Octonionic Planes

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UNIVERSIDADE FEDERAL DE MINAS GERAIS INSTITUTO DE CIENCIASˆ EXATAS DEPARTAMENTO DE MATEMATICA´

OCTONIONIC PLANES

In´acio Augusto Rabelo Pinto

Belo Horizonte - Minas Gerais - Brasil

2021 UNIVERSIDADE FEDERAL DE MINAS GERAIS INSTITUTO DE CIENCIASˆ EXATAS DEPARTAMENTO DE MATEMATICA´

OCTONIONIC PLANES

Vers˜aoFinal

Inacio´ Augusto Rabelo Pinto* Orientador: Nikolai Goussevskii

Disserta¸cão apresentada ao Programa de Pós-Gradua¸cãoem Matemáticada Univer- sidade Federal de Minas Gerais, como requi- sito parcial àobten¸cãodo t´ıtulode Mestre em Matemática.

Belo Horizonte - Minas Gerais - Brasil

2021

Pinto, Inácio Augusto Rabelo

P659o Octonionic planes [manuscrito] / Inácio Augusto Rabelo Pinto. 2021. ix, 90 f. il.

Orientador: Nikolai Alexandrovitch Goussevskii. Dissertação (mestrado) Universidade Federal de Minas Gerais, Instituto de Ciências Exatas, Departamento de Matemática. Referências: f.8790 . 1. Matemática – Teses. 2. Octônios Teses. 3.Espaços hiperbólicos – Teses. 4. Espaços simétricos – Teses. 5. Planos projetivos Teses. 5. Geometria projetiva – Teses. I. Goussevskii, Nikolai Alexandrovitch. II. Universidade Federal de Minas Gerais; Instituto de Ciências Exatas, Departamento de Matemática. III.Título.

CDU 51(043)

Ficha catalográfica elaborada pela bibliotecária Belkiz Inez Rezende Costa CRB 6ª Região nº 1510 Universidade Federal de Minas Gerais Departamento de Matemática Programa de Pós-Graduação em Matemática

FOLHA DE APROVAÇÃO

Octonionic Planes

INÁCIO AUGUSTO RABELO PINTO

Dissertação defendida e aprovada pela banca examinadora constituída pelos Senhores:

______Prof. Nikolai Alexandrovitch Goussevskii UFMG

______Prof. Heleno da Silva Cunha UFMG

______Prof. Maurício Barros Corrêa Júnior UFMG

Belo Horizonte, 09 de março de 2021.

Av. Antônio Carlos, 6627 – Campus Pampulha - Caixa Postal: 702 CEP-31270-901 - Belo Horizonte – Minas Gerais - Fone (31) 3409-5963 e-mail: [email protected] - home page: http://www.mat.ufmg.br/pgmat (...) e nisso eu sou - como a Poesia é e eu sei que meu trabalho éárduo e existe para si mesmo como no cemitérioda cidade a insôniado vigia

Guen´adiAigui

Trecho de Silˆencio (1954-1956) - tradu¸c˜aode Boris Schnaiderman. Abstract

The octonionic projective and hyperbolic planes are studied. We present the constructions based on Jordan Algebras, the Riemannian metrics, the structure of homogeneous spaces, and the classiﬁcation of totally geodesic submanifolds. We also develop a similar description of the real, complex, and quaternionic projective and hyperbolic spaces.

Keywords: Octonions, Octonionic projective plane, Octonionic hyperbolic plane, Projective spaces, Hyperbolic spaces, Riemannian symmetric spaces of rank one.

vi Resumo

Os planos projetivo e hiperbólicooctoniônicossãoestudados. Apresentamos as constru¸cõesbaseadas em álgebrasde Jordan, as métricas Riemannianas, a estrutura de espa¸coshomogêneose a classifica¸cãodas subvariedades totalmente geodésicas. Desen- volvemos tambémuma descri¸cãosimilar dos espa¸cosprojetivos e hiperbólicosreais, complexos e quaterniônicos.

