Orthogonal gyrodecompositions of real inner product gyrogroups

: In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, Möbius, Proper Velocity, and Chen’s gyrogroups. This leads to the study of left (right) coset partition of a real inner product gyrogroup induced from a subgyrogroup that is a ﬁnite dimensional subspace. As a result, we obtain gyroprojectors onto the subgyrogroup and its orthogonal complement. We construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subgyrogroup. With the algebraic structure of the decompositions, we study ﬁber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exempliﬁed for the aforementioned gyrogroups.


Preliminaries
Definition 1 (Gyrogroups). A nonempty set G, together with a binary operation ⊕ on G, is called a gyrogroup if it satisfies the following axioms.
(G1) There exists an element 0 ∈ G such that 0 ⊕ a = a for all a ∈ G. (G2) For each a ∈ G, there exists an element b ∈ G such that b ⊕ a = 0. (G3) For all a, b ∈ G, there is an automorphism gyr[a, b] ∈ Aut G such that for all c ∈ G. It can be proved that the element 0 in (G1) is also a right identity and is unique, called the gyrogroup identity. Further, the element b in (G2) is also a right inverse of a and is unique, called the inverse of a, written a. A gyrogroup (G, ⊕) is said to be gyrocommutative if it satisfies the following gyrocommutative law: for all a, b ∈ G.
To capture useful analogies between gyrogroups and groups there is defined as a second binary operation in G, called the coaddition, which is denoted by and given by for all a, b ∈ G (cf. Definition 2.7 of [2]). We have also that a b = a ( b).
The main cancellation laws in a gyrogroup (G, ⊕) are a ⊕ ( a ⊕ b) = b left cancellation law (3) (b a) a = b first right cancellation law (4) (b a) ⊕ a = b second right cancellation law.
Using the left gyroassociative law given in (G3) and the left cancellation law (3) we can write the following gyrator identity gyr[a, b]c = (a ⊕ b) ⊕ (a ⊕ (b ⊕ c)) (6) from which it follows that gyr[a, b]0 = 0, i.e. gyrations are gyroautomorphisms of G that preserve the identity element. The next theorem shows the unique solution of the two basic gyrogroup equations.
Theorem 1. (see [2]) Let (G, ⊕) be a gyrogroup, and let a, b ∈ G. The unique solution of the equation a ⊕ x = b is x = a ⊕ b, and the unique solution of the equation x ⊕ a = b is x = b a.
Definition 2 (Real inner product gyrovector spaces, [2]). A real inner product gyrovector space (G, ⊕, ⊗) (gyrovector space, in short) is a gyrocommutative gyrogroup (G, ⊕) that obeys the following axioms: (A1) G is a subset of a real inner product vector space V, G ⊆ V, from which it inherits its inner product · and norm · , which are invariant under gyroautomorphisms; that is, for all a, b, u, v ∈ G. (A2) G admits a scalar multiplication ⊗ satisfying for all r, r 1 , r 2 ∈ R, a ∈ G, |r| ⊗ a r ⊗ a = a a (V5) gyr[u, v](r ⊗ a) = r ⊗ gyr[u, v]a (V6) gyr[r 1 ⊗ v, r 2 ⊗ v] = I, I is the identity map (A3) The set G = {± a : a ∈ G} admits a real vector space structure ( G , ⊕, ⊗) such that for all a, b ∈ G, r ∈ R.
Unlike in the vector case, gyroaddition ⊕ does not in general distribute with scalar multiplication. However, gyrovector spaces possesses a monodistributive law given by r ⊗ (r 1 ⊗ a ⊕ r 2 ⊗ a) = r ⊗ (r 1 ⊗ a) ⊕ r ⊗ (r 2 ⊗ a) for all r, r 1 r 2 ∈ R, a ∈ G.
The motions of a gyrovector space (G, ⊕, ⊗) are all its left gyrotranslations L a : x → a ⊕ x, where a, x ∈ G, and its automorphisms τ ∈ Aut(G, ⊕, ⊗) satisfying τ(r ⊗ a) = r ⊗ τ(a) (8) τ(a · b) = τ(a) · τ(b) (9) for all a, b ∈ G, r ∈ R. A gyrovector space is a gyrometric space with a gyrodistance given by that satisfies the gyrotriangle inequality (see Theorem 6.9 in [2]): Moreover, a gyrovector space is also cogyrometric with a cogyrodistance d (a, b) given by that satisfies the cogyrotriangle inequality (see Theorem 6.11 in [2]): Curves on which the gyrotriangle inequality reduces to an equality are called gyrolines, while curves on which the cogyrotriangle inequality reduces to an equality are called cogyrolines. Gyrolines and cogyrolines play an important role in hyperbolic analytic geometry regulated by the gyrovector space structure. For distinct elements a and b in G, the two distinct hyperbolic line expressions are defined by Gyroline or the hyperbolic line, Cogyroline or the hyperbolic dual line. (14) Gyrolines and cogyrolines are uniquely determined by any two distinct points contained by them. Therefore, expressions in (14) can be replaced by Cogyrolines are natural objects in gyrovector spaces and also have important properties. We mention the following: P5. Two cogyrolines that share two distinct points are coincident. (Theorem 6.53) P6. Cogyrolines admit parallelism. (Theorem 6.65) P7. The cogyrotranslation d of the cogyroline (t ⊗ b) ⊕ a is again a cogyroline, that is, ((t ⊗ b) ⊕ a) d is a cogyroline. Moreover, the cogyroline and the cogyrotranslated cogyroline are parallel. (Theorem 6.66) P8. Cogyrolines are cogyrogeodesics. (Theorem 6.78) For the basic theory of gyrogroups and gyrovector spaces the interested reader is referred to [2,3,9]. Next we present the definition of subgyrogroups and L-subgyrogroups. For more details about factorization of gyrogroups by L-subgyrogroups, Cayley's Theorem, and isomorphisms theorems, see [8].

