0, 1]. Since x(·) > is a geodesic in Bq with metric tensor G, the function x˙(t) G(x(t))x˙(t) is constant for t ∈ [0, 1]. > The constant is equal to X1 G(a)X1. At the same time,

dkx(t)k 2 x˙(t)>G(x(t))x˙(t) ≥ kx˙(t)k2m (kx(t)k2) ≥ m (kx(t)k2). 1 dt 1

Therefore, for all t ∈ [0, 1] we have

q Z kx(t)k q > 2 t X1 G(a)X1 ≥ | m1(s )ds|. kx(0)k

Owing to (11) we have kx(t)k < q for all t ∈ [0, 1]. Symmetry 2020, 12, 1634 7 of 30

Following this result, we further assume that condition (11) holds.

Assumption 3. The function m1 satisﬁes the condition

Z q q 2 m1(s )ds = ∞. 0

We now deﬁne a scalar multiplication t ⊗ a in Bq associated with the operation ⊕ [37] such that

(t1 ⊗ a) ⊕ (t2 ⊗ a) = (t1 + t2) ⊗ a, (t1 ⊗ (t2 ⊗ a)) = (t1t2) ⊗ a (14) for all t1, t2 ∈ R and a ∈ Bq. Let h be the function [0, q) → [0, ∞), given by

Z p q 2 h(p) = m1(s )ds. 0

Owing to Assumptions2 and3, the function h is an increasing bijection [0, q) → [0, ∞). Therefore it has an inverse, which is also an increasing bijection, and which is denoted by h−1. The multiplication n by a number is given as follows: for all t ∈ R we have t ⊗ a = 0 if a = 0, and for non zero a ∈ Bq a t ⊗ a = h−1(t h(kak)). (15) kak

It is straightforward that this operation satisﬁes conditions (14)[37]. Notice that kt ⊗ ak → q as |t| → ∞ for every a ∈ Bq\{0} owing to Assumption3. We denote by dist(a, b) the distance between points a and b in Bq with a canonical metric tensor G. That is, Z 1 q dist(a, b) = inf{ x˙(s)>G(x(s))x˙(s)ds}, 0 where the inﬁmum is taken over the set of smooth curves x such that x(0) = a and x(1) = b. The relation between the distance function and the function h is given as follows [37].

Theorem 3. For all a, b ∈ Bq dist(a, b) = h(k(−a) ⊕ bk).

In particular, for both Einstein addition and Möbius addition we have [37]

dist(a, b) = atanh(k(−a) ⊕ bk).

Deﬁnition 3. Let m0, m1: [0, 1) → (0, ∞) be functions that satisfy Assumptions2 and3, and let G be the canonical metric tensor (8) parametrized by (m0, m1). Then we say that the binary operation ⊕, deﬁned in steps 1–3, is generated by G, or generated by the pair (m0, m1).

Lemma 2. Let a ⊕ b be an operation generated by a pair of functions (m0, m1) satisfying Assumptions2 and3. Then for every unitary n × n matrix U and all a, b ∈ Bq we have

(Ua) ⊕ (Ub) = U(a ⊕ b). (16)

Proof. We apply the steps 1–3 of the procedure deﬁning a ⊕ b. Assume X0, X1, and x(·) are the vectors and the function that are calculated on the steps 1–3 for a ⊕ b. Assume X0,U, X1,U and xU(·) are the corresponding values for the sum (Ua) ⊕ (Ub). Then, clearly,

X0,U = UX0, X1,U = UX1, xU = Ux. (17) Symmetry 2020, 12, 1634 8 of 30

Therefore, (Ua) ⊕ (Ub) = xU(1) = Ux(1) = U(a ⊕ b). (18)

4. Isomorphic Operations In this section, we consider necessary and sufﬁcient conditions under which binary operations in the balls Bq and Bp with q, p ∈ (0, ∞] are isomorphic. Assume q, p ∈ (0, ∞]. Let G, G˜ be canonical metric tensors in Bq and Bp generated by the pairs of functions (m0, m1) and (m˜ 0, m˜ 1) respectively. Assume the pairs (m0, m1) and (m˜ 0, m˜ 1) satisfy Assumptions2 and3. Denote by ⊕, ⊕˜ the binary operations generated by the pairs of functions (m0, m1) and (m˜ 0, m˜ 1) respectively.

Deﬁnition 4. Operations ⊕ in Bq and ⊕˜ in Bp are isomorphic if there exists a smooth bijection ϕ: Bq → Bp such that for all a, b ∈ Bq ϕ(a ⊕ b) = ϕ(a)⊕˜ ϕ(b) (19) and ϕ(Ux) = Uϕ(x) (20) for all unitary matrices U and all vectors x ∈ Bq. The function ϕ is said to be the bijection of the isomorphism between the operations ⊕ and ⊕˜ .

Remark 1. Assume ϕ is an isomorphism of operations ⊕ and ⊕˜ . Then the following conditions on ϕ hold.

1. ϕ(−x) = −ϕ(x) for all x ∈ Bq. 2. ϕ(0) = 0. 3. The function −ϕ is also an isomorphism between the operations ⊕ and ⊕˜ .

Proof. Condition (20) with U = −I implies

ϕ(−x) = −ϕ(x). (21)

Therefore, the functions ϕ satisfying (20) are odd. Besides, ϕ(0) = Uϕ(0) for all unitary matrices U. Therefore, ϕ(0) = 0. The last property follows from the fact that (−a) ⊕ (−b) = −(a ⊕ b), for all a, b ∈ Bq, and (−a)⊕˜ (−b) = −(a⊕˜ b) for all a, b ∈ Bp.

Remark 2. The relation of isomorphisms is symmetric: if operations ⊕ and ⊕˜ are isomorphic with a bijection ϕ, then operations ⊕˜ and ⊕ are isomorphic with the bijection ϕ−1.

−1 Proof. Since ϕ is a bijection, there exists an inverse mapping ϕ : Bp → Bq. Equations (3) and (19) are equivalent to the equations:

ϕ−1(c⊕˜ d) = ϕ−1(c) ⊕ ϕ−1(d), (22) ϕ−1(Uy) = Uϕ−1(y) for all c, d, y ∈ Bp and unitary matrices U.

The following theorem asserts that the relation of isomorphism is transitive.

Theorem 4. Assume operations ⊕1 in Bq1 and ⊕2 in Bq2 are isomorphic with an isomorphism ϕ1: Bq1 → Bq2 , and operations ⊕2 and ⊕3 in Bq3 are isomorphic with an isomorphism ϕ2: Bq2 → Bq3 . Then operations ⊕1 and

⊕3 are isomorphic with an isomorphism ϕ2 ◦ ϕ1: Bq1 → Bq3 . Symmetry 2020, 12, 1634 9 of 30

Proof. For all a, b ∈ Bq3 we have

ϕ2(ϕ1(a ⊕1 b)) = ϕ2(ϕ1(a) ⊕2 ϕ1(b)) = ϕ2(ϕ1(a)) ⊕3 ϕ2(ϕ1(b)), and for all unitary matrices U and all x ∈ Bq we have

ϕ2(ϕ1(Ux)) = Uϕ2(ϕ1(x)).

