Abstract
We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic.
1. Introduction
The theory of the binary operation of the Beltrami–Klein ball model, known as Einstein addition, and the binary operation of the Beltrami–Poincare ball model of hyperbolic geometry, known as Möbius addition has been extensively developed for the last twenty years [,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,]. There appeared a theory that may be called gyrogeometry, based on nice algebraic properties of the aforementioned operations and the concepts of gyrogroups, gyrovector spaces, gyrotrigonometry, and gyrogeometric objects [,,,,,,,,,]. This theory has been extended to the space of matrices with indefinite scalar products, which is closely related to the phenomenon of entanglement in quantum physics [,,], and to harmonic analysis [,,].
Recently there appeared a new approach in this theory based on the analysis of local properties of underlying operations, corresponding metric tensors and Riemannian manifolds. It turned out that both Einstein addition and Möbius addition may be recovered from corresponding metric tensors using standard operations of differential geometry: logarithmic mapping, parallel transport, and geodesics []. Basic properties of Einstein and Möbius gyrogroups and gyrovector spaces were derived using this approach [].
Einstein and Möbius additions are special cases of more general binary operations, namely, operations invariant with respect to unitary transformations. A differential geometry approach to the theory of such operations has been developed in [,]. The central object in this approach is a metric tensor in a special form, which was called the canonical form. This form depends on two scalar functions, and , which determine the value of the first fundamental form at x in the directions orthogonal and parallel to x respectively.
We adressed the following problems in this paper.
- Is it true that for every binary operation invariant with respect to unitary transformations the metric tensor has a canonical form (8) parametrized by a pair of functions ?
- What are the necessary and sufficient conditions on functions , such that their canonical metric tensor is smooth and is generated by a binary operation?
- Find necessary and sufficient conditions under which two binary operations having canonical metric tensors are isomorphic.
- Show that every algebraic isomorphism of binary operations generates an isomorphism of corresponding gyrogroups.
- Show that an isomorphism of binary operations is transitive.
- Find necessary and sufficient conditions for binary operations to be isomorphic to Euclidean addition.
- Find necessary and sufficient conditions for binary operations to be isomorphic to Einstein addition.
- Find a binary operation in isomorphic to every binary operation isomorphic to Einstein addition in the unit ball.
- Find necessary and sufficient conditions for two operations having the same function to be isomorphic.
- Describe all binary operations isomorphic to Einstein addition and having the same function .
- Find necessary and sufficient conditions for two operations having the same function to be isomorphic.
The organization of the paper is as follows. Following the introduction, in Section 2, we prove that all the metric tensors of smooth binary operations invariant with respect to unitary transformations have the canonical form (8). In Section 3, we find the necessary and sufficient conditions on functions , to generate metric tensors of binary operations. We give the definition of isomorphism of binary operations and necessary and sufficient conditions for binary operations to be isomorphic in terms of the functions , in Section 4. We also show how to construct binary operations isomorphic to a given binary operation. In Section 5, we show that every algebraic isomorphism between binary operations gives rise to an isomorphism between corresponding gyrogroups. The necessary and sufficient conditions for binary operations to be isomorphic to Euclidean addition are presented in Section 6. The same problems for Einstein addition are solved in Section 7. In this section, a set of binary operations in parametrized by positive numbers and isomorphic to Einstein addition are presented. In Section 8 we describe all the binary operations isomorphic to Einstein addition and having the same function in the representation of their canonical metric tensors. The necessary and sufficient conditions for binary operations, having the same function in the representation of their canonical metric tensors, to be isomorphic are given in Section 9. In Section 10, we prove that every two binary operations, having the same function in the representation of their canonical metric tensors, are isomorphic if and only if they are related by the standard scalar multiplication in a corresponding gyrovector space.
2. Binary Operations and Metric Tensors Invariant with Respect to Unitary Transformations
Let be the open ball in the Euclidean space with radius :
The set is identified with .
We consider smooth binary operations f: invariant with respect to unitary transformations. More precisely, we make the following assumption.
Assumption 1.
The function f is differentiable, and
for all vectors and all unitary matrices U.
For every vector we consider the matrix
and define the Hermitian positive semi-definite matrix function
Lemma 1.
Under Assumption 1 the matrix function G satisfies the following condition: for any vector and a unitary matrix U we have
Proof.
We have
Therefore
□
We consider G as a metric tensor in .
Definition 1.
The metric tensor G in (4) is said to be associated with the binary operation f.