Orthogonal decompositions
Trying to be as general as possible we define next a real inner product gyrogroup contained in a real inner product space using the general addition (15).
Definition 5 (Real inner product gyrogroups). A gyrogroup (G, ⊕) contained in a real inner product space (V, +, ·) is called a real inner product gyrogroup if it contains the zero of V as the gyrogroup identity and the binary operation ⊕ is given by for all a, b ∈ G, where φ i : G × G → R is a map with φ i (a, a) = 0 for all a ∈ G, i = 1, 2, and ⊕ satisfies the gyrogroup axioms. Moreover, we say that the real inner product gyrogroup (G, ⊕) admits orthogonal decompositions if the following additional conditions are satisfied: is the unique solution of the system of equations: We remark that since the general addition a ⊕ b defined in (15) satisfies the gyrogroup axioms of Definition 1 then the functions φ 1 and φ 2 have to satisfy some functional equalities and relations. In particular, since, 0 ⊕ b = b and a ⊕ 0 = a, for all a, b ∈ G, then it implies that φ 1 (a, 0) = 1 and φ 2 (0, b) = 1, for all a, b ∈ G. Regarding the conditions (H1) and (H2), the first condition says essentially that G consists of a ball with finite or infinite radius in the carrier space V, while the second condition allows us to obtain orthogonal decompositions of G, and corresponding left and right cosets of G.
Concrete prominent examples of real inner product gyrogroups that admit orthogonal decompositions are exhibited in Section 4: Euclidean Einstein, Möbius, Proper Velocity, and Chen's gyrogroups.
Throughout the remainder of this article, we assume that (G, ⊕) is a real inner product gyrogroup contained in the carrier inner product space (V, +, ·) satisfying conditions (H1) and (H2).

