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1

Gyrovector Spaces

And Their

Differential Geometry

Dedicated to Professor Grigorios Tsagas in admiration

Abraham A. Ungar

Department of Mathematics

North Dakota State University

Fargo, ND 58105, USA

Email: Abraham.Ungar@ndsu.edu

ABSTRACT This article adds physical appeal to Einstein addition, the Einstein

velocity addition law of relativistically admissible velocities. Einstein addition turns out

to be isomorphic to M¨

obius addition in the sense of isomorphisms between gyrovector

spaces. Gyrovector spaces, in turn, form the setting for hyperbolic geometry just as

vector spaces form the setting for Euclidean geometry. A remarkable link between the

gyrovector spaces that we study in this article and hyperbolic geometry is provided by

differential geometry.

This paper is published in [52] and appears in [53, Chap. 7].

1 INTRODUCTION

Hundred years have passed since Einstein introduced his special theory of

relativity in 1905, and more than hundred ﬁfty years have passed since August

Ferdinand M¨

obius ﬁrst studied the transformations that now bear his name.

Yet, the rich structure Einstein and M¨

obius thereby exposed is still far from

being exhausted.

In the sense of gyrovector spaces, M¨

obius and Einstein additions are isomor-

phic. M¨

obius addition is the M¨

obius transformation of the open complex unit

disc without rotation, and Einstein addition is the Einstein velocity addition

law of relativistically admissible coordinate velocities. Replacing coordinate

velocities by proper velocities, one obtains Ungar addition, a term coined by

Jing-Ling Chen in [7]. The additions of Einstein, M¨

obius, and Ungar, are all

gyrovector space operations that are presented in this article in the context of

the theory of gyrovector spaces and their differential geometry.

The intrinsic beauty and usefulness of M¨

obius transformations is well

known; see, for instance, [17, 18, 19, 20, 49, 51]. In contrast, the general

2

Einstein addition of velocities that need not be parallel is unheard of in mod-

ern literature on special relativity. Among outstanding exceptions we note the

relativity physics books by Fock [14] and by Sexl and Urbantke [39].

The reason for the omission of the general Einstein velocity addition is well

expressed by Robert W. Brehme [4],

The transformation law for the spatial components of the coordinate velocity, known

as the Einstein (or relativistic) velocity addition theorem, is awkward and difﬁcult to

use in any but the very simplest situations [that is, Einstein velocity addition of parallel

velocities].

Robert W. Brehme [4] 1968.

However, following the discovery that (i) Einstein addition is a gyrovector

addition in gyrovector spaces that shares remarkable analogies with vector

addition in vector spaces, and that (ii) it is isomorphic with M¨

obius addition,

it becomes increasingly clear that Einstein addition is an old idea whose time

has come back [48].

Gyrovector spaces algebraically regulate hyperbolic geometry just as vector

spaces algebraically regulate Euclidean geometry. In the same way that vec-

tor spaces are commutative groups of vectors that admit scalar multiplication,

gyrovector spaces are gyrocommutative gyrogroups of gyrovectors that admit

scalar multiplication. In order to elaborate a precise language for dealing with

hyperbolic geometry, which emphasizes analogies with classical notions, we

extensively use the preﬁx “gyro”, giving rise to gyrolanguage, the language

that we use in this article. The preﬁx “gyro” stems from Thomas gyration.

The latter, in turn, is the mathematical abstraction of the peculiar relativistic

effect known as Thomas precession into an operator, called a gyrooperator, and

denoted “gyr′′. The gyrooperator generates special automorphisms called gy-

roautomorphisms. The effects of the gyroautomorphisms are called (Thomas)

gyrations in the same way that the effects of rotation automorphisms are called

rotations.

Owing to its great coherence and the vast amount of supporting results

in hyperbolic geometry and physics [44], gyrogroup theory and gyrovector

space theory ﬁnd their way to the mainstream literature; see, for instance,

[12, 22, 23, 25, 36, 38]. Challenging preconceived notions, like the dogma of

Einsteinian relativity vs. Minkowskian relativity [47], that was not struck down

until the early 21st Century [44, 48, 50], gyrovector spaces provide powerful,

far reaching insights into the relativistic mass problem [50] and Riemannian

geometry [24]. Earlier studies along the line of gyrovector spaces, but on a

higher level of abstraction, are found in Sabinin’s book [37].

2 DEFINITIONS

Several deﬁnitions leading to the concept of the gyrogroup are presented.

Motivation for the gyrogroup deﬁnition comes from Sections 4 – 7, where it will

3

be shown that familiar “additions”, like Einstein addition and M¨

obius addition,

are nothing else but gyrocommutative gyrogroup operations.

Deﬁnition 1 (Binary Operations, Groupoids, and Groupoid Automorphisms).

A binary operation +in a set Sis a function + : S×S→S. We use the notation

a+bto denote +(a, b)for any a, b ∈S. A groupoid (S, +) is a nonempty

set, S, with a binary operation, +. An automorphism φof a groupoid (S, +)

is a bijective (that is, one-to-one) self-map of Swhich preserves its groupoid

operation, that is, φ(a+b) = φ(a) + φ(b)for all a, b ∈S.

Groupoids may have identity elements. An identity element of a groupoid

(S, +) is an element 0∈Ssuch that 0 + s=s+ 0 = sfor all s∈S.

Deﬁnition 2 (Loops).A loop is a groupoid (S, +) with an identity element in

which each of the two equations a+x=band y+a=bfor the unknowns x

and ypossesses a unique solution.

Deﬁnition 3 (Groups).A group is a groupoid (G, +) whose binary operation

satisﬁes the following axioms. In Gthere is at least one element, 0, called a

left identity, satisfying

(G1) 0+a=a

for all a∈G. There is an element 0∈Gsatisfying axion (G1) such that for

each a∈Gthere is an element −a∈G, called a left inverse of a, satisfying

(G2) −a+a= 0

Moreover, the binary operation obeys the associative law

(G3) (a+b) + c=a+ (b+c)

for all a, b, c ∈G.

The binary operation in a given set is known as the set operation. The set of all

automorphisms of a groupoid (S, ⊕), denoted Aut(S, ⊕), forms a group with

group operation given by bijection composition. The identity automorphism is

denoted by I. We say that an automorphism τvanishes if τ=I.

Groups are classiﬁed into commutative and noncommutative groups.

Deﬁnition 4 (Commutative Groups).A group (G, +) is commutative if its

binary operation obeys the commutative law

(G6) a+b=b+a

for all a, b ∈G.

Deﬁnition 5 (Gyrogroups).A groupoid (G, ⊕)is a gyrogroup if its binary

operation satisﬁes the following axioms. In Gthere is at least one element, 0,

called a left identity, satisfying

(G1) 0⊕a=a

for all a∈G. There is an element 0∈Gsatisfying axiom (G1) such that for

each a∈Gthere is an element ⊖a∈G, called a left inverse of a, satisfying

4

(G2) ⊖a⊕a= 0

Moreover, for any a, b, c ∈Gthere exists a unique element gyr[a, b]c∈Gsuch

that the binary operation obeys the left gyroassociative law

(G3) a⊕(b⊕c) = (a⊕b)⊕gyr[a, b]c

The map gyr[a, b] : G→Ggiven by c7→ gyr[a, b]cis an automorphism of

the groupoid (G, ⊕),

(G4) gyr[a, b]∈Aut(G, ⊕)

and the automorphism gyr[a, b]of Gis called the gyroautomorphism of G,

generated by a, b ∈G. The operation gyr : G×G→Aut(G, ⊕)is called

the gyrooperation of G. Finally, the gyroautomorphism gyr[a, b]generated by

any a, b ∈Gpossesses the left loop property

(G5) gyr[a, b] = gyr[a⊕b, b]

The gyrogroup axioms in Deﬁnition 5 are classiﬁed into three classes.

(1) The ﬁrst pair of axioms, (G1) and (G2), is a reminiscent of the group

axioms;

(2) The last pair of axioms, (G4) and (G5), presents the gyrooperation

axioms; and

(3) The middle axioms, (G3), is a hybrid axiom linking the two pairs of

axioms in (1) and (2).

As in group theory, we use the notation

a⊖b=a⊕(⊖b)(1)

in gyrogroup theory as well.

In full analogy with groups, gyrogroups are classiﬁed into gyrocommutative

and non-gyrocommutative gyrogroups.

Deﬁnition 6 (Gyrocommutative Gyrogroups).A gyrogroup (G, ⊕)is gyro-

commutative if its binary operation obeys the gyrocommutative law

(G6) a⊕b= gyr[a, b](b⊕a)

for all a, b ∈G.

Deﬁnition 7 (The Gyrogroup Cooperation).Let (G, ⊕)be a gyrogroup. The

gyrogroup cooperation is a second binary operation, ⊞, in Ggiven by the

equation

a⊞b=a⊕gyr[a, ⊖b]b(2)

for all a, b ∈G. The gyrogroup operation, ⊕, is also called a gyrooperation

and, accordingly, the gyrogroup cooperation, ⊞, is also called a cogyroopera-

tion.

Replacing bby ⊖bin(2) we have

a⊟b=a⊖gyr[a, b]b(3)

5

In the special case when all the gyrations of a (gyrocommutative) gyrogroup

vanish, the (gyrocommutative) gyrogroup reduces to a (commutative) group,

where the gyrogroup operation and cooperation coincide, being reduced to the

group operation.