The quadratic form
is called the first fundamental form. For every curve x: the length of x in the space with the metric tensor G is equal to
We are looking for a general form of symmetric matrix functions G satisfying (5).
Theorem 1.
Let condition (5) be satisfied for all and all unitary matrices U. Then there exist functions , : such that for all non zero we have
Proof.
For an arbitrary number consider a vector . For an arbitrary unitary matrix consider the unitary matrix
Notice that , so that Equation (5) implies
We split up the matrix in a way similar to U, obtaining the block matrix
where , , and C is a symmetric matrix. Then
Following (5) we have
for any unitary matrix . This means that , and there exists a number d such that . Thus,
For an arbitrary vector x such that we consider a unitary matrix such that . Then, this matrix can be split as follows:
where is an matrix. Since the matrix U is unitary, we have
Equation (5) determines the value of :
If we set
then Equation (8) holds. □
Definition 2.
The representation (8) is called [] the canonical representation of the metric tensor G. We say that the pair of functions parametrizes G.
3. Binary Operations Invariant with Respect to Unitary Transformations
In this section, we define binary operations with given smooth metric tensors G satisfying condition (5), and conditions on the function necessary and sufficient for the existence of such operations.
In (8) the matrix function is a metric tensor. Therefore the functions , should be non negative. We impose some smoothness conditions: the matrix function G is differentiable. In terms of the functions , it means that the functions , are differentiable, and . If we multiply G by a positive number, then all properties of the object remain the same. Therefore, we further impose the following assumption.
Assumption 2.
The functions , in (5) are differentiable, , for all , and .
A binary operation ⊕ in is defined as follows []. Consider a manifold with metric tensor G. Denote by the exponential mapping from a point a at vector v, denote by the logarithmic mapping: such that , and is the parallel transport of a vector v from zero to a. Then
In [] it is proved that if the metric tensor G is given by (3) and (4), then the operations ⊕ and f coincide,
for all .
Further we need a procedure for calculating for various functions [].
If , then . If , the . If , then we obtain by the following three steps []:
- Calculate a vector
- Calculate a vector
- Solve the following initial value problem:
The value is equal to ,
The value of is well-defined if and only if the vector belongs to the ball , that is, a solution of the initial value problem (10) exists on , and for all .
Theorem 2.
The value of is well-defined for all if and only if
Proof.
Pick a vector a such that . Let us try to calculate . The first two steps of the procedure give
In step 3, the values of and are parallel. Therefore, a solution has the form , where is a scalar function. In this case
and Equation (10) take the form
Therefore
and
Since and , we have
and
Inequality (12) implies that a solution of the initial value problem (10) on the interval such that for all does not exist. Hence is not defined.
Conversely, assume that equality (11) holds. Consider arbitrary vectors , and a solution x of the initial value problem (10). We need to show that for all . Since is a geodesic in with metric tensor G, the function is constant for . The constant is equal to . At the same time,
Therefore, for all we have
Owing to (11) we have for all . □
Following this result, we further assume that condition (11) holds.
Assumption 3.
The function satisfies the condition
We now define a scalar multiplication in associated with the operation ⊕ [] such that
for all and . Let h be the function , given by
Owing to Assumptions 2 and 3, the function h is an increasing bijection . Therefore it has an inverse, which is also an increasing bijection, and which is denoted by . The multiplication by a number is given as follows: for all we have if , and for non zero
It is straightforward that this operation satisfies conditions (14) []. Notice that as for every owing to Assumption 3.
We denote by the distance between points a and b in with a canonical metric tensor G. That is,
where the infimum is taken over the set of smooth curves x such that and .
The relation between the distance function and the function h is given as follows [].
Theorem 3.
For all
In particular, for both Einstein addition and Möbius addition we have []
Definition 3.
Let , : be functions that satisfy Assumptions 2 and 3, and let G be the canonical metric tensor (8) parametrized by . Then we say that the binary operation ⊕, defined in steps 1–3, is generated by G, or generated by the pair .
Lemma 2.
Let be an operation generated by a pair of functions satisfying Assumptions 2 and 3. Then for every unitary matrix U and all we have
Proof.
We apply the steps 1–3 of the procedure defining . Assume , , and are the vectors and the function that are calculated on the steps 1–3 for . Assume , and are the corresponding values for the sum . Then, clearly,
Therefore,
□
4. Isomorphic Operations
In this section, we consider necessary and sufficient conditions under which binary operations in the balls and with are isomorphic.