Unique decomposition and orthogonal gyroprojections
Let P be a finite-dimensional linear subspace of V and let P ⊥ be its orthogonal complement in V (that is, V = P ⊕ P ⊥ is an orthogonal direct sum). Define Theorem 3. If P is a finite-dimensional subspace of V, then P G and P ⊥ G are subgyrogroups of G.
Proof. Note that P G = ∅ since 0 ∈ P G . Let a, b ∈ P G . Then a, b ∈ P and a, b ∈ G. By (15), a ⊕ b ∈ P since P is closed under vector addition. By the closure property of G, a ⊕ b ∈ G. Hence, a ⊕ b ∈ P G . Note that 0 = a ⊕ ( a) = φ 1 (a, a)a + φ 2 (a, a) a, which implies a = − φ 1 (a, a) φ 2 (a, a) a since φ 2 (a, a) = 0. Hence, a ∈ P and so a ∈ P G . This proves that P G is a subgyrogroup of G. Since the proof does not make use of the assumption of being finite-dimensional, we conclude that P ⊥ G is a subgyrogroup of G as well.
According to Definition 4, P G is an L-subgyrogroup if gyr[a, h]P G = P G , for all a ∈ G and h ∈ P G . Since gyrations are isometries of (G, ⊕, ⊗) that preserve the norm inherited from (V, +, ·), the only gyrations that preserve P G are from the automorphism group of P G . Hence, P G is not an L-subgyrogroup.
A first fundamental result that we are going to show is the conditions under which the addition ⊕ in a real inner product gyrogroup is associative.
Proof. By the left gyroassociative law we have We know that isometries of a real inner product space (V, +, ·) that fixes the origin and an arbitrary element c are the isometries that belong to a linear subspace orthogonal to c or the trivial isometry (identity isometry) that belongs to a one-dimensional linear subspace of V. Therefore, since gyrations of (G, ⊕, ⊗) are isometries in the carrier space V, it follows that gyr[a, b]c = c if and only if a · c = 0 and b · c = 0 or a = λ ⊗ b for some λ ∈ R, i.e., either a and b belong to a linear subspace orthogonal to c or a and b are in a given one dimensional linear subspace. Note that by axiom (V6) in Definition 2 it follows that gyr[a, λ ⊗ a]=I.
The following theorem shows that any real inner product gyrogroup G can be decomposed into gyrosums of P G and P ⊥ G , where P is a finite-dimensional subspace of the carrier space V.
Theorem 5 (Unique Decomposition). Let P be a finite-dimensional subspace of V. For all c ∈ G, there are unique elements a, u ∈ P G , b, v ∈ P ⊥ G such that Proof. Let c ∈ G be arbitrary. Since G ⊆ V and V = P ⊕ P ⊥ , it follows that c = c 1 + c 2 with c 1 ∈ P and c 2 ∈ P ⊥ . Note that c 2 = c 1 + c 2 2 = c 1 2 + c 2 2 . Hence, c 1 ≤ c and c 2 ≤ c . By hypothesis (H1), c 1 ∈ G and c 2 ∈ G. Since c 1 + c 2 = c ∈ G and c 1 · c 2 = 0, we have by hypothesis (H2) that there exist scalars λ 1 = λ 1 (c 1 , c 2 ), λ 2 = λ 2 (c 1 , c 2 ) ∈ R such that (x = λ 1 c 1 , y = λ 2 c 2 ) ∈ G × G is the unique solution of the system of equations: Set a = λ 1 c 1 and b = λ 2 c 2 . Then a ∈ P G and b ∈ P ⊥ G . Furthermore, we obtain To prove uniqueness of the factorization, suppose that c = a ⊕ b with a ∈ P G , b ∈ P ⊥ G . Then and so c 1 = φ 1 (a , b )a and c 2 = φ 2 (a , b )b since the sum P ⊕ P ⊥ is direct. It follows that (a , b ) is a solution of (19). Hence, a = λ 1 c 1 = a and b = λ 2 c 2 = b.
We remark that the proof of the uniqueness of Theorem 5 could also be made using Theorem 4.
Corollary 1 (Orthogonal Decompositions). The following orthogonal decompositions of G hold: Using Theorem 5, we can define orthogonal gyroprojections of the gyrogroup (G, ⊕) onto the subgyrogroups P G and P ⊥ G , with respect to the decompositions G = P G ⊕ P ⊥ G and G = P ⊥ G ⊕ P G . In fact, each element c ∈ G admits the unique decomposition c = c 1 + c 2 = c 2 + c 1 , with c 1 ∈ P G and c 2 ∈ P ⊥ G and according to Theorem 5 four orthogonal gyroprojectors can be defined: such that where the superscripts and r stand for "left" and "right" and indicate that the gyroprojection is on the left or on the right of each decomposition. The following identities hold immediately: Moreover, we obtain the following orthogonal gyrodecompositions: Corollary 2. For all a ∈ P G and b ∈ P ⊥ G , we have Proof. By (15) and Theorem 5 we have which implies by the Unique Decomposition Theorem (Theorem 5) that Hence, using the definition of the gyroprojections (20) and (25) we obtain the identities Using similar reasonings we obtain also By Corollary 1 the gyrogroup (G, ⊕) has two unique decompositions G = P G ⊕ P ⊥ G and G = P ⊥ G ⊕ P G . The relation between them is given more precisely in the next theorem. Theorem 6. Let a, b ∈ (G, ⊕) such that a · b = 0. Then there exist nonzero scalars µ 1 (a, b) and µ 2 (a, b) such that Proof. Let a, b ∈ G such that a ⊥ b. Then by (15) we have Consider now c 1 = φ 1 (a, b)a and c 2 = φ 2 (a, b)b. Then c 1 ⊥ c 2 and by Theorem 5 there exist nonzero scalars Taking we obtain (26).

Corollary 3.
For all a ∈ P G and b ∈ P ⊥ G , we have Proof. From Theorem 6 we have:

Left and right coset spaces
Let H be a subgyrogroup of G. In contrast to groups, the relation obtaining an equivalence relation on G and leading to the notion of L-subgyrogroups. Since in our case P G is not an L-subgyrogroup, we cannot apply the results in [8]. In the case of real inner product gyrogroups that have attached a gyrovector space structure we are going to show that the gyroprojection Q (respectively, Q r ) induces an equivalence relation on G so that G can be written as a disjoint union of left (respectively, right) cosets with representatives from P ⊥ G . Let P be a finite-dimensional linear subspace of V. Define ∼ by for all a, b ∈ G.
Theorem 7. The relation ∼ defined by (29) is an equivalence relation on G.
For each b ∈ G, we denote by [b] the equivalence class containing b determined by the relation ∼ .
Theorem 8. Let P be a finite-dimensional linear subspace of V. Then the collection Proof. According to Lemma 1, we know that for all Denote by (G/P G , ∼ ) the set of all equivalence classes obtained by the equivalence relation ∼ . By Lemma 1 and Theorem 8 the coset space (G/P G , ∼ ) represents the collection {b ⊕ P G : b ∈ P ⊥ G }.

Corollary 4.
The set (G/P G , ∼ ) is a left coset space of G whose cosets are of the form b ⊕ P G with b ∈ P ⊥ G .
Next, we present the right counterpart of Theorem 8 defining first the equivalence relation that leads to the construction of the right cosets.
Let P be a finite-dimensional linear subspace of V. Define ∼ r by ) and stands for the coaddition in G defined in (2).
The relation ∼ r defined by (31) is an equivalence relation on G.
Proof. Let a, b, c ∈ G.
(Reflexive property) Since a = P (a) ⊕ Q r (a), it follows from Theorem 1 that P (a) = a Q r (a). Hence, a Q r (a) ∈ P G and so a ∼ r a.
(Symmetric property) Suppose that a ∼ r b. Then a Q r (b) ∈ P G and so there is an element belongs to P G . Hence, a ∼ r c.
For each b ∈ G, we denote by [b] r the equivalence class containing b determined by the relation ∼ r .
Theorem 10. Let P be a finite-dimensional linear subspace of V. Then the collection and so equality holds.
Denote by (G/P G , ∼ r ) the set of all equivalence classes obtained by the equivalence relation ∼ r .
Corollary 5. The set (G/P G , ∼ r ) is a right coset space of G whose cosets are of the form P G ⊕ b with b ∈ P ⊥ G .