3 FIRST GYROGROUP THEOREMS

Theorem 8. Let (G, +) be a gyrogroup. For any elements a, b, c, x ∈Gwe

have:

(1) If a+b=a+c, then b=c(general left cancellation law; see (9)).

(2) gyr[0, a] = Ifor any left identity 0in G.

(3) gyr[x, a] = Ifor any left inverse xof ain G.

(4) gyr[a, a] = I

(5) There is a left identity which is a right identity.

(6) There is only one left identity.

(7) Every left inverse is a right inverse.

(8) There is only one left inverse of a.

(9) −a+ (a+b) = b(left cancellation law).

(10) gyr[a, b]x=−(a+b) + {a+ (b+x)}

(11) gyr[a, b]0 = 0

(12) gyr[a, b](−x) = −gyr[a, b]x

(13) gyr[a, 0] = I .

Proof.

(1) Let xbe a left inverse of acorresponding to a left identity, 0, in

G. We have x+ (a+b)=x+ (a+c). By left gyroassociativity,

(x+a) + gyr[x, a]b=(x+a) + gyr[x, a]c. Since 0is a left identity,

gyr[x, a]b= gyr[x, a]c. Since automorphisms are bijective, b=c.

(2) By left gyroassociativity we have for any left identity 0of G,a+x=

0 + (a+x)=(0 + a) + gyr[0, a]x=a+ gyr[0, a]x. By (1) we then

have x= gyr[0, a]xfor all x∈Gso that gyr[0, a] = I.

(3) By the left loop property and by (2) above we have gyr[x, a] = gyr[x+

a, a] = gyr[0, a] = I.

(4) Follows from an application of the left loop property and (2) above.

(5) Let xbe a left inverse of acorresponding to a left identity, 0, of G.

Then by left gyroassociativity and (3) above, x+ (a+ 0) =(x+a) +

gyr[x, a]0 = 0 + 0 = 0 = x+a. Hence, by (1), a+ 0 = afor all

a∈Gso that 0is a right identity.

6

(6) Suppose 0and 0∗are two left identities, one of which, say 0, is also a

right identity. Then 0 = 0∗+ 0 = 0∗.

(7) Let xbe a left inverse of a. Then x+ (a+x)=(x+a) + gyr[x, a]x

=0 + x=x=x+ 0, by left gyroassociativity, (G2), (3), (5), and (6)

above. By (1) we have a+x= 0 so that xis a right inverse of a.

(8) Suppose xand yare left inverses of a. By (7) above, they are also right

inverses, so a+x= 0 = a+y. By (1), x=y.

(9) By left gyroassociativity and by (3) above, −a+ (a+b) = (−a+a) +

gyr[−a, a]b=b.

(10) Follows from an application of the left cancellation law (9) to the left

gyroassociative law (G3).

(11) Follows from (10) with z= 0.

(12) Since gyr[a, b]is an automorphism of (G, +) we have from (11)

gyr[a, b](−x) + gyr[a, b]x= gyr[ab(−x+x) = gyr[a, b]0 = 0, and

hence the result.

(13) Follows from (10) with b= 0 and a left cancellation, (9).

The left cancellation law

a⊕(⊖a⊕b) = b(4)

in a gyrogroup (G, ⊕), established in Theorem 8(9), comes with a slightly

different right cancellation law [44]

(b⊟a)⊕a=b(5)

to which the gyrogroup cooperation gives rise.

Theorem 9. Let (G, ⊕)be a gyrogroup. The groupoid (G, ⊞)of the gyrogroup

cooperation is a loop.

Proof. The identity element of the groupoid (G, ⊞)is the identity element, 0,

of the gyrogroup (G, ⊕)since, by Theorem 8 (2) and (13) we have

a⊞0 = 0 ⊞a=a(6)

If x⊞a= 0 then, by the right cancellation law we have x= (x⊞a)⊖aso

that x=⊖ais a left inverse of ain (G, ⊞). Furthermore, ⊖ais also a right

inverse of ain (G, ⊞)since

a⊞(⊖a) = a⊖gyr[a, a]a=a⊖a= 0 (7)

7

The unique solution of the equation

x⊞a=b(8)

is, by a right cancellation,

x=b⊖a(9)

The unique solution of the equation

a⊞x=b(10)

is

x= gyr[b, ⊖a](⊖a⊕b)(11)

as we show below. The equation in (10),

b=a⊞x=a⊕gyr[a, ⊖x]x(12)

implies, by a left cancellation, the equation

⊖a⊕b= gyr[a, ⊖x]x(13)

or, equivalently,

⊖gyr[a, z]z=⊖a⊕b(14)

where we use the notation z=⊖x.

Solving (14) for the unknown z, we have

z=⊖gyr[a, ⊖a⊕b](⊖a⊕b)

=⊖gyr[b, ⊖a](⊖a⊕b)(15)

Replacing zby ⊖xin (15) we ﬁnally have

x= gyr[b, ⊖a](⊖a⊕b)(16)

as desired.

We may note that in the gyrocommutative case the solution (16) of (10)

reduces to x=b⊖a.

4 THE M ¨

OBIUS COMPLEX DISC GYROGROUP

M¨

obius transformation of the complex open unit disc

D={z∈C:|z|<1}(17)

in the complex plane Chas the polar decomposition

z7→ eiθ a+z

1 + az =eiθ(a⊕Mz)(18)

8

It induces the M¨

obius addition ⊕Min the disc, allowing the M ¨

obius transfor-

mation of the disc to be viewed as a M¨

obius left gyrotranslation

z7→ a⊕Mz=a+z

1 + az (19)

followed by a rotation. Here θ∈Ris a real number, a, z ∈D, and ais the

complex conjugate of a.

M¨

obius addition ⊕Mis neither commutative nor associative. The breakdown

of commutativity in M¨

obius addition is "repaired" by the introduction of a

gyro-operation

gyr : D×D→Aut(D,⊕M)(20)

that generates gyroautomorphisms according to the equation

gyr[a, b] = a⊕Mb

b⊕Ma=1 + ab

1 + ab ∈Aut(D,⊕M)(21)

where Aut(D,⊕M)is the automorphism group of the M ¨

obius groupoid (D,⊕M).

The inverse of the automorphism gyr[a, b]is clearly gyr[b, a],

gyr−1[a, b] = gyr[b, a](22)

The gyrocommutative law of M ¨

obius addition ⊕Mthat follows from the

deﬁnition of gyr in (21),

a⊕Mb= gyr[a, b](b⊕Ma)(23)

is not terribly surprising since it is generated by deﬁnition, but we are not

ﬁnished.

Coincidentally, the gyroautomorphism gyr[a, b]that repairs the breakdown

of commutativity of ⊕Min (23), repairs the breakdown of the associativity of

⊕Mas well, giving rise to the respective left and right gyroassociative laws

a⊕M(b⊕Mz) = (a⊕Mb)⊕Mgyr[a, b]z

(a⊕Mb)⊕Mz=a⊕M(b⊕Mgyr[b, a]z)(24)

for all a, b, z ∈D. Moreover, M¨

obius gyroautomorphisms possess the two

elegant identities

gyr[a⊕Mb, b] = gyr[a, b]

gyr[a, b⊕Ma] = gyr[a, b](25)

One can now readily check that the M¨

obius complex disc groupoid (D,⊕M)

is a gyrocommutative gyrogroup.

9

5 M ¨

OBIUS GYROGROUPS

Identifying vectors in the Euclidean plane R2with complex numbers in the

complex plane Cin the usual way we have

R2∋u↔(u1, u2) = u1+iu2=u∈C(26)

The inner product and the norm in R2then become the real numbers

u·v↔Re(¯uv) = ¯uv +u¯v

2

kuk ↔ |u|

(27)

Under the translation (27) of elements of the disc

R2

s=1 ={u∈R2:kuk<1}(28)

of the Euclidean plane R2to elements of the complex unit disc D, M¨

obius

addition (18) in Vs=R2

s=1 takes the form

u⊕Mv=(1 + 2u·v+kvk2)u+ (1 − kuk2)v

1 + 2u·v+kuk2kvk2

↔(1 + ¯uv +u¯v+|v|2)u+ (1 − |u|2)v

1 + ¯uv +u¯v+|u|2|v|2

=(1 + u¯v)(u+v)

(1 + ¯uv)(1 + u¯v)

=u+v

1 + ¯uv

=u⊕Mv

(29)

for all u,v∈R2

s=1, and all u, v ∈D. In (29) we have thus recovered the

M¨

obius addition ⊕Min the open unit disc Dof C, (18).

Suggestively, we introduce the following deﬁnition of M¨

obius addition in

the ball.

Deﬁnition 10 (M¨

obius Addition In The Ball).Let Vbe a real inner product

space [30], and let Vsbe the s-ball of V,

Vs={Vs∈V:kvk< s}(30)

for any ﬁxed s > 0. M¨

obius addition ⊕Mis a binary operation in Vsgiven by

the equation

u⊕Mv=(1 + 2

s2u·v+1

s2kvk2)u+ (1 −1

s2kuk2)v

1 + 2

s2u·v+1

s4kuk2kvk2(31)

10

where ·and k·k are the inner product and norm that the ball Vsinherits from

its space V.