Owing to property (20) the isomorphisms have the following representation.

Lemma 3. Assume ϕ is a bijection of an isomorphism between an operation in Bq generated by a pair of functions (m0, m1) satisfying Assumptions2 and3 and an operation in Bp generated by the pair of functions (m˜ 0, m˜ 1) also satisfying Assumptions2 and3. Then there exists a smooth bijection ϕ˜: [0, q) → [0, p) and a number s ∈ {−1, 1} such that for all x ∈ Bq we have x ϕ(x) = sϕ˜(kxk) . (23) kxk

Proof. Let α ∈ (0, q) and a vector e be the ﬁrst unit vector: e = (1, ... , 0)>. Consider an arbitrary unitary (n − 1) × (n − 1) matrix U1 and an n × n unitary matrix U = blockdiag{1, U1}. Then Ue = e. Apply condition (20): ϕ(αe) = ϕ(Uαe) = Uϕ(αe). (24)

If ϕ(αe) is not parallel to e, then there exists a matrix U1 such that Equation (24) is wrong. Therefore the vectors αe and ϕ(αe) are parallel: there exists a number λ = λ(α) ∈ (0, p) such that ϕ(αe) = sλ(α)e, where s = sign[e> ϕ(αe)]. From (19) it follows that ϕ(0) = 0. Therefore ϕ(αe) 6= 0. Due to continuity of the bijection ϕ the number s is the same for all α. Set

ϕ˜(α) = λ(α).

Then αe ϕ(αe) = sϕ˜(kαek) . kαek

For an arbitrary vector x ∈ Bq we ﬁnd a unitary matrix U such that

x = Ukxke.

Then kxke x ϕ(x) = ϕ(Ukxke) = Uϕ(kxke) = Usϕ˜(kxk) = sϕ˜(kxk) . kxk kxk The function ϕ˜ is smooth since the function ϕ is smooth.

Remark 3. Assume operations ⊕ and ⊕˜ are isomorphic. Then there exists an isomorphism ϕ between these operations that has the representation (23) with s = 1.

Proof. Assume ϕ is an isomorphism, which satisﬁes condition (23) with s = −1. Owing to statement 2 of Remark1 the function −ϕ is also an isomorphism that satisﬁes condition (23) with s = 1.

Further, without loss of generality, we assume that in the representation (23) of isomorphisms ϕ we have s = 1. The following deﬁnition is useful. Symmetry 2020, 12, 1634 10 of 30

Deﬁnition 5. We say that a function ϕ: Bq → Bp is determined by an increasing bijection ϕ˜: [0, q) → [0, p) if for all x ∈ Bq x ϕ(x) = ϕ˜(kxk) . (25) kxk

Assume ϕ is a bijection of an isomorphism between ⊕ and ⊕˜ . How can the functions m˜ 0, m˜ 1 be found in terms of ϕ, m0 and m1?

Theorem 5. Assume binary operations ⊕ in Bp and ⊕˜ in Bq with p, q ∈ (0, ∞] are generated by the pairs of functions (m0, m1) and (m˜ 0, m˜ 1) respectively. Then the operations ⊕˜ , ⊕ are isomorphic if and only if there exists an increasing smooth bijection ϕ˜: [0, q) → [0, p) such that ϕ˜0(0) > 0, and for all r ∈ (0, q)

ϕ˜(r) 2 m˜ (r2) = m (ϕ˜(r)2) , (26) 0 0 ϕ˜0(0)r

ϕ˜0(r) 2 m˜ (r2) = m (ϕ˜(r)2) . (27) 1 1 ϕ˜0(0)

If conditions (26) and (27) are satisﬁed, then the function ϕ: Bq → Bp determined by ϕ˜ is an isomorphism between ⊕ and ⊕˜ .

Proof. First we prove the necessity. Assume the operations ⊕˜ and ⊕ are isomorphic, and ϕ is the corresponding isomorphism satisfying condition (25). For every non zero vector x ∈ Bq consider an n × n matrix g(x) such that

(−x) ⊕ (x + ∆x) = g(x)∆x + o(k∆xk).

Then G(x) = g(x)>g(x) is the metric tensor generated by the pair (m0, m1). A similar relation concerns the operation ⊕˜ : there exists an n × n matrix g˜(x) such that

(−x)⊕˜ (x + ∆x) = g˜(x)∆x + o(k∆xk), and G˜(x) = g˜(x)>g˜(x) is the metric tensor generated by the pair (m˜ 0, m˜ 1). Then

ϕ[(−x)⊕˜ (x + ∆x)] = ϕ(g˜(x)∆x + o(k∆xk)) g˜(x)∆x (28) = ϕ˜(kg˜(x)∆xk) + o(k∆xk) = ϕ˜0(0)g˜(x)∆x + o(k∆xk), kg˜(x)∆xk and

ϕ(−x) ⊕ ϕ(x + ∆x) = (−ϕ(x)) ⊕ (ϕ(x) + ϕ0(x)∆x) + o(k∆xk) (29) = g(ϕ(x))ϕ0(x)∆x + o(k∆xk).

By means of (19), (28) and (29) we have

ϕ˜0(0)g˜(x) = g(ϕ(x))ϕ0(x), Symmetry 2020, 12, 1634 11 of 30 and therefore ϕ˜0(0)2G˜(x) = ϕ0(x)G(ϕ(x))ϕ0(x)>. (30)

Since G(ϕ(x)) > 0, and ϕ0(x) 6≡ 0, we get

ϕ˜0(0) 6= 0.

According to (8) and (25) " # x 0 xx> ϕ˜(kxk) xx> ϕ0(x) = ϕ˜(kxk) = ϕ˜0(kxk) + I − , kxk kxk2 kxk kxk2

" # xx> xx> G(ϕ(x)) = m (ϕ˜(kxk)2) I − + m (ϕ˜(kxk)2) . 0 kxk2 1 kxk2

Hence, " # ϕ˜(kxk)2 xx> xx> ϕ0(x)G(ϕ(x))ϕ0(x)> = m (ϕ˜(kxk)2) I − + m (ϕ˜(kxk)2)ϕ˜0(kxk)2 . (31) 0 kxk2 kxk2 1 kxk2

Since G˜(x) is the metric tensor generated by the pair of functions (m˜ 0, m˜ 1), we have

xx> xx> G˜(x) = m˜ (kxk2) I − + m˜ (kxk2) . (32) 0 kxk2 1 kxk2

Equations (30) and (31) imply " # m (ϕ˜(kxk)2) ϕ˜(kxk)2 xx> m (ϕ˜(kxk)2) xx> G˜(x) = 0 I − + 1 ϕ˜0(kxk)2 (33) ϕ˜0(0)2 kxk2 kxk2 ϕ˜0(0)2 kxk2