Quotient gyrogroups and the isomorphism theorem
The coset spaces (G/P G , ∼ ) and (G/P G , ∼ r ) turn out to be gyrogroups, called left and right quotient gyrogroups by P G , respectively. Lemma 3. Let P be an arbitrary finite-dimensional subspace of V.
From Lemma 3 and the fact that P ⊥ G is a subgyrogroup of G, we obtain that the following operations on the coset spaces (G/P G , ∼ ) and (G/P G , ∼ r ) are well defined: In fact, the left and right coset spaces form gyrogroups, as shown in the following theorems.
Theorem 11. The left coset space (G/P G , ∼ ) forms a gyrogroup under the operation defined by (33).
Proof. The coset 0 ⊕ P G is a left identity in G/P G . For each a ⊕ P G ∈ G/P G with a ∈ P ⊥ G , the coset ( a) ⊕ P G lies in G/P G and is a left inverse of a ⊕ P G . For which proves that gyr[X, Y] is well defined. The proof that gyr[X, Y] fits the definition of a gyrogroup follows the same steps as in the proof of Theorem 29 of [9] with appropriate modifications.
Proceeding in a similar fashion, we obtain the right counterpart of Theorem 11. Definition 6. The gyrogroup (G/P G , ∼ ) in Theorem 11 is called a left quotient gyrogroup and the gyrogroup (G/P G , ∼ r ) in Theorem 12 is called a right quotient gyrogroup. We define the canonical projection mappings π and π r from G to the left and right coset spaces by Theorem 13 (The Isomorphism Theorem). Let P be an arbitrary finite-dimensional linear subspace of V. Then π restricts to a gyrogroup isomorphism from P ⊥ G to (G/P G , ∼ ) and π r restricts to a gyrogroup isomorphism from P ⊥ G to (G/P G , ∼ r ). Therefore, the following gyrogroup isomorphisms hold:

Geometric characterization of cosets
In this section, we assume that (G, ⊕) is a real inner product gyrogroup together with a scalar multiplication ⊗, turning (G, ⊕, ⊗) a gyrovector space . The next theorem gives a geometric characterization of the left equivalence classes b ⊕ P G with b ∈ P ⊥ G in terms of the automorphisms of (P G , ⊕, ⊗) acting on a given gyroline L g . Theorem 14. Let b ∈ P ⊥ G and let 0 = c ∈ P G be fixed. Then where the gyroline L g is the gyrogeodesic given by we know that L g is a gyroline in G in the plane defined by the vectors b and c. Now, we consider a ∈ P G arbitrary. Since it is always possible to find t ∈ R such that t ⊗ c = a , then there exists an automorphism τ ∈ Aut(G, ⊕, ⊗) such that a = τ(t ⊗ c) and τ leaves P G invariant and takes each element of P ⊥ G as a fixed point. This means that τ ∈ Aut(P G , ⊕, ⊗). In particular, since b ∈ P ⊥ G then τb = b, and consequently, τ −1 b = b. Then, by the invariance property P3 of gyrolines (described in In an analogous way, we can characterize the right equivalence classes P G ⊕ b with b ∈ P ⊥ G in terms of the automorphisms of (P G , ⊕, ⊗) acting on a given cogyroline L c . Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 23 May 2020 doi:10.20944/preprints202005.0371.v1 Theorem 15. Let b ∈ P ⊥ G and 0 = c ∈ P G be fixed. Then where the cogyroline L c is the cogyrogeodesic given by we know that L c is a cogyroline in G in the plane defined by the vectors b and c. Now, we consider a ∈ P G arbitrary. Then there exist t ∈ R and τ ∈ Aut(P G , ⊕, ⊗) such that a = τ(t ⊗ c) and τ leaves P G invariant. Thus, by property P3 of gyrolines since Hence, we get