M¨

obius addition ⊕Min the open unit ball Vsof any real inner product space

Vis thus a most natural extension of the M¨

obius addition in the open complex

unit disc. Like the M ¨

obius disc (D,⊕M), the M ¨

obius ball (Vs,⊕M)turns out

to be a gyrocommutative gyrogroup, as one can readily check by computer

algebra.

M¨

obius addition satisﬁes the gamma identity

γu⊕

Mv=γuγvr1 + 2

s2u·v+1

s4kuk2kvk2(32)

for all u,v∈Vs, where γuis the gamma factor

γv=1

r1−kvk2

s2

(33)

The gamma factor appears also in Einstein addition, and it is known in

special relativity theory as the Lorentz factor.

The M¨

obius gyrogroup cooperation (2) is given by M¨

obius coaddition

u⊞Mv=γ2

uu+γ2

vv

γ2

u+γ2

v−1(34)

satisfying the gamma identity

γu⊞Mv=γ2

u+γ2

v−1

q1 + 2γ2

uγ2

v(1 −u·v

s2)−(γ2

u+γ2

v)

(35)

M¨

obius coaddition is commutative.

In earlier studies by Ahlfors [1] and Ratcliffe [34], M¨

obius addition is treated

as a hyperbolic translation. M ¨

obius translation became M¨

obius addition in

[43] following the discovery of the analogies it shares, as a gyrocommutative

gyrogroup operation, with ordinary vector addition. Applications of M ¨

obius

addition and its hyperbolic geometry in quantum mechanics are found in [8,

27, 28, 45, 46].

6 EINSTEIN GYROGROUPS

Attempts to measure the absolute velocity of the earth through the hypothet-

ical ether had failed. The most famous of these experiments is one performed

by Michelson and Morley in 1887 [13]. It was 18 years later before the null

results of these experiments were ﬁnally explained by Einstein in terms of a

11

new velocity addition law that bears his name, that he introduced in his 1905

paper that founded the special theory of relativity [9, 10].

Contrasting Newtonian velocities, which are vectors in the Euclidean three-

space R3, Einsteinian velocities must be relativistically admissible, that is,

their magnitude must not exceed the vacuum speed of light c, which is about

3×105km·sec−1.

Let

R3

c={v∈R3:kvk< c}(36)

be the c-ball of all relativistically admissible velocities of material particles.

It is the open ball of radius c, centered at the origin of the Euclidean three-

space R3, consisting of all vectors vin R3with magnitude kvksmaller than

the vacuum speed of light c. Einstein addition ⊕in the ball is given by the

equation

u⊕Ev=1

1 + u·v

c2u+1

γu

v+1

c2

γu

1 + γu

(u·v)u(37)

for all u,v∈R3

c, where u·vis the inner product that the ball R3

cinherits from

its space R3, and where γuis the gamma factor (33).

Einstein addition (37) of relativistically admissible velocities was introduced

by Einstein in his 1905 paper [10, p. 141]) where the magnitudes of the two

sides of Einstein addition (37) are presented. One has to remember here that the

Euclidean 3-vector algebra wasnot so widely known in 1905 and, consequently,

was not used by Einstein. Einstein calculated in [9] the behavior of the velocity

components parallel and orthogonal to the relative velocity between inertial

systems, which is as close as one can get without vectors to the vectorial

version (37).

In the Newtonian limit, c→ ∞, the ball R3

cof all relativistically admissible

velocities expands to the whole of its space R3, as we see from (36), and

Einstein addition ⊕in R3

creduces to the ordinary vector addition +in R3, as

we see from (37) and (33).

Suggestively, we extend Einstein addition of relativistically admissible ve-

locities by abstraction in the following deﬁnition of Einstein addition in the

ball.

Deﬁnition 11 (Einstein Addition In The Ball).Let Vbe a real inner product

space and let Vsbe the s-ball of V,

Vs={Vs∈V:kvk< s}(38)

Einstein addition ⊕Eis a binary operation in Vsgiven by the equation

u⊕Ev=1

1 + u·v

s2u+1

γu

v+1

s2

γu

1 + γu

(u·v)u(39)

12

where γuis the gamma factor, and where ·and k·k are the inner product and

norm that the ball Vsinherits from its space V.

Like M¨

obius addition in the ball, one can show by computer algebra that

Einstein addition in the ball is a gyrocommutative gyrogroup operation, giving

rise to the Einstein ball gyrogroup (Vs,⊕E).

Einstein addition satisﬁes the gamma identity

γu⊕

Ev=γuγv1 + u·v

s2(40)

for all u,v∈Vs.

Einstein gyrogroup cooperation (2) in an Einstein gyrogroup (Vs,⊕E)is

given by Einstein coaddition

u⊞Ev=γu+γv

γ2

u+γ2

v+γuγv(1 + u·v

s2)−1(γuu+γvv)

= 2⊗E

γuu+γvv

γu+γv

(41)

where the scalar multiplication by the factor 2 is deﬁned by the equation

2⊗Ev=v⊕Ev. A more general deﬁnition of the scalar multiplication by any

real number will be presented in Section 8.

Einstein coaddition is commutative, satisfying the gamma identity

γu⊞Ev=γ2

u+γ2

v+γuγv(1 + u·v

s2)−1

γuγv(1 −u·v

s2) + 1 (42)

The gamma identity (40) written in its equivalent form

γu⊖v=γuγv1−u·v

s2(43)

signaled the emergence of hyperbolic geometry in special relativity when it was

ﬁrst studied by Sommerfeld [40] and Variˇ

cak [54] in terms of rapidities. The

rapidity φvof a relativistically admissible velocity vis deﬁned by the equation

[29]

φv= tanh−1kvk

s(44)

so that,

cosh φv=γv

sinh φv=γvkvk

s

(45)

13

In the years 1910–1914, the period which experienced a dramatic ﬂower-

ing of creativity in the special theory of relativity, the Croatian physicist and

mathematician Vladimir Variˇ

cak (1865–1942), professor and rector of Zagreb

University, showed in Ref. [54], that this theory has a natural interpretation in

the hyperbolic geometry of Bolyai and Lobachevski [2] [35]. Indeed, written

in terms of rapidities, identity (43) takes the form

cosh φu⊖v= cosh φucosh φv−sinh φusinh φvcos A(46)

where, according to J.F. Barrett [3], the angle Ahas been interpreted by Som-

merfeld [40], and Variˇ

cak [54], as a hyperbolic angle in the relativistic “triangle

of velocities” in the Beltrami ball model of hyperbolic geometry. The role of

Carath´

eodory in this approach to special relativity and hyperbolic geometry has

been described by J.F. Barrett [3], emphasizing that (46) is the “cosine rule” in

hyperbolic geometry.

7 UNGAR GYROGROUPS

The term Ungar gyrogroup was coined by Jing-Ling Chen in [7].

Deﬁnition 12 (Ungar Addition).Let (V,+,·)be a real inner product space

with addition, +, and inner product, ·. The Ungar gyrovector space (V,⊕U,⊗E)

is the real inner product space Vequipped with addition ⊕U, given by

u⊕Uv=u+v+βu

1 + βu

u·v

s2+1−βv

βvu(47)

where βv, called the beta factor, is given by the equation

βv=1

r1 + kvk2

s2

(48)

Ungar addition is the relativistic addition of proper velocities rather than

coordinate velocities as in Einstein addition [44, p. 143]. It can be shown by

computer algebra that Ungar addition is a gyrocommutative gyrogroup addition,

giving rise to the Ungar gyrogroup (Vs,⊕U).

Ungar addition satisﬁes the beta identity

1

βu⊕

Uv

=1

βu

1

βv

+u·v

s2(49)

or, equivalently,

βu⊕

Uv=βuβv

1 + βuβv

u·v

s2

(50)

14

The Ungar gyrogroup cooperation (2) is commutative, given by Ungar coad-

dition

u⊞Uv=βu+βv

1 + βuβv(1 −u·v

s2)(u+v)(51)

8 DEFINITION AND FIRST GYROVECTOR SPACE

THEOREMS

Gyrogroups, both gyrocommutative and non-gyrocommutative, ﬁnite and

inﬁnite, abound in group theory [15, 16]. Some gyrocommutative gyrogroups

admit scalar multiplication, turning themselves into gyrovector spaces. The

latter, in turn, are analogous to vector spaces just as gyrogroups are analogous

to groups. Indeed, gyrovector spaces form the setting for hyperbolic geometry

just as vector spaces form the setting for Euclidean geometry.

Deﬁnition 13 (Inner Product Gyrovector Spaces).A(n inner product) gyrovec-

tor space (G, ⊕,⊗)is a gyrocommutative gyrogroup (G, ⊕)that obeys the

following axioms:

(1) Gadmits an inner product ,·,(i) which gives rise to a positive deﬁnite norm

kak, that is, kak2=a·a,kak ≥ 0and kak= 0 if and only if a=0,

|a·b| ≤ kakkbk; and (ii) which is invariant under gyroautomorphisms,

that is,

gyr[u,v]a·gyr[u,v]b=a·b

for all points a,b,u,v∈G.

(2) Gadmits a scalar multiplication, ⊗, satisfying the following properties.