Comparing Equations (32) and (33), we get

ϕ˜(kxk) 2 m˜ (kxk2) = m (ϕ˜(kxk)2) , 0 0 ϕ˜0(0)kxk

ϕ˜0(kxk) 2 m˜ (kxk2) = m (ϕ˜(kxk)2) . 1 1 ϕ˜0(0) The necessity is thus proved. We now prove the sufﬁciency. Assume ϕ˜: [0, q) → [0, p) is a smooth bijection, ϕ˜0(0) > 0, and conditions (26) and (27) hold. Deﬁne a function ϕ by Equation (25). Then ϕ is a smooth bijection Bq → Bp. Denote by ⊕ˆ the binary operation in Bp given by

a⊕ˆ b = ϕ−1(ϕ(a) ⊕ ϕ(b)). (34) for all a, b ∈ Bq. The theorem will be proved if we show that ⊕ˆ = ⊕˜ . To this end it is sufﬁcient from [37] to prove that the metric tensors associated with these two operations are equal. We have

ϕ(−x) ⊕ ϕ(x + ∆x) ϕ˜(kx + ∆xk) ϕ˜(kxk) ϕ˜(kx + ∆xk) = − x + ∆x kx + ∆xk kxk kx + ∆xk (35) ! ϕ˜(kxk) xx> xx> = I − + ϕ˜0(kxk) ∆x + o(kxk). kxk kxk2 kxk2 Symmetry 2020, 12, 1634 12 of 30

Therefore (−x)⊕ˆ (x + ∆x) = = ϕ−1 g(ϕ(x)) (ϕ(x + ∆x) − ϕ(x)) + o(k∆xk)

! 1 ϕ˜(kxk) xx> xx> = g(ϕ(x)) I − + ϕ˜0(kxk) ∆x + o(k∆xk). ϕ˜0(0) kxk kxk2 kxk2

Taking into account that the metric tensor G associated with the operation ⊕ is equal to

xx> xx> G(x) = g(x)>g(x) = m (kxk2) I − + m (kxk2) , 0 kxk2 1 kxk2 we can calculate the metric tensor Gˆ associated with the operation ϕˆ:

Gˆ(x) = h(x)>G(ϕ(x))h(x), where ! ϕ˜(kxk) xx> ϕ˜0(kxk) xx> h(x) = I − + . ϕ˜0(0)kxk kxk2 ϕ˜0(0) kxk2

It is straightforward to check that the metric tensor Gˆ has the form (8) parametrized by the functions ϕ˜(r) 2 mˆ (r2) = m (ϕ˜(r)2) , 0 0 ϕ˜0(0)r

ϕ˜0(r) 2 mˆ (r2) = m (ϕ˜(r)2) . 1 1 ϕ˜0(0)

These functions coincide with the functions m˜ 0 and m˜ 1 respectively. Hence, the metric tensors associated with the operations ⊕˜ and ⊕ˆ coincide. According to the uniqueness of binary operations associated with the same metric tensor [37], we have ⊕ˆ = ⊕˜ . Following deﬁnition (34) of the operation ⊕ˆ , this operation is isomorphic to the operation ⊕ with the isomorphism ϕ.

Assume ϕ˜t is a family of smooth increasing bijections [0, q) → [0, p) with t ∈ (0, ∞) and such that 0 1 ϕ˜ (0) = t. Assume there exists a limit in C [0, q): limt→0 ϕ˜t, which we denote by ϕ˜0(s). Assume the function ϕ˜0 is a smooth increasing bijection [0, q) → [0, p). Assume an operation ⊕ is generated by a pair (m0, m1). For all t ≥ 0 consider the functions x ϕ (x) = ϕ˜ (kxk) t t kxk and operations ⊕t deﬁned as follows: for all a, b ∈ Bq

−1 a ⊕t b = ϕt (ϕt(a) ⊕ ϕt(b)).

Then the operations ⊕t are isomorphic to the operation ⊕ for all t > 0. Further, we show that the limiting operation ⊕0 may not be isomorphic to the operation ⊕.

5. Gyrogroups In this section we consider a simple way to get gyrogroups via bijections. We use the following deﬁnitions [34]. Symmetry 2020, 12, 1634 13 of 30

Deﬁnition 6. A set S with a binary operation ⊕ is called a groupoid. A groupoid (S, ⊕) is called a gyrogroup if its binary operation satisﬁes the following axioms.

1. There is at least one element, 0, in S such that

0 ⊕ a = a (36)

for all a ∈ S. 2. There is an element 0 ∈ S satisfying (36) such that for every a ∈ S there is an element −a ∈ S (called a left inverse of a) such that (−a) ⊕ a = 0. (37)

3. For every a, b ∈ S there is an automorphism gyr[a, b]: S → S of the groupoid S such that for every c ∈ S we have a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ (gyr[a, b]c). (38)

4. The operator gyr: S × S → Aut(S, ⊕) possesses the following property:

gyr[a ⊕ b, b] = gyr[a, b] (39)

for all a, b ∈ S.

Notice [30] that the left identity 0 is also a right identity, and the left inverse of a is also a right inverse of a: a ⊕ 0 = a, a ⊕ (−a) = 0 for all a ∈ S.

Theorem 6. Assume (S1, ⊕1) is a gyrogroup, S2 is a set of elements, and ϕ: S2 → S1 is a bijection. Then for the binary operation given by −1 a ⊕2 b = ϕ (ϕ(a) ⊕1 ϕ(b)) (40) the groupoid (S2, ⊕2) is a gyrogroup.

Proof. We have to show that axioms 1–4 in Deﬁnition6 are satisﬁed.

1. Pick an element 0 ∈ S1 satisfying condition 2. For every a ∈ S2 we have

−1 −1 a ⊕2 ϕ (0) = ϕ (ϕ(a) ⊕1 0) = a. (41)

−1 Hence, ϕ (0) is an element satisfying axiom 1 for the groupoid (S2, ⊕2). 2. For every a ∈ S2 we have

−1 −1 −1 a ⊕2 ϕ (−ϕ(a)) = ϕ (ϕ(a) ⊕1 (−ϕ(a))) = ϕ (0). (42)

Therefore, axiom 2 is satisﬁed for the groupoid (S2, ⊕2). 3. For every x, y ∈ S1 let gyr1[x, y] be the automorphism of S1 from the property 3. For every a, b, c ∈ S2 set −1 gyr2[a, b]c = ϕ (gyr1[ϕ(a), ϕ(b)]ϕ(c)). (43) Symmetry 2020, 12, 1634 14 of 30

Then

−1 gyr2[a, b](c1 ⊕2 c2) =ϕ {gyr1[ϕ(a), ϕ(b)]ϕ(c1 ⊕2 c2)} −1 =ϕ {[gyr1[ϕ(a), ϕ(b)]ϕ(c1) ⊕1 ϕ(c2)} −1 =ϕ {(gyr1[ϕ(a), ϕ(b)]ϕ(c1)) ⊕1 (gyr1[ϕ(a), ϕ(b)]ϕ(c2))} −1 −1 −1 =ϕ {ϕ(ϕ (gyr1[ϕ(a), ϕ(b)]ϕ(c1))) ⊕1 ϕ(ϕ (gyr1[ϕ(a), ϕ(b)]ϕ(c2)))} −1 =ϕ {ϕ(gyr2[a, b]c1) ⊕1 ϕ(gyr2[a, b]c2)}

=(gyr2[a, b]c1) ⊕2 (gyr2[a, b]c2). (44)

Therefore, gyr2[a, b] is an automorphism of S2. Besides,

−1 a ⊕2 (b ⊕2 c) =ϕ {ϕ(a) ⊕1 [ϕ(b) ⊕1 ϕ(c)]} −1 =ϕ {(ϕ(a) ⊕1 ϕ(b)) ⊕1 (gyr1[ϕ(a), ϕ(b)]ϕ(c))} −1 −1 (45) =ϕ (ϕ(a) ⊕1 ϕ(b)) ⊕2 ϕ (gyr1[ϕ(a), ϕ(b)]ϕ(c))

=(a ⊕2 b) ⊕2 (gyr2[a, b]c).