Fiber bundles and sections of real inner product gyrogroups
We denote (G, X, π, Y) as a fiber bundle with base space X, fiber Y, and bundle map π : According with the two unique decompositions G = P G ⊕ P ⊥ G and G = P ⊥ G ⊕ P G , and using Lemma 1, Lemma 2, (24), and (28), we can define four different fiber bundle structures on (G, ⊕) with fiber bundle mappings given by We remark that the mappings π 3 and π 4 defined in (38) correct the definitions presented in [5].
It is easy to see that the first and the second bundles are trivial ones. The first bundle π 1 is isomorphic to the trivial bundle (P G × P ⊥ G , P ⊥ G , Q r , P G ), where Q r is the projection onto the second factor defined by: Hence, the following diagram commutes: where Φ 1 is the isomorphism between (G/P G , ∼ r ) and P ⊥ G given by Φ 1 (P G ⊕ b) = b, for any b ∈ P ⊥ G . All global sections of the first fiber bundle are given by for any continuous map g : The second bundle π 2 is isomorphic to the trivial bundle (P ⊥ G × P G , P ⊥ G , Q , P G ), where Q is the projection onto the first factor defined by: Indeed, the following diagram commutes where Φ 2 is the isomorphism between (G/P G , ∼ ) and P ⊥ G given by Φ 2 (b ⊕ P G ) = b, for any b ∈ P ⊥ G . All global sections of the second fiber bundle are given by for any continuous map g : In the third and fourth bundles we will consider the sections obtained from the quotient spaces (G/P ⊥ G , ∼ r ) and (G/P ⊥ G , ∼ ). In the third case if we consider for any a ∈ P G fixed the map τ with b ∈ P ⊥ G , which means that τ (1) a (G/P G , ∼ r ) = a ⊕ P ⊥ G is a left coset in (G/P ⊥ G , ∼ ), we obtain a global section. In fact, π 3 (τ (1) G . This means that the left cosets of (G/P ⊥ G , ∼ ) are global sections for (G/P G , ∼ r ). However, if we consider the map τ (2) a defined for any a ∈ P G \{0} by for b ∈ P ⊥ G , which means that τ (2) a (G/P G , ∼ r ) = P ⊥ G ⊕ a is a right coset in (G/P ⊥ G , ∼ r ), then by (38) we have given by (27) with the order of a and b being changed. Depending on the properties of the mapping µ 2 we can have global or local sections. If for a ∈ P G fixed we have a is only a local section for the fiber bundle defined by π 3 . The case a = 0 gives a global section since The conclusions obtained for the sections τ (1) a and τ (2) a are valid for the fiber bundles associated to the projection mappings π 1 and π 3 .
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 23 May 2020 doi:10.20944/preprints202005.0371.v1 In the fourth case we consider the sections τ (3) a and τ (4) a defined for any a ∈ P G by and τ (4) It is easy to see that τ a is a global section for π 4 since for any b ∈ P ⊥ G . Since τ a (G/P G , ∼ ) = P ⊥ G ⊕ a is a right coset of (G/P ⊥ G , ∼ r ) then the right cosets of (G/P ⊥ G , ∼ r ) are global sections for (G/P G , ∼ ). Concerning τ a we have for any a ∈ P G , where µ 1 (a, b) is given by (27). If for a ∈ P G fixed we have {µ 1 (a, b) a is only a local section for the fiber bundle π 4 . The case a = 0 gives a global section since for any b ∈ P ⊥ G . The conclusions obtained for the sections τ a and τ (4) a are valid for the fiber bundles associated to the projection mappings π 2 and π 4 .
We summarize in the next theorem the duality relations between left and right cosets obtained from the orthogonal factorization of G by P G and by P ⊥ G .
Theorem 16. Let P G be a subgyrogroup of (G, ⊕). The following relations hold: 1. The cosets of (G/P ⊥ G , ∼ ) are global sections for the quotient space (G/P G , ∼ r ). 2. The cosets of (G/P ⊥ G , ∼ r ) are global sections for the quotient space The cosets of (G/P ⊥ G , ∼ r ) are global sections for the quotient space (G/P G , ∼ ). 4. The cosets of (G/P ⊥ G , ∼ ) are global sections for the quotient space

Examples of real inner product gyrogroups
In this section, we show that four standard gyrogroups known in the literature are indeed real inner product gyrogroups that admit orthogonal decompositions.