For all real numbers r, r1, r2∈Rand all points a∈G:

(V1) 1⊗a=a

(V2) (r1+r2)⊗a=r1⊗a⊕r2⊗aScalar Distributive Law

(V3) (r1r2)⊗a=r1⊗(r2⊗a)Scalar Associative Law

(V4) |r|⊗a

kr⊗ak=a

kakScaling Property

(V5) gyr[u,v](r⊗a) = r⊗gyr[u,v]aGyroautomorphism Property

(V6) gyr[r1⊗v, r2⊗v] = IIdentity Automorphism

(3) Real vector space structure (kGk,⊕,⊗)for the set kGkof one-dimensional

‘vectors’

kGk={±kak:a∈G} ⊂ R

15

with vector addition ⊕and scalar multiplication ⊗, such that for all r∈Rand

a,b∈G,

(V7) kr⊗ak=|r|⊗kakHomogeneity Property

(V8) ka⊕bk ≤ kak⊕kbkGyrotriangle inequality

Owing to the scalar distributive law, the condition for 1⊗ais equivalent to

the condition

n⊗a=a⊕... ⊕a(n times)(52)

and

a⊗(−t) = ⊖a⊗t(53)

In the special case when all the gyrations of a gyrovector space vanish, the

gyrovector space reduces to a vector space. A gyrovector space possesses a

weak distributive law, called the monodistributive law.

Theorem 14 (The Monodistributive Law).A gyrovector space (G, ⊕,⊗)pos-

sesses the monodistributive law

r⊗(r1⊗a⊕r2⊗a) = r⊗(r1⊗a)⊕r⊗(r2⊗a)(54)

Proof. The proof follows from (V2) and (V3),

r⊗(r1⊗a⊕r2⊗a) = r⊗{(r1+r2)⊗a}

= (r(r1+r2))⊗a

= (rr1+rr2)⊗a

= (rr1)⊗a⊕(rr1)⊗a

=r⊗(r1⊗a)⊕r⊗(r1⊗a)

(55)

9 GYROLINES

In full analogy with (i) the two identical line expressions

a+btThe Euclidean Line

bt+aThe Euclidean Line (56)

a,b∈G,t∈R, in Euclidean analytic geometry, which is regulated by the

(associative) algebra of vector spaces (G, +,·), (ii) the two distinct hyperbolic

line expressions

a⊕b⊗tGyroline,The Hyperbolic Line

b⊗t⊕aCogyroline,The Hyperbolic Dual Line (57)

16

t∈R, of hyperbolic analytic geometry are regulated by the (nonassociative)

algebra of gyrovector spaces (G, ⊕,⊗).

In order to emphasize that the Euclidean line is uniquely determined by any

two distinct points that it contains, one may replace the expressions in (56) by

a+ (−a+b)tThe Euclidean Line

(b−a)t+aThe Euclidean Line (58)

The ﬁrst line in (58) is the unique Euclidean line that passes through the

points aand b. Considering the line parameter tas “time”, the line passes

through the point aat time t= 0, and owing to a left cancellation, it passes

through the point bat time t= 1.

Similarly, the second line in (58) is the unique Euclidean line that passes

through the points aand b. It passes through the point aat time t= 0, and

owing to a right cancellation, it passes through the point bat time t= 1. In

vector spaces, of course, left cancellations and right cancellations coincide.

This is, however, not the case in gyrovector spaces.

In full analogy with (58) , in order to emphasize that the hyperbolic lines

are uniquely determined by any two distinct points that they contain, one may

replace the expressions in (57) by

a⊕(⊖a⊕b)⊗tGyroline,The Hyperbolic Line

(b⊟a)⊗t⊕aCogyroline,The Hyperbolic Dual Line (59)

The ﬁrst line in (59) is the unique gyroline that passes through the points

aand b. It passes through the point aat time t= 0, and owing to a left

cancellation, it passes through the point bat time t= 1.

Similarly, the second line in (59) is the unique cogyroline that passes through

the points aand b. It passes through the point aat time t= 0, and owing to

a right cancellation, it passes through the point bat time t= 1. Unlike left

cancellations and right cancellations in vector spaces, where they coincide, left

cancellations and right cancellations in gyrovector spaces are distinct, forcing us

to employ the cooperation in the second expression of (59). It is the presence of

the cooperation in the second expression in (59) that allows a right cancellation

when t= 1.

The formal deﬁnition of the gyroline follows. The formal deﬁnition of its

associated cogyroline will be presented in Section 10.

Deﬁnition 15 (Gyrolines, Gyrosegments).Let a,bbe any two distinct points

in a gyrovector space (G, ⊕,⊗). The gyroline in Gthat passes through the

points aand bis the set of all points

Lg=a⊕(⊖a⊕b)⊗t(60)

17

in G,t∈R. The gyrovector space expression in (60) is called the representation

of the gyroline Lgin terms of the two points aand bthat it contains.

A gyrosegment with endpoints aand bis the set of all points in (60) with

0≤t≤1.

Considering the real parameter tas “time”, the gyroline (60) passes through

the point aat time t= 0 and, owing to the left cancellation law, it passes

thought the point bat time t= 1.

It is anticipated in Deﬁnition 15 that the gyroline is uniquely represented by

any two given points that it contains. The following theorem shows that this is

indeed the case.

Theorem 16. Two gyrolines that share two distinct points are coincident.

Proof. Let

a⊕(⊖a⊕b)⊗t(61)

be a gyroline that contains two given distinct points p1and p2in a gyrovector

space (G, ⊕,⊗). Then, there exist real numbers t1, t2∈R,t16=t2, such that

p1=a⊕(⊖a⊕b)⊗t1

p2=a⊕(⊖a⊕b)⊗t2

(62)

A gyroline containing the points p1and p2has the form

p1⊕(⊖p1⊕p2)⊗t(63)

which, by means of (62) is reducible to (61) with a reparametrization. Indeed,

by (62), the Gyrotranslation Theorem [44], scalar distributivity and associativ-

ity, and left gyroassociativity, we have

p1⊕(⊖p1⊕p2)⊗t

= [a⊕(⊖a⊕b)⊗t1]⊕{⊖[a⊕(⊖a⊕b)⊗t1]⊕[a⊕(⊖a⊕b)⊗t2]}⊗t

= [a⊕(⊖a⊕b)⊗t1]⊕gyr[a,(⊖a⊕b)⊗t1]{⊖(⊖a⊕b)⊗t1⊕(⊖a⊕b)⊗t2}⊗t

= [a⊕(⊖a⊕b)⊗t1]⊕gyr[a,(⊖a⊕b)⊗t1]{(⊖a⊕b)⊗(−t1+t2)}⊗t

= [a⊕(⊖a⊕b)⊗t1]⊕gyr[a,(⊖a⊕b)⊗t1](⊖a⊕b)⊗((−t1+t2)t)

=a⊕{(⊖a⊕b)⊗t1⊕(⊖a⊕b)⊗((−t1+t2)t)}

=a⊕(⊖a⊕b)⊗(t1+ (−t1+t2)t)

(64)

thus obtaining the gyroline (61) with a reparametrization . It is a reparametriza-

tion in which the original gyroline parameter tis replaced by the new gyroline

parameter t1+ (−t1+t2)t,t2−t16= 0.

Hence, any gyroline (61) that contains the two points p1and p2coincides

with the gyroline (63).

18

10 COGYROLINES

Following the discussion leading to Deﬁnition 15 of the gyroline, we now

present the deﬁnition of the cogyroline.

Deﬁnition 17 (Cogyrolines, Cogyrosegments).Let a,bbe any two distinct

points in a gyrovector space (G, ⊕,⊗). The cogyroline in Gthat passes

through the points aand bis the set of all points

Lc= (b⊟a)⊗t⊕a(65)

t∈R. The gyrovector space expression in (65) is called the representation of

the cogyroline Lcin terms of the two points aand bthat it contains.

A cogyrosegment with endpoints aand bis the set of all points in (65) with

0≤t≤1.

Considering the real parameter tas “time”, the cogyroline (65) passes

through the point aat time t= 0 and, owing to the right cancellation law,

it passes thought the point bat time t= 1.

It is anticipated in Deﬁnition 17 that the cogyroline is uniquely represented

by any two given points that it contains. The following theorem shows that this

is indeed the case.

Theorem 18. Two cogyrolines that share two distinct points are coincident.

Proof. Let

(b⊟a)⊗t⊕a(66)

be a cogyroline that contains the two distinct points p1and p2. Then, there

exist real numbers t1, t2∈R,t16=t2, such that

p1= (b⊟a)⊗t1⊕a

p2= (b⊟a)⊗t2⊕a(67)

A cogyroline containing the points p1and p2has the form

(p2⊟p1)⊗t⊕p1(68)

which, by means of (67) is reducible to (66) with a reparametrization. Indeed,

by (67), the Cogyrotranslation Theorem [44], scalar distributivity and associa-

tivity, and left gyroassociativity with Axiom (V6) of gyrovector spaces, we

19

have

(p2⊟p1)⊗t⊕p1

={[(b⊟a)⊗t2⊕a]⊟[(b⊟a)⊗t1⊕a]}⊗t⊕[(b⊟a)⊗t1⊕a]

={(b⊟a)⊗t2⊖(b⊟a)⊗t1}⊗t⊕[(b⊟a)⊗t1⊕a]

={(b⊟a)⊗(t2−t1)}⊗t⊕[(b⊟a)⊗t1⊕a]

= (b⊟a)⊗((t2−t1)t)⊕[(b⊟a)⊗t1⊕a]

={(b⊟a)⊗((t2−t1)t)⊕(b⊟a)⊗t1}⊕a]

= (b⊟a)⊗((t2−t1)t+t1)⊕a

(69)

obtaining a reparametrization for the cogyroline (66) in which the original

cogyroline parameter tis replaced by the new cogyroline parameter (t2−

t1)t+t1,t2−t16= 0.