Therefore, axiom 3 is satisﬁed for the groupoid (S2, ⊕2). 4. For every a, b, c ∈ S2 we have

−1 gyr2[a ⊕2 b, b]c =ϕ {gyr1[ϕ(a ⊕2 b), ϕ(b)]ϕ(c)} −1 =ϕ {gyr1[ϕ(a) ⊕1 ϕ(b), ϕ(b)]ϕ(c)} −1 (46) =ϕ {gyr1[ϕ(a), ϕ(b)]ϕ(c)}

=gyr2[a, b]c.

Therefore, axiom 4 is satisﬁed for the groupoid (S2, ⊕2).

Deﬁnition 7. A gyroproup (S, ⊕) is said to be gyrocommutative if for all a, b ∈ S

a ⊕ b = gyr[a, b](b ⊕ a). (47)

Theorem 7. Assume (S1, ⊕1) is a gyrocommutative gyrogroup, S2 is a set of elements, and ϕ: S2 → S1 is a bijection. Then for the binary operation ⊕2, given by

−1 a ⊕2 b = ϕ (ϕ(a) ⊕1 ϕ(b)), (48) the gyroproup (S2, ⊕2) is gyrocommutative.

Proof. For all a, b ∈ S2 we have

−1 a ⊕2 b =ϕ (ϕ(a) ⊕1 ϕ(b)) −1 =ϕ {gyr1[ϕ(a), ϕ(b)](ϕ(b) ⊕1 ϕ(a))} −1 (49) =ϕ {gyr1[ϕ(a), ϕ(b)]ϕ(b ⊕2 a)}

=gyr2[a, b](b ⊕2 a).

Hence, axiom (47) is satisﬁed. Symmetry 2020, 12, 1634 15 of 30

We can apply these results to balls Bq with q ∈ (0, ∞]. Notice that for Euclidean addition (and hence for all additions isomorphic to Euclidean addition) the operation gyr[a, b] is equal to identity. For Einstein addition (and hence for all additions isomorphic to Einstein addition) the operation gyr[a, b] is a multiplication by a unitary matrix. Here we encounter the following open problem. Does there exist a binary operation generated by a pair of functions (m0, m1) satisfying Assumptions2 and3, which is not isomorphic to Einstein addition and not isomorphic to Euclidean addition?

6. Operations Isomorphic to Euclidean Addition

n The metric tensor of Euclidean addition in R = B∞ is equal to the identity. This means that m0 ≡ 1, m1 ≡ 1 in the canonical representation (8) of the Euclidean metric tensor. The following theorem gives necessary and sufﬁcient conditions for a binary operation generated by a pair of functions (m0, m1) to be isomorphic to Euclidean addition.

Theorem 8. Let G be a canonical metric tensor in Bq parametrized by a pair of functions (m0, m1) satisfying Assumptions2 and3. Let ⊕ be a binary operation generated by G. Then the operation ⊕ is isomorphic to Euclidean addition if and only if for all u ∈ [0, q2) we have

0 2 [(um0(u)) ] m1(u) = . (50) m0(u)

If an operation ⊕ is isomorphic to Euclidean addition, then there exists a positive number t such that the n bijection ϕ: Bq → R of the isomorphism between ⊕ and Euclidean addition is given by q 2 ϕ(x) = m0(kxk )tx. (51)

Proof. According to Theorem5 the operation ⊕ is isomorphic to Euclidean addition if and only if there exists a differentiable bijection ϕ˜: [0, q) → [0, ∞) such that ϕ˜0(0) > 0, and for all r ∈ [0, q) we have

ϕ˜(r) 2 m (r2) = (52) 0 ϕ˜0(0)r and ϕ˜0(r) 2 m (r2) = . (53) 1 ϕ˜0(0) Assume such a function ϕ˜ exists, and let t = ϕ˜0(0). Solving Equation (52) for ϕ˜ yields q 2 ϕ˜(r) = tr m0(r ).

Therefore p d( r2m (r2)) d(r2m (r2)) 1 ϕ˜0(r) = t 0 = t 0 . 2 p 2 dr d(r ) m0(r ) Denote u = r2. Then (53) implies

2 1 d(um0(u)) m1(u) = p , m0(u) du which coincides with (50). Symmetry 2020, 12, 1634 16 of 30

Now assume that (50) holds. Then for all r ∈ [0, q) we have p q 1 d(r2m (r2)) d(r m (r2)) m (r2) = 0 = 0 . 1 p 2 2 m0(r ) d(r ) dr

For an arbitrary positive number t deﬁne q 2 ϕ˜(r) = tr m0(r ).

0 Since m0(0) = 1, we have t = ϕ˜ (0). Therefore

ϕ˜(r) 2 m (r2) = , 0 ϕ˜0(0)

ϕ˜0(r) 2 m (r2) = , 1 ϕ˜0(0) and Equations (52) and (53) are satisﬁed. n According to Theorem5 the isomorphism ϕ: Bq → R is given by

x q ϕ(x) = ϕ˜(kxk) = m (kxk2)tx. kxk 0

Example 1. Let ⊕ be a binary operation with a canonical metric tensor G parametrized by a pair of functions n (m0, m1) satisfying Assumptions2 and3. Let ⊕ be isomorphic to Euclidean addition in R . If

1 m (u) = , (54) 0 1 − u then, according to Theorem8, we have 1 m (u) = . (55) 1 (1 − u)3 If 1 m (u) = , (56) 0 (1 − u)2 then, according to Theorem8, we have (1 + u)2 m (u) = . (57) 1 (1 − u)4

The pairs of functions (m0, m1) in (54)–(57) appear [38] when we study the sets of cogyrolines for Einstein addition and Möbius addition.

7. Operations Isomorphic to Einstein Addition

The metric tensor of Einstein addition ⊕E in B1 is canonical, having the form (8) with parameters

1 1 m (u) = , m (u) = . 0,E 1 − u 1,E (1 − u)2

Assume q ∈ (0, ∞] and a pair of functions (m0, m1) satisﬁes Assumptions2 and3. Consider a canonical metric tensor G parametrized by this pair. Let ⊕ be the binary operation generated by (m0, m1). Then we raise the following question. Symmetry 2020, 12, 1634 17 of 30

What are the conditions on a pair of functions (m0, m1) necessary and sufﬁcient for the operation ⊕ to be isomorphic to Einstein addition ⊕E?