Assume . Let G, be canonical metric tensors in and generated by the pairs of functions and respectively. Assume the pairs and satisfy Assumptions 2 and 3. Denote by ⊕, the binary operations generated by the pairs of functions and respectively.
Definition 4.
Operations ⊕ in and in are isomorphic if there exists a smooth bijection φ: such that for all
and
for all unitary matrices U and all vectors .
The function φ is said to be the bijection of the isomorphism between the operations ⊕ and .
Remark 1.
Assume φ is an isomorphism of operations ⊕ and . Then the following conditions on φ hold.
- 1.
- for all .
- 2.
- .
- 3.
- The function is also an isomorphism between the operations ⊕ and .
Proof.
Condition (20) with implies
Therefore, the functions satisfying (20) are odd. Besides, for all unitary matrices U. Therefore, . The last property follows from the fact that , for all , and for all . □
Remark 2.
The relation of isomorphisms is symmetric: if operations ⊕ and are isomorphic with a bijection φ, then operations and ⊕ are isomorphic with the bijection .
Proof.
The following theorem asserts that the relation of isomorphism is transitive.
Theorem 4.
Assume operations in and in are isomorphic with an isomorphism : , and operations and in are isomorphic with an isomorphism : . Then operations and are isomorphic with an isomorphism : .
Proof.
For all we have
and for all unitary matrices U and all we have
□
Owing to property (20) the isomorphisms have the following representation.
Lemma 3.
Assume φ is a bijection of an isomorphism between an operation in generated by a pair of functions satisfying Assumptions 2 and 3 and an operation in generated by the pair of functions also satisfying Assumptions 2 and 3. Then there exists a smooth bijection : and a number such that for all we have
Proof.
Let and a vector e be the first unit vector: . Consider an arbitrary unitary matrix and an unitary matrix . Then . Apply condition (20):
If is not parallel to e, then there exists a matrix such that Equation (24) is wrong. Therefore the vectors and are parallel: there exists a number such that , where . From (19) it follows that . Therefore . Due to continuity of the bijection the number s is the same for all . Set
Then
For an arbitrary vector we find a unitary matrix U such that
Then
The function is smooth since the function is smooth. □
Remark 3.
Assume operations ⊕ and are isomorphic. Then there exists an isomorphism φ between these operations that has the representation (23) with .
Proof.
Further, without loss of generality, we assume that in the representation (23) of isomorphisms we have .
The following definition is useful.
Definition 5.
We say that a function φ: is determined by an increasing bijection : if for all
Assume is a bijection of an isomorphism between ⊕ and . How can the functions be found in terms of , and ?
Theorem 5.
Assume binary operations ⊕ in and in with are generated by the pairs of functions and respectively.
Then the operations , ⊕ are isomorphic if and only if there exists an increasing smooth bijection : such that , and for all
Proof.
First we prove the necessity. Assume the operations and ⊕ are isomorphic, and is the corresponding isomorphism satisfying condition (25). For every non zero vector consider an matrix such that
Then
is the metric tensor generated by the pair . A similar relation concerns the operation : there exists an matrix such that
and
is the metric tensor generated by the pair . Then
and
Since , and , we get
Hence,
Since is the metric tensor generated by the pair of functions , we have
The necessity is thus proved.
We now prove the sufficiency. Assume : is a smooth bijection, , and conditions (26) and (27) hold. Define a function by Equation (25). Then is a smooth bijection . Denote by the binary operation in given by
for all . The theorem will be proved if we show that . To this end it is sufficient from [] to prove that the metric tensors associated with these two operations are equal. We have
Therefore
Taking into account that the metric tensor G associated with the operation ⊕ is equal to
we can calculate the metric tensor associated with the operation :
where
It is straightforward to check that the metric tensor has the form (8) parametrized by the functions
These functions coincide with the functions and respectively. Hence, the metric tensors associated with the operations and coincide. According to the uniqueness of binary operations associated with the same metric tensor [], we have .
Following definition (34) of the operation , this operation is isomorphic to the operation ⊕ with the isomorphism . □
Assume is a family of smooth increasing bijections with and such that . Assume there exists a limit in : , which we denote by . Assume the function is a smooth increasing bijection . Assume an operation ⊕ is generated by a pair . For all consider the functions
and operations defined as follows: for all
Then the operations are isomorphic to the operation ⊕ for all . Further, we show that the limiting operation may not be isomorphic to the operation ⊕.