Hence, any cogyroline (66) that contains the two points p1and p2is identical

to the cogyroline (68).

Cogyrolines admit parallelism in hyperbolic geometry, suggesting the fol-

lowing

Deﬁnition 19 (Gyroparallelism).The two cogyrolines

(b⊟a)⊗t⊕a

(b′⊟a′)⊗t⊕a′(70)

in a gyrovector space (G, ⊕,⊗)are gyroparallel if the two points

b⊟a

b′⊟a′(71)

in Gare related by the equation

b′⊟a′=λ⊗(b⊟a)(72)

for some real number λ∈R.

11 M ¨

OBIUS GYROVECTOR SPACES

M¨

obius gyrogroups (Vs,⊕M)admit scalar multiplication ⊗M, turning them-

selves into M¨

obius gyrovector spaces (Vs,⊕M,⊗M).

Deﬁnition 20 (M¨

obius Scalar Multiplication).Let (Vs,⊕M)be a M¨

obius gy-

rogroup. The M¨

obius scalar multiplication r⊗Mv=v⊗Mrin Vsis given by the

20

a, t = 0

b, t = 1

Gyroline (in algebra)

Geodesic (in geometry)

a⊕(⊖a⊕b)⊗t

−∞ < t < ∞

Figure 1. The unique geodesic (gyroline) in

the M¨

obius gyrovector plane passing through

the points aand b.

a, t = 0

b, t = 1

b⊟a

Cogyroline (in algebra)

Cogeodesic (in geometry)

(b⊟a)⊗t⊕a

−∞ < t < ∞

Figure 2. The unique cogeodesic (cogyro-

line) in the M¨

obius gyrovector plane passing

through the points aand b.

equation

r⊗Mv=s1 + kvk

sr

−1−kvk

sr

1 + kvk

sr

+1−kvk

sr

v

kvk

=stanh(rtanh−1kvk

s)v

kvk

(73)

where r∈R,v∈Vs,v6=0; and r⊗M0=0.

The unique M¨

obius gyroline and cogyroline that pass through two given

points aand bare represented by the equations

Lg

ab =a⊕M(⊖Ma⊕Mb)⊗Mt

Lc

ab = (b⊟Ma)⊗Mt⊕Ma(74)

t∈R, in a M ¨

obius gyrovector space (Vs,⊕M,⊗M). Gyrolines in a M ¨

obius

gyrovector space coincide with the well-known geodesics of the Poincar´

e ball

model of hyperbolic geometry, as we will prove in Section 17. M ¨

obius gyrolines

in the disc are Euclidean circular arcs that intersect the boundary of the disc

orthogonally, Fig. 1. In contrast, M¨

obius cogyrolines in the disc are Euclidean

circular arcs that intersect the boundary of the disc diametrically, that is, on the

opposite sides of a diameter called the supporting diameter, Fig. 2.

21

α

β

γ

akAk2=cos α+cos(β+γ)

cos α+cos(β−γ)

α+β+γ < π b

kBk2=cos β+cos(α+γ)

cos β+cos(α−γ)

c

kCk2=cos γ+cos(α+β)

cos γ+cos(α−β)

A

B

C

A=⊖b⊕c, a =kAk

B=⊖c⊕a, b =kBk

C=⊖a⊕b, c =kCk

cos α=⊖a⊕b

k⊖a⊕bk·⊖a⊕c

k⊖a⊕ck

Figure 3. ⊕=⊕

M. A hyperbolic triangle ∆abc in the M ¨

obius gyrovector plane D=

(R2

c,⊕

M,⊗). It shares visual and symbolic analogies with its Euclidean counterpart. Unlike

the Euclidean triangle, angle sum of the hyperbolic triangle is less than π, and its hyperolic side

lengths are uniquely determined by its hyperbolic angles [44].

A hyperbolic triangle in the M¨

obius gyrovector plane (R2

c,⊕M,⊗M)is shown

in Fig. 3. It shares visual and symbolic analogies with its Euclidean counterpart.

Unlike the Euclidean triangle, angle sum of the hyperbolic triangle is less than

π, and its side lengths are uniquely determined by its hyperbolic angles. For

more about the hyperbolic angle in gyrovector spaces see [44].

M¨

obius gyrolines do not admit parallelism. Given a gyroline Lg

0=ab

and a point cnot on the gyroline, there exist inﬁnitely many gyrolines that

pass through the point cand do not intersect the gyroline Lg

0, two of which,

Lg

1=c1c2and Lg

2=c3c4, are shown in Fig. 4 for the M¨

obius gyrovector

plane.

In contrast, cogyrolines do admit parallelism. Given a cogyroline Lc

0=ab

and a point cnot on the cogyroline, there exists a unique cogyroline that passes

through the point cand does not intersect the cogyroline Lc

0. It is the cogyroline

Lc

1=a′b′shown in Fig. 5 for the M¨

obius gyrovector plane.

We note that (i) the two parallel cogyrolines Lc

0=ab and Lc

1=a′b′in

Fig. 5 share their supporting diameters, and that (ii) their associated points

22

a

b

c

c1

c2

c4

c3

M¨obius GyrolinesM¨obius Gyrolines

do not admit parallelism

Figure 4. Through the point c, not on the

gyroline ab, there are inﬁnitely many gyro-

lines, like c1c2and c3c4, that do not intersect

gyroline ab. Hence, the Euclidean parallel

postulate is not satisﬁed.

a

b

c

a′

b′

b⊟a

b′⊟a′

M¨obius Cogyrolines

admit parallelism

Figure 5. Through the point c, not on the

cogyroline ab, there is a unique cogyroline

a′b′that does not intersect the cogyroline ab.

Hence, the Euclidean parallel postulate is sat-

isﬁed.

b⊟aand b′⊟a′lie on the common supporting diameter. Hence, these points

in Vs⊂Vrepresent two Euclidean vectors in Vthat are Euclidean parallel

to the supporting diameter, so that there exists a real number r6= 0 such that

b′⊟a′=r(b⊟a). Equivalently, there exists a real number λ6= 0 such that

b′⊟a′=λ⊗(b⊟a)(75)

as we see from Deﬁnition 20 of scalar multiplication. Hence, by Deﬁnition 19,

the cogyrolines Lc

0and Lc

1are gyroparallel.

12 EINSTEIN GYROVECTOR SPACES

Einstein gyrogroups (Vs,⊕E)admit scalar multiplication ⊗E, turning them-

selves into Einstein gyrovector spaces (Vs,⊕E,⊗E).

Deﬁnition 21 (Einstein Scalar Multiplication).Let (Vs,⊕E)be a M¨

obius gy-

rogroup. The M¨

obius scalar multiplication r⊗Ev=v⊗Erin Vsis given by the

equation

r⊗Ev=s(1 + kvk/s)r−(1 − kvk/s)r

(1 + kvk/s)r+ (1 − kvk/s)r

v

kvk

=stanh(rtanh−1kvk

s)v

kvk

(76)

23

a, t = 0

b, t = 1

The Einstein Gyroline

through the points aand b

a⊕E(⊖Ea⊕Eb)⊗Et

−∞ < t < ∞

Figure 6. The unique gyroline in an Ein-

stein gyrovector space (Vs,⊕

E,⊗

E)through

two given points aand b. The case of the

Einstein gyrovector plane, when Vs=R2

s=1

is the real open unit disc, is shown graphically.

a, t = 0

b, t = 1

b⊟Ea

The Cogyroline

through the points aand b

(b⊟Ea)⊗Et⊕Ea

−∞ < t < ∞

Figure 7. The unique cogyroline in

(Vs,⊕

E,⊗

E)through two given points aand

b. The case of the Einstein gyrovector plane,

when Vs=R2

s=1 is the real open unit disc, is

shown graphically.

where r∈R,v∈Vs,v6=0; and r⊗E0=0.

Interestingly, the scalar multiplication that M¨

obius and Einstein addition

admit coincide. This stems from the fact that for parallel vectors in the ball,

M¨

obius addition and Einstein addition coincide as well.

Einstein scalar multiplication can also be written as

r⊗Ev=1−(γv−pγ2

v−1)2r

1 + (γv−pγ2

v−1)2r

γv

pγ2

v−1v(77)

v6=0.

The unique Einstein gyroline and cogyroline that pass through two given

points aand bare represented by the equations

Lg

ab =a⊕E(⊖Ea⊕Eb)⊗Et

Lc

ab = (b⊟Ea)⊗Et⊕Ea(78)

t∈R, in an Einstein gyrovector space (Vs,⊕E,⊗E). Gyrolines in an Einstein

gyrovector space coincide with the well-known geodesics of the Beltrami (also

known as Klein) ball model of hyperbolic geometry, as we will prove in Section

19. Einstein gyrolines in the disc are Euclidean straight lines, Fig. 6. In contrast,

Einstein cogyrolines in the disc are Euclidean elliptical arcs that intersect the

24

Plane Origin

a, t = 0

b, t = 1

The Ungar Gyroline

through the points aand b

a⊕U(⊖Ua⊕Ub)⊗Ut

−∞ < t < ∞

Figure 8. The unique gyroline in an Un-

gar gyrovector space (V,⊕

U,⊗

U)through two

given points aand b. The case of the Ungar

gyrovector plane, when V=R2is the Eu-

clidean plane, is shown graphically.

