Gyrovector spaces and their differential geometry

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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

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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

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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

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(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)

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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.

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(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)

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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)

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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.

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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)

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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

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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
c2u+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
s2u+1
γu
v+1
s2
γu
1 + γu
(u·v)u(39)

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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=γuγv1 + 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=γuγv1−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)

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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
βvu(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=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
(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
sv
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⊖Mx1
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⊟Mx1
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⊖Ex1
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) = c1
R+x2
1
R2(c+R),x1x2
R2(c+R)
X2(x2, x2) = cx1x2
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⊟Ex1
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⊖Ux1
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⊟Ux1
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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