5. Gyrogroups
In this section we consider a simple way to get gyrogroups via bijections. We use the following definitions [].
Definition 6.
A set S with a binary operation ⊕ is called a groupoid. A groupoid is called a gyrogroup if its binary operation satisfies the following axioms.
- 1.
- There is at least one element, 0, in S such thatfor all .
- 2.
- There is an element satisfying (36) such that for every there is an element (called a left inverse of a) such that
- 3.
- For every there is an automorphism : of the groupoid S such that for every we have
- 4.
- The operator : possesses the following property:for all .
Notice [] that the left identity 0 is also a right identity, and the left inverse of a is also a right inverse of a: , for all .
Theorem 6.
Assume is a gyrogroup, is a set of elements, and φ: is a bijection. Then for the binary operation given by
the groupoid is a gyrogroup.
Proof.
We have to show that axioms 1–4 in Definition 6 are satisfied.
- Pick an element satisfying condition 2. For every we haveHence, is an element satisfying axiom 1 for the groupoid .
- For every we haveTherefore, axiom 2 is satisfied for the groupoid .
- For every let be the automorphism of from the property 3. For every setThenTherefore, is an automorphism of . Besides,Therefore, axiom 3 is satisfied for the groupoid .
- For every we haveTherefore, axiom 4 is satisfied for the groupoid . □
Definition 7.
A gyroproup is said to be gyrocommutative if for all
Theorem 7.
Assume is a gyrocommutative gyrogroup, is a set of elements, and φ: is a bijection. Then for the binary operation , given by
the gyroproup is gyrocommutative.
Proof.
For all we have
Hence, axiom (47) is satisfied. □
We can apply these results to balls with . Notice that for Euclidean addition (and hence for all additions isomorphic to Euclidean addition) the operation is equal to identity. For Einstein addition (and hence for all additions isomorphic to Einstein addition) the operation is a multiplication by a unitary matrix.
Here we encounter the following open problem. Does there exist a binary operation generated by a pair of functions satisfying Assumptions 2 and 3, which is not isomorphic to Einstein addition and not isomorphic to Euclidean addition?
6. Operations Isomorphic to Euclidean Addition
The metric tensor of Euclidean addition in is equal to the identity. This means that , in the canonical representation (8) of the Euclidean metric tensor.
The following theorem gives necessary and sufficient conditions for a binary operation generated by a pair of functions to be isomorphic to Euclidean addition.
Theorem 8.
Let G be a canonical metric tensor in parametrized by a pair of functions satisfying Assumptions 2 and 3. Let ⊕ be a binary operation generated by G. Then the operation ⊕ is isomorphic to Euclidean addition if and only if for all we have
If an operation ⊕ is isomorphic to Euclidean addition, then there exists a positive number t such that the bijection φ: of the isomorphism between ⊕ and Euclidean addition is given by
Proof.
According to Theorem 5 the operation ⊕ is isomorphic to Euclidean addition if and only if there exists a differentiable bijection : such that , and for all we have
and
Assume such a function exists, and let . Solving Equation (52) for yields
Therefore
Now assume that (50) holds. Then for all we have
For an arbitrary positive number t define
According to Theorem 5 the isomorphism : is given by
□
Example 1.
Let ⊕ be a binary operation with a canonical metric tensor G parametrized by a pair of functions satisfying Assumptions 2 and 3. Let ⊕ be isomorphic to Euclidean addition in . If
then, according to Theorem 8, we have
If
then, according to Theorem 8, we have
7. Operations Isomorphic to Einstein Addition
The metric tensor of Einstein addition in is canonical, having the form (8) with parameters
Assume and a pair of functions satisfies Assumptions 2 and 3. Consider a canonical metric tensor G parametrized by this pair. Let ⊕ be the binary operation generated by . Then we raise the following question.
What are the conditions on a pair of functions necessary and sufficient for the operation ⊕ to be isomorphic to Einstein addition ?
Theorem 9.
An operation ⊕ generated by a pair of functions satisfying Assumptions 2 and 3 is isomorphic to Einstein addition if and only if and there exists a positive number C such that
for all .
Proof.
First we prove the necessity. Assume the operation ⊕ is isomorphic to Einstein addition. By Theorem 5 with , , there exists a differentiable bijection : such that and
for all . Since is a bijection , we have , and therefore
Now we prove the sufficiency. According to Theorem 5, it suffices to show the existence of a bijection : such that Equations (60) and (61) hold. Set . Define
Owing to Assumption 2 we have and for all . Then and (58) implies that the function is strictly increasing. Since and , the function is an increasing bijection . Now we check Equation (60):
and Equation (61):
The sufficiency is proved. The isomorphism is given by (63). □
Denote
Lemma 4.