Theorem 9. An operation ⊕ generated by a pair of functions (m0, m1) satisfying Assumptions2 and3 is isomorphic to Einstein addition if and only if limu→q2 um0(u) = ∞ and there exists a positive number C such that 0 2 [(um0(u)) ] m1(u) = (58) m0(u)(1 + Cum0(u)) for all u ∈ [0, q2). If Equation (58) holds, then the corresponding isomorphism between ⊕E and ⊕ is given by a function ϕ having representation (25) with p Cr2m (r2) ϕ˜(r) = 0 . (59) p 2 2 1 + Cr m0(r )

Proof. First we prove the necessity. Assume the operation ⊕ is isomorphic to Einstein addition. By Theorem5 with m0 = m0,E, m1 = m1,E, there exists a differentiable bijection ϕ˜: [0, q) → [0, 1) such that ϕ˜0(0) > 0 and 1 ϕ˜(r) 2 m (r2) = , (60) 0 1 − ϕ˜(r)2 rϕ˜0(0)

1 ϕ˜0(r) 2 m (r2) = (61) 1 (1 − ϕ˜(r)2)2 ϕ˜0(0) for all r ∈ [0, q). Since ϕ˜ is a bijection [0, q) → [0, 1), we have limr→q ϕ˜(r) = 1, and therefore √ 1 ϕ˜( u) 2 lim um0(u) = limu→q2 √ 0 = ∞. u→q2 1 − ϕ˜( u)2 ϕ˜ (0)

Let t = ϕ˜0(0) and u = r2. Then √ √ √ ϕ˜( u)ϕ˜0( u) ϕ˜( u)2 m0 (u) = √ − √ , 0 (1 − ϕ˜( u)2)2t2u3/2 (1 − ϕ˜( u)2)t2u2

√ 0 √ 0 ϕ˜( u)ϕ˜ ( u) m0(u) + um (u) = √ , 0 (1 − ϕ˜( u)2)2t2u1/2

2 1 1 + t um0(u) = √ , 1 − ϕ˜( u)2 and √ [(um (u))0]2 ϕ˜0( u)2 0 = √ = m (u) 2 2 2 2 1 (62) m0(u)(1 + t um0(u)) (1 − ϕ˜( u) ) t with C = t2. Equation (62) coincides with Equation (58). The necessity is proved. Now we prove the sufficiency. According to Theorem5, it suffices√ to show the existence of a bijection ϕ˜: [0, q) → [0, 1) such that Equations (60) and (61) hold. Set t = C. Define p tr m (r2) ϕ˜(r) = 0 . (63) p 2 2 2 1 + t r m0(r ) Symmetry 2020, 12, 1634 18 of 30

2 0 Owing to Assumption2 we have m0(0) = 1 and m1(u) > 0 for all u ∈ [0, q ). Then ϕ˜ (0) = t > 0 and (58) implies that the function um0(u) is strictly increasing. Since ϕ˜(0) = 0 and limu→q2 um0(u) = ∞, the function ϕ˜ is an increasing bijection [0, q) → [0, 1). Now we check Equation (60):

1 ϕ˜(r) 2 = m (r2), 1 − ϕ˜(r)2 rϕ˜0(0) 0 and Equation (61):

2 p 2 1 ϕ˜0(r) 1 d(r m (r2)) = 0 2 2 0 2 2 2 (1 − ϕ˜(r) ) ϕ˜ (0) 1 + t r m0(r ) dr

1 1 d(um (u)) 2 = 0 | = ( 2) 2 2 2 2 u=r2 m1 r . 1 + t r m0(r ) m0(r ) du The sufﬁciency is proved. The isomorphism is given by (63).

Denote m2(u) = um0(u).

Lemma 4. Identity (58) holds if and only if for all r ∈ (0, q) we have s Z r q 1 Cm (r2) ( 2) = √ 2 m1 s ds atanh 2 . (64) 0 C 1 + Cm2(r )

Proof. We use the deﬁnition of the function m2 to write Equation (58) in the following equivalent form:

q s m0 (s2) m (s2) = 2 (65) 1 p 2 2 m2(s )(1 + Cm2(s )) for all s ∈ (0, q). Equation (64) follows from integration of Equation (65) over [0, r] using the following two formulas, d m (s2) = 2sm0 (s2), ds 2 2 and s Z y dx 1 Cy p = √ . 0 x(1 + Cx) C 1 + Cy

We recall that the function Z r q 2 h(r) = m1(s )ds 0 is used in the deﬁnition of the operation ⊗ of multiplication by numbers [37]. In particular, for all numbers t and vectors x ∈ Bq we have x t ⊗ x = h−1(t h(kxk)) . kxk

Lemma 5. Condition (64) is equivalent to the condition √ 2 2 tanh ( Ch(r)) m2(r ) = √ (66) C(1 + tanh2( Ch(r))) for all r ∈ [0, q). Symmetry 2020, 12, 1634 19 of 30

2 Proof. Solving (64) for m2(r ) yields (66).

Corollary 1. Let ⊕ be an operation generated by a pair of functions (m0, m1) satisfying Assumptions2 and3. Then the operation ⊕ is isomorphic to Einstein addition ⊕E if and only if limu→q2 m0(u) = ∞ and there exists a positive number C such that condition (65) holds or condition (66) holds.

Proof. The statement follows from Lemmas4 and5, and Theorem9.

Example 2. Let 1 m (u) = , C = 1. (67) 1 (1 − u)2 Then Z r q 2 m1(s )ds = atanh(r), 0 and Equation (64) implies m (r2) 2 = 2 r 2 , 1 + m2(r ) which leads to r2 1 m (r2) = , m (u) = . (68) 2 1 − r2 0 1 − u

The functions m0 and m1 in (67) and (68) generate Einstein addition.

Example 3. Let 1 m (u) = , C = 4. (69) 1 (1 − u)2 Then Z r q 2 m1(s )ds = atanh(r), 0 and Equation (66) implies

2 2 2 tanh (2 atanh(r)) r 1 m2(r ) = = , m0(u) = . (70) 4(1 − tanh2(2 atanh(r))) (1 − r2)2 (1 − u)2

The functions m0 and m1 in (69) and (70) generate Möbius addition.

Theorem 10. Let ⊕ be a binary operation in Bq generated by a pair of functions (m0, m1) satisfying Assumptions2 and3. Then the operation ⊕ is isomorphic to Einstein addition ⊕E in B1 if and only if the operation ⊕ is isomorphic to an operation ⊕˜ in Rn generated by the functions

1 m˜ (u) = 1, m˜ (u) = . 0 1 1 + u

The operation ⊕E is isomorphic to the operation ⊕˜ with an isomorphism ϕ: Bq → B1 determined by the scalar function r ϕ˜(r) = √ . 1 + r2

Proof. First, we show that the operation ⊕˜ is isomorphic to Einstein addition ⊕E in B1, which is generated by the functions

1 1 m (u) = , m (u) = . 0,E 1 − u 1,E (1 − u)2 Symmetry 2020, 12, 1634 20 of 30

We deﬁne a bijection ϕ˜: [0, ∞) → [0, 1) as follows. Let r ϕ˜(r) = √ . 1 + r2

Then ϕ˜(r) 2 m (ϕ˜(r)2) = 1 = m˜ (r2), 0,E ϕ˜0(0)r 0

ϕ˜0(r) 2 1 m (ϕ˜(r)2) = = m˜ (r2). 1,E ϕ˜0(0) 1 + r2 1

According to Theorem5 the operations ⊕˜ and ⊕E are isomorphic. Following Theorem4 the operations ⊕ and ⊕E are isomorphic if and only if the operations ⊕ and ⊕˜ are isomorphic.