Euclidean Einstein gyrogroup
The (Euclidean) Einstein gyrogroup [10] consists of the open unit ball in R n , for all u, v ∈ B, where , denotes the inner product in R n and γ u is the Lorentz factor given by Einstein addition satisfies the gamma identity In view of (46), we define for all u, v ∈ B. The gyrogroup identity of (B, ⊕ E ) is the zero vector 0. Further, E u = −u for all u ∈ B. Let u, v ∈ B. By the Cauchy-Schwarz inequality, Hence, 1 + u, v > 0. This implies that φ 1 (u, v) > 0 and φ 2 (u, v) > 0. Clearly, the Einstein gyrogroup satisfies (H1). Suppose that u, v ∈ B, u + v ∈ B, and u, v = 0. Then by (47) it is easy to see that the unique solution of the system of equations is given by λ 1 = λ 1 (u, v) = 1 and λ 2 = λ 2 (u, v) = γ u . Thus, (x = u, y = γ u v) is the solution to the system (16), where c 1 = u and c 2 = v. We claim that y ∈ B. To prove the claim, suppose to the contrary that y ∈ B. Hence, γ u v ≥ 1. This implies γ u v ≥ 1 and so v 2 1 − u 2 ≥ 1. Hence, u + v 2 = u 2 + v 2 ≥ 1, contrary to the assumption that u + v ∈ B. Thus, y ∈ B. Hence, the Einstein gyrogroup satisfies (H2). Now, let P be a linear subspace of R n , P B = P ∩ B, and P ⊥ B = P ⊥ ∩ B. Then R n = P ⊕ P ⊥ and P B and P ⊥ B are subgyrogroups of the Einstein gyrogroup (B, ⊕ E ). The next three theorems are consequences of Theorems 5, 8, 10, 13 obtained in Section 2.
Theorem 17. Let P B be a subgyrogroup of (B, ⊕ E ). For all w ∈ B such that w = u + v, with u ∈ P B and v ∈ P ⊥ B , the unique orthogonal decompositions of w according to ⊕ E are given by where λ 1 (u, v) = 1 and λ 2 (u, v) = γ u , Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 23 May 2020 doi:10.20944/preprints202005.0371.v1 As a consequence B = P B ⊕ E P ⊥ B and B = P ⊥ B ⊕ E P B . By (20) the orthogonal gyroprojectors of the Einstein gyrogroup (B, ⊕ E ) onto the subgyrogroups P B and P ⊥ B are given by Theorem 18. Let P B be a subgyrogroup of (B, ⊕ E ).
Theorem 19. Let P B be a subgyrogroup of (B, ⊕ E ). Then the following are gyrogroup isomorphisms: In order to see a remarkable connection between gyroprojections and Euclidean projections, let us refer to a version of Einstein addition defined on an open ball of R n of radius t, where t is a positive number. Set Recall that the t-Einstein addition is defined by for all u, v ∈ B t and γ u is redefined by γ u = 1 − u 2 t 2 − 1 2 . When t is arbitrarily large (that is, t → +∞), B t expands to the whole space R n , ⊕ E t reduces to ordinary vector addition of R n , and γ u → 1. Therefore, the orthogonal gyroprojections of the t-Einstein gyrogroup reduce to ordinary projections of R n : for all u ∈ P and v ∈ P ⊥ . Finally, we analyze the fiber bundles and sections of the Einstein gyrogroup arisen from the two orthogonal decompositions B = P B ⊕ E P ⊥ B and B = P ⊥ B ⊕ E P B . First we give the relation between these two decompositions according to Theorem 6.
Theorem 20. Let u, v ∈ (B, ⊕ E ) such that u, v = 0. Then where Now, we analyze the sections τ (2) u and τ (4) u defined by (40) and (43), respectively, with the changes a ↔ u ∈ P B and b ↔ v ∈ P ⊥ B . Since for any u ∈ P B \{0} it turns out that µ 1 Then for any u ∈ P B \{0}, the section τ (2) u is a local section for the fiber bundles π 1 and π 3 defined in (38). In the case when u = 0 the section τ (2) 0 is a global section as seen in (41). Concerning the section τ (4) u , since for each u ∈ P B we have that Hence, for any u ∈ P B , it follows that τ (4) u is a global section for the fiber bundles π 2 and π 4 defined in (38).
In Figure 1 we show the plots of µ 1 (u, v) v and µ 2 (v, u) v , with u , v < 1, that explain the different behavior between local and global sections. By the results obtained in Section 3 and the previous conclusions we summarize in the next theorem the duality relations between left and right cosets obtained from the orthogonal factorization of B by P B and by P ⊥ B , for the Einstein gyrogroup.
Theorem 21. Let P B be a subgyrogroup of (B, ⊕ E ). The following duality relations hold: 1. The cosets of (B/P ⊥ B , ∼ r ) are global sections for the quotient spaces (B/P B , ∼ ) and (B/P B , ∼ r ), and vice versa. To visualize left and right cosets we restrict now to the 3-dimensional space R 3 . In R 3 the nontrivial linear subspaces are of dimension 1 (straight lines passing through the origin) or dimension 2 (planes passing through the origin). Let L e 3 = {(0, 0, x 3 ) : x 3 ∈ R} be the straight line that passes through the origin and the North Pole e 3 = (0, 0, 1) and let D e 3 = {(x 1 , x 2 , 0) : x 1 , x 2 ∈ R} be the plane that passes through the origin and is perpendicular to e 3 . The restriction of L e 3 and D e 3 to B 3 = {x ∈ R 3 : x < 1} will be denoted by L   For higher dimensions (n > 3) we can have cosets with higher codimension, which are surfaces of revolution obtained from the action of the automorphism group of (P B , ⊕, ⊗) on the gyrolines and cogyrolines shown in Figures 2 and 3. Thus, all the information of these surfaces is encoded in the projection on an adequate plane.