Identity (58) holds if and only if for all we have
Proof.
We recall that the function
is used in the definition of the operation ⊗ of multiplication by numbers []. In particular, for all numbers t and vectors we have
Lemma 5.
Corollary 1.
Proof.
The statement follows from Lemmas 4 and 5, and Theorem 9. □
Example 2.
Let
Example 3.
Let
Theorem 10.
Let ⊕ be a binary operation in generated by a pair of functions satisfying Assumptions 2 and 3. Then the operation ⊕ is isomorphic to Einstein addition in if and only if the operation ⊕ vis isomorphic to an operation in generated by the functions
The operation is isomorphic to the operation with an isomorphism φ: determined by the scalar function
Proof.
First, we show that the operation is isomorphic to Einstein addition in , which is generated by the functions
We define a bijection : as follows. Let
Then
According to Theorem 5 the operations and are isomorphic. Following Theorem 4 the operations ⊕ and are isomorphic if and only if the operations ⊕ and are isomorphic. □
Notice that the operation is isomorphic to Einstein addition with an isomorphism determined by the function
Theorem 11.
The operation in generated by the functions
is isomorphic to the operations in generated by the functions
for all . The corresponding isomorphism φ: is determined by the function
Proof.
We have
The statement of the theorem follows from Theorem 5. □
Theorem 12.
The operation in generated by the functions
is isomorphic to Einstein addition with an isomorphism φ: determined by the scalar function
Proof.
The proof follows from Theorems 4, 10 and 11. □
For every positive number t let the function : be given by
Then for all
Direct calculations show that
The operations with are isomorphic to each other and to Einstein addition.
Thus, any study of hyperbolic geometry in with Einstein addition is equivalent to a corresponding study of hyperbolic geometry in with the binary operation
The limit in (71) yields Euclidean addition, which is not isomorphic to Einstein addition (since, for instance, the fact that Einstein addition is not commutative).
The operation of scalar multiplication is defined according to general rules []. Since
we get
For , a not parallel to b, the co-gyroline
is a Euclidean line in :
where is a vector orthogonal to a, and is a two dimensional plane containing both a and b.
For , , the gyroline
is a hyperbola in that lies on a plane containing a and b. In particular, if , , then
The ratio of the coefficients of a and b tends to as :
These hyperbolas tend to lines as .
Theorem 13.
Let ⊕ be a binary operation in generated by a pair of functions satisfying Assumptions 2 and 3 and such that for all u. Then the operation ⊕ is isomorphic to Einstein addition if and only if and there exists a positive number t such that
for all .
Proof.
Following Theorem 5, it is sufficient to show that there exists a differentiable increasing bijection : such that for all
if and only if the function has the form (72). Let . Then Equation (74) can be integrated,
If , then
Remark 4.
Operations in generated by the functions and satisfying condition (72) are isomorphic for different positive numbers t to the operation with . The isomorphism is a simple stretching: .
8. Operations Isomorphic to Einstein Addition and Having the Same Function
Then
and
Let be the binary operation generated by the canonical metric tensor G parametrized by the functions , given in (77) and (78). Following Theorem 9, the operations are isomorphic to Einstein addition for all . Moreover, for the function
we have
where is Einstein addition. The goal of this section is to find a closed form for the operation . From (79) we get
For every vector let
Then
Taking into account that
and
we get
For the case we have and is Einstein addition:
According to Theorem 9, the operations are isomorphic to Einstein addition for all .
In the limit of (80) as , we have the operation :
where for all . The operation is isomorphic to Euclidean addition with an isomorphism : , given by
9. Isomorphic Operations with the Same Function
Let . As in Section 8, we consider
- (i)
- a pair of functions satisfying Assumptions 2 and 3, a canonical metric tensor G and a binary operation ⊕ in generated by , and
- (ii)
- a pair of functions satisfying Assumptions 2 and 3, a canonical metric tensor and a binary operation in generated by .
Our goal is to answer the following question. How to characterize metric tensors G, if the operations ⊕, are isomorphic, and ? Further in this section we assume .