Plane Origin

a, t = 0

b, t = 1

The Ungar Cogyroline

through the points aand b

(b⊟Ua)⊗Ut⊕Ua

−∞ < t < ∞

Figure 9. The unique cogyroline in

(V,⊕

U,⊗

U)through two given points aand b.

The case of the Ungar gyrovector plane, when

V=R2is the Euclidean plane, is shown

graphically.

boundary of the disc diametrically, that is, on the opposite sides of a diameter

called the supporting diameter, Fig. 7.

13 UNGAR GYROVECTOR SPACES

Ungar gyrogroups (V,⊕U)admit scalar multiplication ⊗U, turning them-

selves into Ungar gyrovector spaces (V,⊕U,⊗U).

Deﬁnition 22 (Ungar Scalar Multiplication).Let (V,⊕E)be an Ungar gy-

rogroup. The Ungar scalar multiplication r⊗Uv=v⊗Urin Vis given by the

equation

r⊗Uv=s

2( r1 + kvk2

s2+kvk

s!r

− r1 + kvk2

s2−kvk

s!r)v

kvk

=ssinh rsinh−1kvk

sv

kvk(79)

where r∈R,v∈V,v6=0; and r⊗U0=0.

25

Plane Origin

a, t = 0

b, t = 1

c

A cogyroline and a point

not on the cogyroline

(b⊟Ua)⊗Ut⊕Ua

−∞ < t < ∞

Figure 10. A cogyroline L

c

ab and a point c

not on the cogyroline in an Ungar gyrovector

plane (R2,⊕

U,⊗

U).

a, t = 0

b, t = 1

c, t = 0

d, t = 1

Gyroparallel Cogyrolines

(b⊟Ua)⊗Ut⊕Ua

(d⊟Uc)⊗Ut⊕Uc

−∞ < t < ∞

Figure 11. The unique cogyroline L

c

cd that

passes through the given point and is gyropar-

allel to the given cogyroline in Fig. 10.

The unique Ungar gyroline and cogyroline that pass through two given points

aand bare represented by the equations

Lg

ab =a⊕U(⊖Ua⊕Ub)⊗Ut

Lc

ab = (b⊟Ua)⊗Ut⊕Ua(80)

t∈R, in an Ungar gyrovector space (V,⊕U,⊗U). Ungar gyrolines in the space

Vare Euclidean hyperbolas with asymptotes that intersect at the origin of the

space V, Fig. 8. In contrast, Ungar cogyrolines in the space are Euclidean

straight lines, Fig. 9.

Let cbe a point not on the cogyroline Lc

ab that passes through the two given

points aand bin an Ungar gyrovector space (V,⊕U,⊗U), Fig. 10. In order to

ﬁnd a point d∈Vsuch that the resulting cogyroline

Lc

cd = (d⊟Uc)⊗Ut⊕Uc(81)

is gyroparallel to the given cogyroline Lc

ab, we impose on dthe condition

d⊟Uc=b⊟Ua(82)

that follows from Deﬁnition 19 as the gyroparallelism condition.

Solving (82) for dby a right cancellation we have

d= (b⊟Ua)⊕Uc(83)

26

thus determining the unique cogyroline Lc

cd, in (82), that passes through the

given point cand is gyroparallel to the given cogyroline Lc

ab.

Since cogyrolines in Ungar gyrovector spaces are Euclidean straight lines,

gyroparallelism in these gyrovector spaces coincides with Euclidean paral-

lelism, as shown in Fig. 11.

14 LINKING GYROVECTOR SPACES TO

DIFFERENTIAL GEOMETRY

In Sections 15 — 22 we will uncover the link between gyrovector spaces

embedded in the Euclidean n-space Rn,n≥2, and differential geometry.

Accordingly, we explore the differential geometry of M¨

obius gyrovector spaces

(Rn

c,⊕M,⊗M), Einstein gyrovector spaces (Rn

c,⊕E,⊗E), and Ungar gyrovector

spaces (Rn,⊕U,⊗U), where Rn

cis the c-ball of the Euclidean n-space,

Rn

c={v∈Rn:kvk< c}(84)

In Rnwe use the vector notation

r= (x1, x2,...,xn)

dr= (dx1, dx2,...,dxn)

r2=r·r=krk2=

n

X

i=1

x2

i,r4= (r2)2

dr2=dr·dr=kdrk2=

n

X

i=1

dx2

i

r·dr=

n

X

i=1

xidxi

(r×dr)2=r2dr2−(r·dr)2

(85)

noting that (r×dr)2is deﬁned in Rnfor any dimension n.

15 THE RIEMANNIAN LINE ELEMENT OF

EUCLIDEAN METRIC

To set the stage for the study of the gyroline and the cogyroline element of

the gyrovector spaces (Rn

c,⊕,⊗)and (Rn,⊕,⊗)in Sections 17 – 22 we begin

with the study of the Riemannian line element ds2of the Euclidean vector

space Rnwith its standard metric given by the distance function

d(a,b) = kb−ak(86)

The norm of the differential

∆s= (v+ ∆v)−v= ∆v(87)

27

gives the distance between the two neighboring points vand v+ ∆vin Rn,

where ∆vis of sufﬁciently small length, k∆vk< ε for some ε > 0.

Let v,∆v∈Rn

cor Rnbe represented by their components relative to rectan-

gular Cartesian coordinates as v= (x1, . . . xn)and ∆v= (∆x1, . . . ∆xn).

The differential ∆scan be written as

∆s=∂∆s

∂∆x1∆v=0

∆x1+... +∂∆s

∂∆xn∆v=0

∆xn

+ε1∆x1+... +εn∆xn

(88)

where ε1, . . . , εn→0as ε→0.

We write (88) as

ds=∂∆s

∂∆x1∆v=0

dx1+... +∂∆s

∂∆xn∆v=0

dxn(89)

and use the notation ds2=kdsk2.

Since ∂∆s

∂∆xk

= (0,...,1,...,0) (90)

(a 1 in the kth position), (89) gives

ds= (dx1,...,dxn)(91)

so that the Riemannian line element of the Euclidean n-space Rnwith its

standard metric (86) is

ds2=

n

X

i=1

dx2

i=dr2(92)

Following the calculation of the Riemannian line element (92) of the Eu-

clidean n-space Rnwith its metric given by the Euclidean distance function

(86) the stage is set for the presentation in Section 16 and the calculation of the

(1) gyroline element of each of the gyrovector spaces in Sections 17 – 22 with

their gyrometrics given by their respective gyrodistance functions; and the

(2) cogyroline element of each of the gyrovector spaces in Sections 17 – 22 with

their cogyrometrics given by their respective cogyrodistance functions.

16 THE GYROLINE AND THE COGYROLINE

ELEMENT

The gyrometric and the cogyrometric of a gyrovector space (G, ⊕,⊗)is

given by its gyrodistance and cogyrodistance function

d⊕(b⊖a) = kb⊖ak

d⊞(b⊟a) = kb⊟ak(93)

28

respectively.

To determine the line element ds2of the n-dimensional Riemannian man-

ifold which corresponds to a gyrovector space gyrometric and cogyrometric,

we consider the gyrodifferential and the cogyrodifferential given, respectively,

by the equations

∆s= (v+ ∆v)⊖v

∆s= (v+ ∆v)⊟v(94)

in a gyrovector space (G, ⊕,⊗), where G=Rn

cor G=Rn.

The norm of the gyrodifferential and the cogyrodifferential in Rngives,

respectively, the gyrodistance and the cogyrodistance

k∆sk=d⊕(v+ ∆v,v) = k(v+ ∆v)⊖vk

k∆sk=d⊞(v+ ∆v,v) = k(v+ ∆v)⊟vk(95)

between the two neighboring points vand v+ ∆vof Rn

cor Rn. Here +is

vector addition in Rn, and ∆vis an element of Rn

cor Rnof sufﬁciently small

length, k∆vk< ε for some ε > 0.

Let v,∆v∈Rn

cor Rnbe represented by their components relative to rectan-

gular Cartesian coordinates as v= (x1, . . . xn)and ∆v= (∆x1, . . . ∆xn).

The differential ∆scan be written as

∆s=∂∆s

∂∆x1∆v=0

∆x1+... +∂∆s

∂∆xn∆v=0

∆xn

+ε1∆x1+... +εn∆xn

(96)

where ε1, . . . , εn→0as ε→0.

We write (96) as

ds=∂∆s

∂∆x1∆v=0

dx1+... +∂∆s

∂∆xn∆v=0

dxn(97)

and use the notation ds2=kdsk2. Following the origin of dsfrom a gyrod-

ifferential or a cogyrodifferential, we call dsthe element of arc gyrolength or

cogyrolength, and call ds2=kdsk2the gyroline or cogyroline element. Each

gyroline and cogyroline element forms a Riemannian line element.