Notice that the operation ⊕˜ is isomorphic to Einstein addition with an isomorphism ϕ determined by the function r ϕ˜−1(r) = √ . 1 − r2

Theorem 11. The operation ⊕˜ in Rn generated by the functions

1 m˜ (u) = 1, m˜ (u) = 0 1 1 + u

n is isomorphic to the operations ⊕˜ t in R generated by the functions

1 m˜ (u) = 1, m˜ (u) = . 0,t 1,t 1 + t2u for all t > 0. The corresponding isomorphism ϕ: Rn → Rn is determined by the function

ϕ˜(r) = tr.

Proof. We have ϕ˜(r) 2 m˜ (ϕ˜(r)2 = 1 = m˜ (r2), 0 ϕ˜0(0)r 0,t

ϕ˜0(r) 2 1 m˜ (ϕ˜(r)2) = = m˜ (r2). 1 ϕ˜0(0) 1 + t2r2 1,t The statement of the theorem follows from Theorem5.

Theorem 12. The operation ⊕˜ in Rn generated by the functions

1 m˜ (u) = 1, m˜ (u) = 0 1 1 + t2u

n is isomorphic to Einstein addition ⊕E with an isomorphism ϕ: R → B1 determined by the scalar function tr ϕ˜t(r) = √ . t2r2 + 1

Proof. The proof follows from Theorems4, 10 and 11.

n For every positive number t let the function ϕt: R → B1 be given by tw ϕt(w) = p . 1 + t2kwk2 Symmetry 2020, 12, 1634 21 of 30

Then for all a, b ∈ Rn −1 a ⊕t b = ϕt (ϕt(a) ⊕E ϕt(b)). Direct calculations show that " # q 2 > 2 2 t a b a ⊕t b = 1 + t kbk + p a + b. (71) 1 + 1 + t2kak2

The operations ⊕t with t > 0 are isomorphic to each other and to Einstein addition. Thus, any study of hyperbolic geometry in B1 with Einstein addition is equivalent to a corresponding study of hyperbolic geometry in Rn with the binary operation

q > 2 a b a ⊕1 b = 1 + kbk + p a + b. 1 + 1 + kak2

The limit t → 0 in (71) yields Euclidean addition, which is not isomorphic to Einstein addition (since, for instance, the fact that Einstein addition is not commutative). The operation of scalar multiplication is deﬁned according to general rules [37]. Since

Z p q Z p 2 ds 1 ht(p) = m1,t(s )ds = √ = asinh(tp), 0 0 1 + t2s2 t we get a a 1 r ⊗ a = h−1(r h (kak)) = sinh(r asinh(tkak). t kak t t kak t For a, b ∈ Rn, a not parallel to b, the co-gyroline

Q(a, b) = {(sa) ⊕t b : s ∈ R} is a Euclidean line in Rn:

> > Q(a, b) = {x ∈ P(a, b) : (a⊥) x = (a⊥) b} where a⊥ is a vector orthogonal to a, and P(a, b) is a two dimensional plane containing both a and b. For a, b ∈ Rn, b 6= 0, the gyroline

S(a, b) = {a ⊕t (sb) : s ∈ R}

> is a hyperbola in Rn that lies on a plane containing a and b. In particular, if a b = 0, a 6= 0, then q 2 2 2 a ⊕t (sb) = 1 + s t kbk a + sb.

The ratio of the coefﬁcients of a and b tends to ±tkbk as s → ±∞: p 1 + s2t2kbk2 lim = ±tkbk. s→±∞ s

These hyperbolas tend to lines as t → 0. Symmetry 2020, 12, 1634 22 of 30

n Theorem 13. Let ⊕ be a binary operation in R generated by a pair of functions (m0, m1) satisfying Assumptions2 and3 and such that m1(u) = 1 for all u. Then the operation ⊕ is isomorphic to Einstein addition if and only if limu→∞um0(u) = ∞ and there exists a positive number t such that

2 2 tanh (tr) 1 m0(r ) = (72) 1 − tanh2(tr) t2r2 for all r ∈ (0, ∞).

Proof. Following Theorem5, it is sufﬁcient to show that there exists a differentiable increasing bijection ϕ˜: (0, 1) → (0, ∞) such that for all p ∈ (0, 1)

1 ϕ˜(p) 2 = m (ϕ˜(p)2) , (73) 1 − p2 0 ϕ˜0(0)p

1 ϕ˜0(p) 2 = (74) (1 − p2)2 ϕ˜0(0) 0 −1 if and only if the function m0 has the form (72). Let t = (ϕ˜ (0)) . Then Equation (74) can be integrated,

ϕ˜(p) = t−1 atanh(p). (75)

If r = ϕ˜(p), then p = tanh(rt). (76)

Substituting the value of p from (76) into (73) and solving it for m0, we get (72).

n Remark 4. Operations in R generated by the functions m1 = 1 and m0 satisfying condition (72) are isomorphic for different positive numbers t to the operation with t = 1. The isomorphism is a simple stretching: x → t−1x.

8. Operations Isomorphic to Einstein Addition and Having the Same Function m0 −1 Set m0(u) = (1 − u) . Consider functions m1 satisfying Equation (58):

[(um (u))0]2 ( ) = 0 ∀ ∈ [ ) m1 u 2 , u 0, 1 , m0(u)[1 + µ u m0(u)] with positive numbers µ. According to (59) we set rµ ϕ˜µ(r) = p . 1 − r2 + µ2r2

Then  2 ϕ˜ (r) 1 µ = = ( 2) ∀ ∈ [ )  q  2 m0 r r 0, 1 , (77) 0 2 1 − r ϕ˜µ(0) r 1 − ϕ˜µ(r) and " #2 ϕ˜0 (r) µ = ( 2) ∀ ∈ [ ) 0 2 m1 r r 0, 1 . (78) ϕ˜µ(0)(1 − ϕ˜µ(r) ) Symmetry 2020, 12, 1634 23 of 30

Let ⊕µ be the binary operation generated by the canonical metric tensor G parametrized by the functions m0, m1 given in (77) and (78). Following Theorem9, the operations ⊕µ are isomorphic to Einstein addition for all µ > 0. Moreover, for the function x ϕ (x) = ϕ˜ (kxk) (79) µ µ kxk we have −1 a ⊕µ b = ϕµ (ϕµ(a) ⊕E ϕµ(b)), where ⊕E is Einstein addition. The goal of this section is to ﬁnd a closed form for the operation ⊕µ. From (79) we get µx ϕµ(x) = p ∀x ∈ B1, 1 + (µ2 − 1)kxk2

−1 x ϕµ (x) = p ∀x ∈ B1. µ2 + kxk2(1 − µ2)

For every vector x ∈ B1 let 1 λµ,x = p . 1 + (µ2 − 1)kxk2 Then ϕµ(x) = µλµ,xx. Taking into account that