According to Theorem 5 the operations ⊕ and are isomorphic, if and only if there exists an increasing smooth bijection : such that , and
for all . We can solve Equation (81) for the function , and substitute this function into Equation (82) to find a relation between the functions and . Thus, Theorem 5 for our case when has the following form.
Theorem 14.
Assume the pairs of functions , and satisfy Assumptions 2 and 3. Denote by ⊕ and the binary operations in generated by the pairs of functions and respectively. Assume a differentiable bijection : is such that , and
for all .
Then the operations ⊕ and are isomorphic if and only if
for all .
The positive number can be chosen arbitrarily. For each such t Equation (83) defines a function . Then from (84) we get the corresponding function . Hence, we get a one parameter set of functions such that the binary operations generated by pairs are isomorphic. It is remarkable that in the limit we get operations isomorphic to Euclidean addition. Let us consider two special cases.
Special case 1. Assume
Denote . Then Equation (83) may be solved for :
Theorem 14 states that the operations ⊕ and are isomorphic if and only if
Assume, as for Einstein addition,
Then
If , then we obviously get and . If , then for all , and for the corresponding limit we have:
The functions satisfy condition (50). According to Theorem 8, a binary operation generated by the canonical metric tensor (8) with the functions is isomorphic to Euclidean addition.
Assume
Then
The binary operations generated by pairs are isomorphic to Einstein addition, and not isomorphic to Euclidean addition.
Special case 2. Assume
Solve Equation (83) for :
Calculate the factor on the right hand side of (84):
According to Theorem 14 an operation generated by the pair is isomorphic to an operation generated by the pair if and only if there exists a positive number t such that
for all .
For Möbius addition we have
Direct calculations show that the binary operation generated by a pair is isomorphic to the binary operation generated by the pair if and only if there exists a non-zero number t such that
If then obviously and . The corresponding binary operations are isomorphic to Möbius addition (and hence isomorphic to each other and isomorphic to Einstein addition) for all . Therefore these operations are not isomorphic to Euclidean addition in . We denote the limit by :
According to Theorem 8, the operation generated by the pair is isomorphic to Euclidean addition, since
for all .
10. Isomorphic Operations with the Same Function
Let and be pairs of functions satisfying Assumptions 2 and 3. Let ⊕, be binary operations in generated by the pairs and respectively. Notice that the second function in both pairs is the same. The distance functions in the space with operations ⊕ and are completely determined by the same function , and therefore they coincide.
Now we consider the problem of finding a relation between ⊕ and .
Theorem 15.
Canonical metric tensors of binary operations ⊕ and have the same function and the binary operations ⊕ and are isomorphic if and only if there exists a positive number t such that
for all , where ⊗ is a scalar multiplication in the space with a binary operation ⊕.
Proof.
Since , we have , where is the inverse of h. According to the definition of multiplication by numbers [] we have
for all . Let
Then and
Direct calculations show that
Therefore the metric tensor of the operation has the form (8) parametrized by the functions
According to Theorem 14 the operations ⊕ and are isomorphic. By definition,
Here we employed the change of variable: . Now differentiate identity (91) with respect to p to get
for all .
Conversely, assume that the operations ⊕ and are isomorphic, and that the functions and coincide. Then according to Theorem 5, Definition 4 and Remarks 1 and 2 there exists a differentiable bijection : such that ,
for all and
for all , where
for all . Set . Define the function
Then
Therefore,
for all . Hence,
for all . Since the function is invertible, we have
for all . Now we apply Equation (93) obtaining
for all . □
Example. If ⊕ is Einstein addition, and is Möbius addition, then
In this case Equation (88) holds with .
11. Conclusions
In this paper we consider smooth binary operations invariant with respect to unitary transformations. We prove that such binary operations have tensors (8) parametrized by non negative functions , . It is shown that binary operations are well-defined iff a special additional constraint (11) on functions hold. We present necessary and sufficient conditions for two operations to be isomorphic. We give necessary and sufficient conditions for binary operations to be isomorphic to Einstein addition, and, separately, to Euclidean addition. We also pointed out necessary and sufficient conditions for two operations having the same function , or, separately, the same function , to be isomorphic.
Future research may concern a development of calculus (integration, differentiation, differential equations) in gyrospaces, finding necessary and sufficient conditions for binary operations to give rise of gyrogroups, an integration of geodesic equations in the polar coordinates, and a description of binary operations with respect to which the Maxwell equations are invariant.
Funding
This research received no external funding.
Conflicts of Interest
The author declares no conflict of interest.
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