For the sake of simplicity, further details are given explicitly for the special

case of n= 2, but the generalization to any integer n > 2is obvious. In the

special case when n= 2, (97) reduces to

ds=∂∆sM

∂∆x1(∆x1= 0

∆x2= 0 )dx1+∂∆sM

∂∆x2(∆x1= 0

∆x2= 0 )dx2

=X1(x1, x2)dx1+X2(x1, x2)dx2

(98)

29

where X1,X2:R2

c→R2or X1,X2:R2→R2are given by

Xk(x1, x2) = ∂∆s

∂∆xk(∆x1= 0

∆x2= 0 )(99)

k= 1,2.

Following standard notation in differential geometry [6, p. 92], the metric co-

efﬁcients of the gyrometric or cogyrometric of the gyrovector plane (R2

c,⊕,⊗)

or (R2,⊕,⊗)in the Cartesian x1x2-coordinates are

E=X1·X1

F=X1·X2

G=X2·X2

(100)

These metric coefﬁcients give rise to the Riemannian line element

ds2=Edx2

1+ 2F dx1dx2+Gdx2

2(101)

The gyrovector plane (R2

c,⊕,⊗)or (R2,⊕,⊗), with its gyrometric or co-

gyrometric, results in a Riemannian line element ds2. The latter, in turn, gives

rise to the Riemannian surface (R2

c, ds2)or (R2, ds2). The Gaussian curvature

Kof this surface is given by the equation [6, p. 237] [31, p. 155] [33, p. 105]

K=−1

2√EG (∂

∂x2

∂E

∂x2

√EG +∂

∂x1

∂G

∂x1

√EG )(102)

EG > 0.

17 THE RIEMANNIAN LINE ELEMENT OF M ¨

OBIUS

GYROMETRIC

In this section we uncover the Riemannian line element to which the gyro-

metric of the M¨

obius gyrovector plane (Rn

c,⊕M,⊗M)gives rise.

Let us consider the gyrodifferential (94),

∆sM= (v+ ∆v)⊖Mv

=x1+ ∆x1

x2+ ∆x2⊖Mx1

x2(103)

in the M¨

obius gyrovector plane (R2

c,⊕M,⊗M)where, ambiguously, + is the

Euclidean addition in R2and in R. To calculate X1and X2we have

dsM=∂∆sM

∂∆x1(∆x1= 0

∆x2= 0 )dx1+∂∆sM

∂∆x2(∆x1= 0

∆x2= 0 )dx2

=X1(x1, x2)dx1+X2(x1, x2)dx2(104)

30

where X1,X2:R2

c→R2, obtaining

X1(x1, x2) = c2

c2−r2(1,0) ∈R2

c

X2(x1, x2) = c2

c2−r2(0,1) ∈R2

c

(105)

where r2=x2

1+x2

2.

The metric coefﬁcients of the gyrometric of the M¨

obius gyrovector plane in

the Cartesian x1x2-coordinates are therefore

E=X1·X1=c4

(c2−r2)2

F=X1·X2= 0

G=X2·X2=c4

(c2−r2)2

(106)

Hence, the gyroline element of the M¨

obius gyrovector plane (R2

c,⊕M,⊗M)is

the Riemannian line element

ds2

M=kdsMk2

=Edx2

1+ 2F dx1dx2+Gdx2

2

=c4

(c2−r2)2(dx2

1+dx2

2)

(107)

An interesting elementary study of the Riemannian structure (107) in the

context of the hyperbolic plane is presented in the introductory chapter of [21].

The Riemannian line element ds2

Mis described in [11, p. 216] as a Riemannian

metric on the Riemann surface Dc=1, where Dc=1 is Poincar´

e complex open

unit disc.

Following Riemann [41, p. 73] we note that E,Gand EG −F2=EG

are all positive in the open disc R2

c, so that the quadratic form (107) is positive

deﬁnite [26, p. 84].

The Gaussian curvature Kof the surface with the line element (107) is a

negative constant,

K=−4

c2(108)

as one can calculate from (102).

31

Extension of (107) from n= 2 to n≥2is obvious, resulting in

ds2

M=c4

(c2−r2)2dr2

=dr2

(1 + 1

4Kr2)2

(109)

The Riemannian line element ds2

Mreduces to its Euclidean counterpart in

the limit of large c,

lim

c→∞ ds2

M=dr2(110)

as expected.

18 THE RIEMANNIAN LINE ELEMENT OF M ¨

OBIUS

COGYROMETRIC

In this section we uncover the Riemannian line element to which the cogy-

rometric of the M¨

obius gyrovector plane (Rn

c,⊕M,⊗M)gives rise.

Let us consider the cogyrodifferential (94),

∆sCM = (v+ ∆v)⊟Mv

=x1+ ∆x1

x2+ ∆x2⊟Mx1

x2(111)

in the M¨

obius gyrovector plane (R2

c,⊕M,⊗M), where + is the Euclidean addition

in R2and R. To calculate X1and X2we have

dsCM =∂∆sCM

∂∆x1(∆x1= 0

∆x2= 0 )dx1+∂∆sCM

∂∆x2(∆x1= 0

∆x2= 0 )dx2

=X1(x1, x2)dx1+X2(x1, x2)dx2

(112)

where X1,X2:R2

c→R2, obtaining

X1(x1, x2) = c2

c4−r4(c2+x2

1−x2

2,2x1x2)∈V

c=R2

c

X2(x1, x2) = c2

c4−r4(2x1x2, c2+x2

1−x2

2)∈V

c=R2

c

(113)

where r2=x2

1+x2

2.

32

The metric coefﬁcients of the cogyrometric of the M ¨

obius gyrovector plane

in the Cartesian x1x2-coordinates are therefore

E=X1·X1=c4

(c4−r4)2(c2+r2)2−4c2x2

2

F=X1·X2=c6

(c4−r4)2x1x2

G=X2·X2=c4

(c4−r4)2(c2+r2)2−4c2x2

1

(114)

Hence, the cogyroline element of the M¨

obius gyrovector plane (R2

c,⊕M,⊗M)

is the Riemannian line element

ds2

CM =kdsCMk2

=Edx2

1+ 2F dx1dx2+Gdx2

2

=c4

(c4−r4)2{(c2+r2)2(dx2

1+dx2

2)−4c2(x1dx2−x2dx1)2}

(115)

Following Riemann [41, p. 73]), we note that E,Gand

EG −F2=c8

(c4−r4)2(116)

are all positive in the open disc R2

c, so that the quadratic form (115) is positive

deﬁnite [26, p. 84]).

In vector notation the Riemannian line element (115), extended to ndimen-

sions, takes the form

ds2

CM =c4

(c4−r4)2{(c2+r2)2dr2−4c2(r×dr)2}(117)

in Cartesian coordinates.

As expected, the Riemannian line element ds2

CM reduces to its Euclidean

counterpart in the limit of large c,

lim

c→∞ ds2

CM =dr2(118)

The Gaussian curvature Kof this surface is positive variable,

K=8c6

(c2+r2)4(119)

as one can calculate from (102).

33

19 THE RIEMANNIAN LINE ELEMENT OF

EINSTEIN GYROMETRIC

In this section we uncover the Riemannian line element to which the gyro-

metric of the Einstein gyrovector plane (Rn

c,⊕E,⊗E)gives rise.

Let us consider the gyrodifferential (94),

∆sE= (v+ ∆v)⊖Ev

=x1+ ∆x1

x2+ ∆x2⊖Ex1

x2(120)

in the Einstein gyrovector plane (R2

c,⊕E,⊗E)where + is the Euclidean addition

in R2and in R. To calculate X1and X2we have

dsE=∂∆sE

∂∆x1(∆x1= 0

∆x2= 0 )dx1+∂∆sE

∂∆x2(∆x1= 0

∆x2= 0 )dx2

=X1(x1, x2)dx1+X2(x1, x2)dx2(121)

where X1,X2:R2

c→R2, obtaining

X1(x1, x2) = c1

R+x2

1

R2(c+R),x1x2

R2(c+R)

X2(x2, x2) = cx1x2

R2(c+R),1

R+x2

2

R2(c+R)

(122)

where R2=c2−r2,r2=x2

1+x2

2.

The metric coefﬁcients of the gyrometric of the Einstein gyrovector plane in

the Cartesian x1x2-coordinates are therefore

X1·X1=E=c2c2−x2

2

(c2−r2)2

X1·X2=F=c2x1x2

(c2−r2)2

X2·X2=G=c2c2−x2

1

(c2−r2)2

(123)

34

Hence, the gyroline element of the Einstein gyrovector plane (R2

c,⊕E,⊗E)is

the Riemannian line element

ds2

E=kdsEk2

=Edx2

1+ 2F dx1dx2+Gdx2

2

=c2dx2

1+dx2

2

c2−r2+c2(x1dx1+x2dx2)2

(c2−r2)2.

(124)

Following Riemann [41, p. 73], we note that E,Gand

EG −F2=c6

(c2−r2)3(125)

r2=x2

1+x2

2, are all positive in the open disc R2

c, so that the quadratic form

(124) is positive deﬁnite [26, pp. 84 – 85].

The Riemannian line element ds2

Eof Einstein gyrometric in the disc turns

out to be the line element of the Beltrami (or Klein) disc model of hyperbolic

geometry. The Beltrami line element is presented, for instance, in McCleary

[31, p. 220], for n= 2, and in Cannon et al [5, ds2

K, p. 71], for n≥2.