2 2 (1 − kϕµ(a)k )(1 − kϕµ(b)k ) k (a) ⊕ (b)k2 = − ϕµ E ϕµ 1 > 2 , (1 + ϕµ(a) ϕµ(b)) and µ2kxk2 1 − kϕ(x)k2 = 1 − = (1 − kxk2)λ2 , 1 + (µ2 − 1)kxk2 µ,x we get ϕµ(a) ⊕E ϕµ(b) a ⊕µ b = q (80) 2 2 2 µ + (1 − µ )kϕµ(a) ⊕E ϕµ(b)k

ϕ (a) ⊕ ϕ (b) = µ E µ r 2 2 2 2 (1−kϕµ(a)k )(1−kϕµ(b)k ) µ + (1 − µ )[1 − > 2 ] (1+ϕµ(a) ϕµ(b))

> (1 + ϕµ(a) ϕµ(b))(ϕµ(a) ⊕E ϕµ(b)) = q > 2 2 2 2 (1 + ϕµ(a) ϕµ(b)) − (1 − µ )(1 − kϕµ(a)k )(1 − kϕµ(b)k )

> > ( + a b ) + p − k k2( − a b ) λµ,a λµ,b kak2 a λµ,aλµ,b 1 a b kak2 a = q . k + k2 + 2 2 [ 2( > )2 − 2k k2k k2 + ( − k k2)( − k k2)] λµ,aa λµ,bb λµ,aλµ,b µ a b µ a b 1 a 1 b

For the case µ = 1 we have λ1,x = 1 and ⊕1 is Einstein addition:

> > ( + a b ) + p − k k2( − a b ) 1 kak2 a 1 a b kak2 a a ⊕E b = a ⊕ b = . 1 1 + a>b

According to Theorem9, the operations ⊕µ are isomorphic to Einstein addition ⊕E for all µ > 0. Symmetry 2020, 12, 1634 24 of 30

In the limit of (80) as µ → 0, we have the operation ⊕coE:

aγa + bγb a ⊕coE b = , p 2 1 + kaγa + bγbk

2 −1/2 where γx = (1 − kxk ) for all x ∈ B1. The operation ⊕coE is isomorphic to Euclidean addition n with an isomorphism ϕ: B1 → R , given by

ϕ(x) = xγx.

9. Isomorphic Operations with the Same Function m0 Let q ∈ (0, ∞]. As in Section8, we consider

(i) a pair of functions (m0, m1) satisfying Assumptions2 and3, a canonical metric tensor G and a binary operation ⊕ in Bq generated by (m0, m1), and (ii) a pair of functions (m˜ 0, m˜ 1) satisfying Assumptions2 and3, a canonical metric tensor G˜ and a binary operation ⊕˜ in Bq generated by (m˜ 0, m˜ 1).

Our goal is to answer the following question. How to characterize metric tensors G, G˜ if the operations ⊕, ⊕˜ are isomorphic, and m0 = m˜ 0? Further in this section we assume m0 = m˜ 0. According to Theorem5 the operations ⊕ and ⊕˜ are isomorphic, if and only if there exists an increasing smooth bijection ϕ˜: [0, q) → [0, q) such that ϕ˜0(0) > 0, and

ϕ˜(r) 2 m (r2) = m (ϕ˜(r)2) , (81) 0 0 ϕ˜0(0)r

ϕ˜0(r) 2 m˜ (r2) = m (ϕ˜(r)2) (82) 1 1 ϕ˜0(0) for all r ∈ [0, 1). We can solve Equation (81) for the function ϕ˜, and substitute this function into Equation (82) to ﬁnd a relation between the functions m1 and m˜ 1. Thus, Theorem5 for our case when m0 = m˜ 0 has the following form.

Theorem 14. Assume the pairs of functions (m0, m1), and (m0, m˜ 1) satisfy Assumptions2 and3. Denote by ⊕ and ⊕˜ the binary operations in B1 generated by the pairs of functions (m0, m1) and (m0, m˜ 1) respectively. Assume a differentiable bijection ϕ˜: [0, 1) → [0, 1) is such that ϕ˜0(0) > 0, and

ϕ˜(r) 2 m (r2) = m (ϕ˜(r)2) (83) 0 0 ϕ˜0(0)r for all r ∈ [0, 1). Then the operations ⊕ and ⊕˜ are isomorphic if and only if

ϕ˜0(r) 2 m˜ (r2) = m (ϕ˜(r)2) (84) 1 1 ϕ˜0(0) for all r ∈ [0, 1).

The positive number t = ϕ˜0(0) can be chosen arbitrarily. For each such t Equation (83) deﬁnes a function ϕ˜. Then from (84) we get the corresponding function m1. Hence, we get a one parameter set of functions m1 such that the binary operations generated by pairs (m0, m1) are isomorphic. It is remarkable that in the limit t → 0 we get operations isomorphic to Euclidean addition. Let us consider two special cases. Symmetry 2020, 12, 1634 25 of 30

Special case 1. Assume 1 m (u) = . 0 1 − u Denote t = ϕ˜0(0). Then Equation (83) may be solved for ϕ˜:

tr ϕ˜(r) = p . (85) 1 + (t2 − 1)r2

Theorem 14 states that the operations ⊕ and ⊕˜ are isomorphic if and only if

1 m˜ (r2) = m (ϕ˜(r)2) . (86) 1 1 (1 + (t2 − 1)r2)3

Assume, as for Einstein addition,

1 m (u) = . 1 (1 − u)2

Then 1 m˜ (r2) = . 1 (1 − r2)2(1 + (t2 − 1)r2)

If t = 1, then we obviously get ϕ˜(r) = r and m˜ 1 = m1. If t → 0, then ϕ˜(r) → 0 for all r ∈ [0, 1), and for the corresponding limit mˆ 1 = limt→0m˜ 1 we have:

2 2 1 mˆ 1(r ) = lim m˜ 1(r ) = . (87) t→0 (1 − r2)3

The functions m0, mˆ 1 satisfy condition (50). According to Theorem8, a binary operation generated by the canonical metric tensor (8) with the functions m0, mˆ 1 is isomorphic to Euclidean addition. Assume 1 m (u) = . 1 (1 − u)4 Then 1 + (t2 − 1)r2 m˜ (r2) = . 1 (1 − r2)4

The binary operations generated by pairs (m0, m˜ 1) are isomorphic to Einstein addition, and not isomorphic to Euclidean addition. Special case 2. Assume 1 m (u) = . 0 (1 − u)2 Solve Equation (83) for ϕ˜: √ 2tr ϕ˜(r) = . q p 2t2r2 + (1 − r2)2 + (1 − r2) (1 − r2)2 + 4t2r2

Calculate the factor on the right hand side of (84):

ϕ˜(r)0 2 2(1 + r2)2 = p . t [2t2r2 + (1 − r2)2 + (1 − r2) (1 − r2)2 + 4t2r2][(1 − r2)2 + 4t2r2] Symmetry 2020, 12, 1634 26 of 30

According to Theorem 14 an operation generated by the pair (m0, m1) is isomorphic to an operation generated by the pair (m0, m˜ 1) if and only if there exists a positive number t such that

ϕ˜(r)0 2 m˜ (r2) = m (ϕ˜(r)2) . 1 1 t for all r ∈ [0, 1). For Möbius addition we have 1 m (r2) = . 1,M (1 − r2)2

Direct calculations show that the binary operation generated by a pair (m0, m˜ 1) is isomorphic to the binary operation generated by the pair (m0, m1,M) if and only if there exists a non-zero number t such that

2 2 " 2 2 # 2 (1 + r ) 2 2t r m˜ 1(r ) = 1 − r + p . (1 − r2)2[(1 − r2)2 + 4t2r2] 1 − r2 + (1 − r2)2 + 4t2r2

If t = 1 then obviously ϕ˜(r) = r and m˜ 1 = m1,M. The corresponding binary operations are isomorphic to Möbius addition (and hence isomorphic to each other and isomorphic to Einstein addition) for all t 6= 0. Therefore these operations are not isomorphic to Euclidean addition in Rn. We denote the limit t → 0 by mˆ 1:

2 2 2 2 1 + r mˆ 1(r ) = lim m˜ 1(r ) = . t→0 (1 − r2)2

According to Theorem8, the operation generated by the pair (m0, mˆ 1) is isomorphic to Euclidean addition, since 0 2 [(um0(u)) ] mˆ 1(u) = m0(u) for all u ∈ [0, 1).