Euclidean Möbius gyrogroup
The (Euclidean) Möbius gyrogroup [11] consists of the open unit ball B in R n endowed with Möbius addition ⊕ M defined by for all u, v ∈ B. The gyrogroup identity of (B, ⊕ M ) is the zero vector 0, and M u = −u for all u ∈ B. A generalization of the Möius addition into the ball of any real inner product space was obtained in [12]. Möbius addition satisfies the gamma identity In view of (55), we define for all u, v ∈ B. Let u, v ∈ B. Using the Cauchy-Schwarz inequality, we have Clearly, the Möbius gyrogroup satisfies (H1). Suppose that u, v ∈ B, u + v ∈ B, and u, v = 0.
Theorem 22. Let P B be a subgyrogroup of (B, ⊕ M ). For all w ∈ B such that w = u + v, with u ∈ P B and v ∈ P ⊥ B , the unique orthogonal decompositions of w according to ⊕ M are given by Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 23 May 2020 doi:10.20944/preprints202005.0371.v1 where We remark that the special cases u = 0 or v = 0 are included in the general solution (63). As a consequence of Theorem 22, for each w ∈ B, if w = u + v, with u ∈ P and v ∈ P ⊥ is the unique decomposition of w, according to the orthogonal direct sum decomposition R n = P ⊕ P ⊥ , then the orthogonal gyroprojections of the Möbius gyrogroup (B, ⊕ M ) onto the subgyrogroups P B and P ⊥ B are given by Theorem 23. Let P B be a subgyrogroup of (B, ⊕ M ).
Theorem 24. Let P B be a subgyrogroup of (B, ⊕ M ). Then the following are gyrogroup isomorphisms: For each positive number t, recall that the t-Möbius addition is defined by for all u, v ∈ B t . Let P be a subspace of R n . For each w ∈ B t , if w = u + v is the unique expression of w, according to the orthogonal direct sum decomposition R n = P ⊕ P ⊥ , with u ∈ P and v ∈ P ⊥ , then the orthogonal gyroprojections of the t-Möbius gyrogroup after rescaling u → u t , v → v t , w → w t are given by When t → +∞, ⊕ M t reduces to ordinary vector addition of R n , P M t and P r M t reduce to the ordinary projection P(u + v) = u, and Q M t and Q r M t reduce to the ordinary projection Q(u + v) = v. Further, the equivalence relation (29) reflects the Euclidean left coset relation: if and only if − v + u ∈ P and the equivalence relation (31) reflects the Euclidean right coset relation: Now, we analyze the fiber bundles and the sections of the Möbius gyrogroup arisen from the two orthogonal decompositions B = P B ⊕ M P ⊥ B and B = P ⊥ B ⊕ M P B . First we give the relation between these two decompositions according to Theorem 6.
Concerning µ 2 (v, u) v it turns out that for each u ∈ P B , the function µ 2 (v, u) v is strictly increasing such that µ 2 (0, u) 0 = 0 and lim v →1 µ 2 (v, u) v = 1. Therefore, we conclude that and, consequently, for any u ∈ P B the section τ (4) u is a global section for the fiber bundles π 2 and π 4 defined in (38).
In Figure 4 we show the plots of µ 1 (u, v) v and µ 2 (v, u) v , with u , v < 1, for the case of the Möbius gyrogroup.

Proper velocity gyrogroup
The (Euclidean) Proper Velocity (PV) gyrogroup [3] consists of the n-dimensional Euclidean space R n endowed with PV addition ⊕ U defined by for all u, v ∈ R n , where β u is the beta factor given by β u = 1 1 + u 2 for all u ∈ R n . In view of (70), we define for all u, v ∈ R n . The gyrogroup identity of (R n , ⊕ U ) is the zero vector 0. Further, U u = −u for all u ∈ R n . Let u, v ∈ R n . The PV addition satisfies the beta identity given by It is clear that the PV gyrogroup satisfies (H1). Suppose that u, v ∈ R n and that u, v = 0. It is easy to see that the unique solution of the system of equations is given by λ 1 = β v and λ 2 = 1. Thus, the PV gyrogroup satisfies (H2) and we obtain the next three theorems immediately as consequences of Theorems 5,8,10,13 in Section 2.
Theorem 26. Let P be a subgyrogroup of (R n , ⊕ U ). For all w ∈ R n such that w = u + v, with u ∈ P and v ∈ P ⊥ , the unique orthogonal decompositions of w according to ⊕ U are given by where For each w ∈ R n , if w = u + v is the unique expression of w, according to the orthogonal direct sum decomposition R n = P ⊕ P ⊥ , then the orthogonal gyroprojections of the PV gyrogroup (R n , ⊕ U ) onto the subgyrogroups P and P ⊥ are given by Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 23 May 2020 doi:10.20944/preprints202005.0371.v1 Theorem 27. Let P be a linear subspace of R n . Then the sets {b ⊕ U P : b ∈ P ⊥ } and {P ⊕ U b : b ∈ P ⊥ } are disjoint partitions of R n , that is, Moreover, (R n /P, ∼ ) = {b ⊕ U P : b ∈ P ⊥ } and (R n /P, ∼ r ) = {P ⊕ U b : b ∈ P ⊥ }.
Recall that t-PV addition is defined by for all u, v, ∈ R n , where β u is redefined by β u = 1 When t is arbitrarily large (that is, t → +∞), ⊕ U t reduces to ordinary vector addition of R n and β w → 1. Therefore, the orthogonal gyroprojections of the t-PV gyrogroup reduce to ordinary projection of R n : for all u ∈ P, v ∈ P ⊥ . Now, we analyze the fiber bundles and sections of the PV gyrogroup arisen from the two orthogonal decompositions R n = P ⊕ U P ⊥ and R n = P ⊥ ⊕ U P. First we give the relation between these two decompositions according to Theorem 6.
We can finally conclude that the same duality relations as in Theorem 21 happens for the case of the PV gyrogroup.
Theorem 30. Let P be a subgyrogroup of (R n , ⊕ U ). The following duality relations hold: 1. The cosets of (R n /P ⊥ , ∼ r ) are global sections for the quotient spaces (R n /P, ∼ ) and (R n /P, ∼ r ), and vice versa. 2. The cosets of (R n /P ⊥ , ∼ ) are global sections for the quotient space (R n /P, ∼ r ), and vice versa. 3. The cosets of (R n /P ⊥ , ∼ ) are local sections for the quotient space (R n /P, ∼ ) except the identity coset 0 ⊕ U P ⊥ = P ⊥ that is a global section, and vice versa.
To visualize the left and right cosets we restrict now to the 3-dimensional space R 3 and we show in Figures 8 and 9 the cosets obtained from the orthogonal decompositions of the gyrogroup (R 3 , ⊕ V ) by the subgyrogroups L e 3 and D e 3 .