An account of the ﬁrst ﬁfty years of hyperbolic geometry that emphasizes

the contributions of Beltrami, who prepared the background for Poincar´

e and

Klein, is found in [32].

The Gaussian curvature of the surface with the line element (124) is

K=−1

c2(126)

as one can calculate from (102).

The extension of (124) from n= 2 to n≥2is obvious, resulting in

ds2

E=c2

c2−r2dr2+c2

(c2−r2)2(r·dr)2(127)

in Cartesian coordinates. As expected, the hyperbolic Riemannian line element

(127) reduces to its Euclidean counterpart in the limit of large c,

lim

c→∞ ds2

E=dr2.(128)

Interestingly, the Beltrami-Riemannian line element (127) can be written as

1

c2ds2

B3=c2dr2−(r×dr)2

(c2−r2)2(129)

as noted by Fock [14, p. 39].

35

The line element ds2

Ein (124) is the line element of Einstein gyrometric.

It turns out to be the metric that the Italian mathematician Eugenio Beltrami

introduced in 1868 in order to study hyperbolic geometry by a Euclidean disc

model, now known as the Beltrami disc [31, p. 220]. An English translation of

his historically signiﬁcant 1868 essay on the interpretation of non-Euclidean

geometry is found in [42]. The signiﬁcance of Beltrami’s 1868 essay rests on

the generally known fact that it was the ﬁrst to offer a concrete interpretation of

hyperbolic geometry by interpreting ‘straight lines’ as geodesics on a surface

of a constant negative curvature.

Using the metric (124), Beltrami constructed a Euclidean disc model of the

hyperbolic plane [31] [42], which now bears his name.

20 THE RIEMANNIAN LINE ELEMENT OF

EINSTEIN COGYROMETRIC

In this section we uncover the Riemannian line element to which the cogy-

rometric of the Einstein gyrovector plane (Rn

c,⊕E,⊗E)gives rise.

Let us consider the cogyrodifferential (94),

∆sCE = (v+ ∆v)⊟Ev

=x1+ ∆x1

x2+ ∆x2⊟Ex1

x2(130)

in the Einstein gyrovector plane (R2

c,⊕E,⊗E), where + is the Euclidean addition

in R2and R. To calculate X1and X2we have

dsCE ="∂∆sCE

∂∆x1#(∆x1= 0

∆x2= 0 )dx1+∂∆sCE

∂∆x2(∆x1= 0

∆x2= 0 )dx2

=X1(x1, x2)dx1+X2(x1, x2)dx2

(131)

where X1,X2:R2

c→R2, obtaining

X1(x1, x2) = 1

c2−r2(c2−r2+x2

1, x1x2)

X2(x2, x2) = 1

c2−r2(x1x2, c2−r2+x2

2)

(132)

where r2=x2

1+x2

2.

36

The metric coefﬁcients of the cogyrometric of the Einstein gyrovector plane

in the Cartesian x1x2-coordinates are therefore

X1·X1=E= 1 + 2c2−r2

(c2−r2)2x2

1

X1·X2=F=2c2−r2

(c2−r2)2x1x2

X2·X2=G= 1 + 2c2−r2

(c2−r2)2x2

2

(133)

Hence, the cogyroline element of the Einstein gyrovector plane (R2

c,⊕E,⊗E)

is the Riemannian line element

ds2

CE =kdsCE k2

=Edx2

1+ 2F dx1dx2+Gdx2

2

=dx2

1+dx2

2+(2c2−r2)

(c2−r2)2(x1dx1+x2dx2)2

(134)

where r2=x2

1+x2

2. In the limit of large c,c→ ∞, the Riemannian dual line

element ds2

CE reduces to its Euclidean counterpart.

Following Riemann (p. 73 in [41]), we note that E,Gand

EG −F2=c4

(c2−r2)2(135)

are all positive in the open disc R2

c, so that the quadratic form (134) is positive

deﬁnite (p. 84 in [26]).

The Gaussian curvature of the Riemannian surface (Dc, ds2

CE )is a positive

variable,

K= 2c2−r2

c4(136)

as one can calculate from (102).

Extension of (134) from n= 2 to n≥2is obvious, resulting in

ds2

CE =dr2+2c2−r2

(c2−r2)2(r·dr)2(137)

in Cartesian coordinates. As expected, the hyperbolic Riemannian line element

reduces to its Euclidean counterpart in the limit of large c,

lim

c→∞ ds2

CE =dr2(138)

37

21 THE RIEMANNIAN LINE ELEMENT OF UNGAR

GYROMETRIC

In this section we uncover the Riemannian line element to which the gyro-

metric of the Ungar gyrovector plane (Rn,⊕U,⊗U)gives rise.

Let us consider the gyrodifferential (94),

∆sU= (v+ ∆v)⊖Uv

=x1+ ∆x1

x2+ ∆x2⊖Ux1

x2(139)

in the Ungar in gyrovector plane (R2

c,⊕U,⊗U)where + is the Euclidean addition

in R2and in R. To calculate X1and X2we have

dsU="∂∆sU

∂∆x1#(∆x1= 0

∆x2= 0 )dx1+"∂∆sU

∂∆x2#(∆x1= 0

∆x2= 0 )dx2

=X1(x1, x2)dx1+X2(x1, x2)dx2

(140)

where X1,X2:R2→R2, obtaining

X1(x1, x2) = 1

c2+r2+c√c2+r2(c2+r2+cpc2+r2−x2

1,−x1x2)

X2(x1, x2) = 1

c2+r2+c√c2+r2(−x1x2, c2+r2+cpc2+r2−x2

2)

(141)

The metric coefﬁcients of the gyrometric of the Ungar gyrovector plane in

the Cartesian x1x2-coordinates are therefore

E=X1·X1=c2+x2

2

c2+r2

F=X1·X2=−x1x2

c2+r2

G=X2·X2=c2+x2

1

c2+r2

(142)

Hence, the gyroline element of the Ungar gyrovector plane (R2,⊕U,⊗U)is

the Riemannian line element

ds2

U=kdsUk2

=Edx2

1+ 2F dx1dx2+Gdx2

2

=dx2

1+dx2

2−1

c2+r2(x1dx1+x2dx2)2

(143)

38

where r2=x2

1+x2

2.

Following Riemann (p. 73 in [41]), we note that E,Gand

EG −F2=c2

c2+r2(144)

are all positive in the R2, so that the quadratic form (143) is positive deﬁnite.

The Gaussian curvature Kof the surface with the line element (143) is a

negative constant,

K=−1

c2(145)

as one can calculate from (102).

Extension of (143) from n= 2 to n≥2is obvious, resulting in

ds2

U=dr2−1

c2+r2(r·dr)2(146)

and, as expected, the hyperbolic Riemannian line element reduces to its Eu-

clidean counterpart in the limit of large c,

lim

c→∞ ds2

U=dr2(147)

22 THE RIEMANNIAN LINE ELEMENT OF UNGAR

COGYROMETRIC

In this section we uncover the Riemannian line element to which the cogy-

rometric of the Ungar gyrovector plane (Rn,⊕U,⊗U)gives rise.

Let us consider the cogyrodifferential (94),

∆sCU = (v+ ∆v)⊟Uv

=x1+ ∆x1

x2+ ∆x2⊟Ux1

x2(148)

in the Ungar gyrovector plane (R2

c,⊕E,⊗E), where + is the Euclidean addition

in R2and R. To calculate X1and X2we have

dsCU ="∂∆sCU

∂∆x1#(∆x1= 0

∆x2= 0 )dx1+"∂∆sCU

∂∆x2#(∆x1= 0

∆x2= 0 )dx2

=X1(x1, x2)dx1+X2(x1, x2)dx2

(149)

39

where X1,X2:R2

c→R2, obtaining

X1(x1, x2) = c2

c2+r2(1,0)

X2(x1, x2) = c2

c2+r2(0,1)

(150)

The metric coefﬁcients of the cogyrometric of the Ungar gyrovector plane

in the Cartesian x1x2-coordinates are therefore

E=X1·X1=c4

(c2+r2)2

F=X1·X2= 0

G=X2·X2=c4

(c2+r2)2

(151)

Hence, the cogyroline element of the Ungar gyrovector plane (R2,⊕U,⊗U)

is the Riemannian line element

ds2

CU =kdsCU k2

=Edx2

1+ 2F dx1dx2+Gdx2

2

=c4

(c2+r2)2(dx2

1+dx2

2)

(152)

where r2=x2

1+x2

2. In vector notation, (85), the Riemannian line element

(152), extended to ndimensions, takes the form

ds2

CU =c4

(c2+r2)2dr2(153)

and, as expected, the hyperbolic Riemannian line element reduces to its Eu-

clidean counterpart in the limit of large c,

lim

c→∞ ds2

CU =dr2(154)

The metric (153) has the form ds2=λ(r)dr2,λ(r)>0, giving rise to an

isothermal Riemannian surface (R2, ds2

CU )[6].

The Riemannian metric ds2

CU in (152) is similar to the Riemannian metric

ds2

Min (107). It is described in [11, p. 214], as a Riemannian metric on the

Riemann surface M,Mbeing the entire complex plane C∪ {∞}.

The Gaussian curvature Kof this surface is a positive constant,

K=16

c2(155)

as one can calculate from (102).

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