10. Isomorphic Operations with the Same Function m1

Let (m0, m1) and (m˜ 0, m1) be pairs of functions satisfying Assumptions2 and3. Let ⊕, ⊕˜ be binary operations in Bq generated by the pairs (m0, m1) and (m˜ 0, m1) respectively. Notice that the second function in both pairs is the same. The distance functions in the space with operations ⊕ and ⊕˜ are completely determined by the same function m1, and therefore they coincide. Now we consider the problem of ﬁnding a relation between ⊕ and ⊕˜ .

Theorem 15. Canonical metric tensors of binary operations ⊕ and ⊕˜ have the same function m1 and the binary operations ⊕ and ⊕˜ are isomorphic if and only if there exists a positive number t such that

1 a⊕˜ b = ⊗ [(t ⊗ a) ⊕ (t ⊗ b)] (88) t for all a, b ∈ Bq, where ⊗ is a scalar multiplication in the space with a binary operation ⊕.

Proof. Assume that identity (88) holds for a positive number t. Let h be the function given by

Z p q 2 h(p) = m1(s )ds. 0 Symmetry 2020, 12, 1634 27 of 30

0 −1 0 −1 Since m1(0) = 1, we have 1 = h (0) = (h ) (0), where h is the inverse of h. According to the deﬁnition of multiplication by numbers [37] we have x t ⊗ x = h−1(t h(kxk)) kxk for all x ∈ Bq. Let −1 ϕ˜t(p) = h (t h(p)). 0 Then t = ϕ˜t(0) and x t ⊗ x = ϕ˜ (kxk). kxk t Direct calculations show that

1 ⊗ [(t ⊗ (−x)) ⊕ (t ⊗ (x + ∆x))] = o(k∆xk) t " # ! q ϕ˜ (kxk) xx> q ϕ˜0 (kxk) xx> + m (ϕ˜ (kxk)2) t I − + m (ϕ˜ (kxk)2) t ∆x. (89) 0 t tkxk kxk2 1 t t kxk2

Therefore the metric tensor of the operation ⊕˜ has the form (8) parametrized by the functions

h i2 ( 2) = ( ( )2) ϕ˜t(kxk) m˜ 0 r m0 ϕ˜ r tkxk , h 0 i2 (90) 2 2 ϕ˜t(kxk) m˜ 1(r ) = m1(ϕ˜(r) ) t .

According to Theorem 14 the operations ⊕ and ⊕˜ are isomorphic. By deﬁnition,

Z p q Z ϕ˜t(p) q 2 1 1 2 m1(s )ds = h(p) = h(ϕ˜t(p)) = m1(s )ds 0 t t 0 (91) Z p q Z p q 1 2 0 2 = m1(ϕ˜(u) )ϕ˜ (u)du = m˜ 1(u )du. t 0 0

Here we employed the change of variable: s = ϕ˜(u). Now differentiate identity (91) with respect to p to get 2 2 m1(p ) = m˜ 1(p ) for all p ∈ [0, q). Conversely, assume that the operations ⊕ and ⊕˜ are isomorphic, and that the functions m1 and m˜ 1 coincide. Then according to Theorem5, Deﬁnition4 and Remarks1 and2 there exists a differentiable bijection ϕ˜: [0, q) → [0, q) such that ϕ˜0(0) > 0,

h i2 ( 2) = ( ( )2) ϕ˜(r) m˜ 0 r m0 ϕ˜ r rϕ˜0(0) , h 0 i2 (92) ( 2) = ( 2) = ( ( )2) ϕ˜ (r) m1 r m˜ 1 r m1 ϕ˜ r ϕ˜0(0) for all r ∈ [0, q) and a⊕˜ b = ϕ−1(ϕ(a) ⊕ ϕ(b)) (93) for all a, b ∈ Bq, where x ϕ(x) = ϕ˜(kxk) kxk 0 for all x ∈ Bq. Set t = ϕ˜ (0). Deﬁne the function

Z p q 2 h(p) = m1(s )ds. 0 Symmetry 2020, 12, 1634 28 of 30

Then Z p q Z ϕ˜t(p) q 1 2 0 1 2 1 h(p) = m1(ϕ˜(u) )ϕ˜ (u)du = m1(s )ds = h(ϕ˜(p)). t 0 t 0 t Therefore, ϕ˜(p) = h−1(t h(p)) for all p ∈ [0, q). Hence, x x t ⊗ a = h−1(th(kxk)) = ϕ˜(kxk) = ϕ(x) kxk kxk for all x ∈ Bq. Since the function ϕ is invertible, we have

x = ϕ−1(t ⊗ x) for all x ∈ Bq. Now we apply Equation (93) obtaining

a⊕˜ b = =ϕ−1[ϕ(a) ⊕ ϕ(b)] 1 =ϕ−1 t ⊗ ⊗ [(t ⊗ a) ⊕ (t ⊗ b)] (94) t 1 = ⊗ [(t ⊗ a) ⊕ (t ⊗ b)] t for all a, b ∈ Bq.

Example. If ⊕ is Einstein addition, and ⊕˜ is Möbius addition, then

1 1 m (u) = , m (u) = m˜ (u) = m˜ (u) = . 0 1 − u 1 1 0 (1 − u)2

In this case Equation (88) holds with t = 2.

11. Conclusions In this paper we consider smooth binary operations invariant with respect to unitary transformations. We prove that such binary operations have tensors (8) parametrized by non negative functions m0, m1. It is shown that binary operations are well-defined iff a special additional constraint (11) on functions m1 hold. We present necessary and sufficient conditions for two operations to be isomorphic. We give necessary and sufficient conditions for binary operations to be isomorphic to Einstein addition, and, separately, to Euclidean addition. We also pointed out necessary and sufficient conditions for two operations having the same function m0, or, separately, the same function m1, to be isomorphic. Future research may concern a development of calculus (integration, differentiation, differential equations) in gyrospaces, finding necessary and sufficient conditions for binary operations to give rise of gyrogroups, an integration of geodesic equations in the polar coordinates, and a description of binary operations with respect to which the Maxwell equations are invariant.

Funding: This research received no external funding. Conﬂicts of Interest: The author declares no conﬂicts of interest.

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