The SL(2, C) general addition and Chen's gyrogroup
Einstein, Möbius, and PV gyrogroups are three different realizations of hyperbolic geometry associated to the Lorentz group. In [13] it was shown that the algebra of the group SL(2, C) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its hyperbolic geometry. A general addition for real inner product gyrogroups extended from the group SL(2, C) is given in the next definition.
Theorem 33. Let P be a linear subspace of V. Then the following are gyrogroup isomorphisms: Finally, when t → +∞, ⊕ C reduces to ordinary vector addition of V. Surprisingly, P C and P r C reduce again to the ordinary projection P(u + v) = u, and Q C and Q r C reduce to the ordinary projection Q(u + v) = v. Further, the equivalence relation (29) reflects the left coset relation in (V, +): if and only if − v + u ∈ P and the equivalence relation (31) reflects the right coset relation in (V, +): Since lim v →+∞ µ 1 (u, v) v = t 2 t 2 +2 u 2 u √ t 2 + u 2 − 2 then for all u ∈ P\{0} we have that µ 1 (u, v) v ∈ 0, t 2 t 2 +2 u 2 u √ t 2 + u 2 − 2 [0, +∞[, for all v ∈ P ⊥ . This implies that τ (2) u is only a local section for the fiber bundles π 1 and π 3 defined in (38). In the case when u = 0 the section τ (2) 0 is a global section as seen in (41).
Regarding µ 2 (v, u) v we have that for each u ∈ P it is a strictly increasing function in the variable v such that µ 2 (0, u) 0 = 0 and lim v →+∞ µ 2 (v, u) v = +∞. Therefore, we conclude that {µ 2 (v, u)v : v ∈ P ⊥ } = P ⊥ . Hence, for any u ∈ P, it follows that τ (4) u is a global section for the fiber bundles π 2 and π 4 defined in (38).
In the case of Chen's gyrogroup we have the same duality relations as in Theorem 30. To visualize the left and right cosets in this case we consider V the 3-dimensional space R 3 and we show in Figures  11 and 12 the cosets obtained from the orthogonal decompositions of the gyrogroup (R 3 , ⊕ C ) by the subgyrogroups L e 3 and D e 3 .

Conclusion
We generalized the study of factorization of Möbius gyrogroups to that of real inner product gyrogroups and proved the Unique Decomposition Theorem. This is the main theorem that leads to other remarkable results proved in this work. It resembles the standard theorem in linear algebra that every inner product space has an orthogonal direct sum decomposition associated to its finite-dimensional subspace. Because of the nonassociativity and the noncommutativity, we defined suitable equivalence relations on real inner product gyrogroups. With the equivalence relations we could partition a real inner product gyrogroup into left and right coset spaces. The four gyrogroups studied to confirm the general theory allow explicit calculations of the gyroprojectors and the left and right cosets. There are several possible applications of our results. We mention, for example, the construction of orthogonal gyroexpansions with respect to an orthogonal basis in a real inner product gyrogroup, or the construction of integral transforms such as the wavelet transform on some manifolds, such as the sphere, the ball, or the hyperboloid (cf. [6,14]). Finally, it would be interesting to generalize these results to complex gyrogroups and to the novel bi-gyrogroups (see [15]), that give a parametrization of generalized Lorentz groups SO(m, n), m, n ∈ N, in pseudo-Euclidean spaces of signature (m, n).
Author Contributions: All authors contributed equally to this article.

Symbol
Description Aut G automorphism group of G β u beta factor ⊕ C Chen's addition coaddition L c cogyroline cosubtraction ⊕ E Einstein addition [b] equivalence class containing b determined by the relation ∼ [b] r equivalence class containing b determined by the relation ∼ r ∼ equivalence relation defined from the left orthogonal gyroprojector Q ∼ r equivalence relation defined from the right orthogonal gyroprojector Q r gyr [a, b] gyroautomorphism generated by a and b ⊕ gyrogroup addition gyrogroup subtraction L g gyroline · inner product in an arbitrary inner product space ·, · inner product in R n π left canonical projection a ⊕ P G left coset of P G with representative a (G/P G , ∼ ) left coset space of G L a left gyrotranslation P left orthogonal gyroprojector associated with P G ⊕ P ⊥ orthogonal complement in a vector space D e 3 plane that passes through the origin and is perpendicular to the North Pole ⊕ U PV addition π r right canonical projection P G ⊕ a right coset of P G with representative a (G/P G , ∼ r ) right coset space of G P r right orthogonal gyroprojector associated with P ⊥ G ⊕ P G Q r right orthogonal gyroprojector associated with P G ⊕ P ⊥ G ⊗ scalar multiplication in a gyrovector space L e 3 straight line that passes through the origin and the North Pole ⊕ E t t-Einstein addition ⊕ M t t-Möbius addition Table S1: List of symbols used in the paper.