Palavras-chave: Octônios,Plano projetivo octoniônico,Plano hiperbólicooctoniônico, Espa¸cosprojetivos, Espa¸cos hiperbólicos,Espa¸cossimétricosRiemannianos de posto um.

vii Agradecimentos

A` minha fam´ılia,em particular minha m˜ae,pelo constante e irrestrito apoio;

A` Dindi, pelo suporte, companheirismo e paciˆenciainabal´aveis;

Ao Nikolai, pela forma¸cãoacadêmicae apresenta¸cãodo tema;

Aos irm˜aosBruno e Toni;

Ao Maur´ıcioe o Heleno, por comporem a banca;

Aos demais colegas e amigos de departamento, pelas impag´aveis conversas de corre- dor.

viii Contents

Introduction1

Introdu¸c˜ao2

Index of Notations3

0 Preliminaries4 0.1 Division Algebras...... 4 0.1.1 Quaternions...... 6 0.1.2 Octonions...... 8 0.1.3 Further Results...... 11 0.2 Lie groups and Lie Algebras...... 11 0.2.1 Basic Theory...... 11 0.2.2 Unitary Groups...... 12 0.2.3 Killing Form...... 14 0.2.4 Classiﬁcation of Simple Lie Algebras...... 16 0.3 Spin Groups...... 17 0.3.1 Cliﬀord Algebras...... 17 0.3.2 Spin Groups...... 18 0.3.3 The Group Spin(8)...... 20 0.3.4 The Group Spin(9)...... 20 0.4 Principal Bundles...... 21 0.5 Symmetric Spaces...... 22 0.5.1 Introduction...... 22 0.5.2 Symmetric Pairs...... 23 0.5.3 Orthogonal Symmetric Lie Algebras...... 24 0.5.4 Lie Triple Systems and Totally Geodesic Submanifolds...... 24 0.5.5 Duality and Rank...... 25

1 F-Projective Spaces 28 1.1 Introduction...... 28 1.2 Collineation Groups...... 31 1.3 Riemannian Metric...... 33 1.4 The Projective Spaces as Symmetric Spaces...... 36 1.4.1 The Real Case...... 36 1.4.2 The Complex Case...... 36 1.4.3 The Quaternionic Case...... 37 1.5 The Matrix Model...... 37 1.5.1 The Real Case...... 38

ix CONTENTS x

1.5.2 The Complex and Quaternionic Cases...... 38 1.6 Totally Geodesic Submanifolds...... 39

2 The Octonionic Projective Plane 41 2.1 The Octonionic Projective Line...... 41 2.2 The Jordan Algebra H3(O)...... 43 2.3 The Octonionic Projective Plane...... 47 2.4 Riemannian Metric...... 50 2 2.5 OP as a Homogeneous Space...... 53 2.6 Totally Geodesic Submanifolds...... 57 2.7 Non-existence of Higher-Dimensional Spaces...... 59

3 F-Hyperbolic Spaces 62 3.1 Introduction...... 62 3.2 Riemannian Metric...... 63 3.3 The Hyperbolic Spaces as Symmetric Spaces...... 65 3.3.1 The Real Case...... 66 3.3.2 The Complex Case...... 67 3.3.3 The Quaternionic Case...... 68 3.4 Totally Geodesic Submanifolds...... 69

4 The Octonionic Hyperbolic Plane 71 4.1 The Jordan Algebra H1,2(O)...... 71 4.2 The Octonionic Hyperbolic Plane...... 72 4.3 Riemannian Metric...... 76 4.4 2 as a Homogeneous Space...... 78 HO 4.5 Totally Geodesic Submanifolds...... 80

A Automorphisms of H3(F) and H1,2(F) 82 A.1 Compact Case...... 82 A.2 Non-compact Case...... 84

Bibliography 87 Introduction

The real numbers, complex numbers, quaternions, and octonions form the four normed division algebras. While the quaternions are not commutative, octonions are not even associative. Despite this peculiarity, these numbers are related to exceptional structures founded by E. Cartan and W. Killing at the end of the 20th century and have a growing interest in Physics. In this work, we study the octonionic projective and hyperbolic planes. They are also called Cayley projective and hyperbolic planes. In addition to containing all the geometry of the real, complex, and quaternionic planes, these spaces also represent the relations mentioned above. However, the construction using projectivization of a vector space is not applicable because of non-associativity, so new tools are introduced to play the role of a vector structure and describe the octonionic spaces. Chapter 0 introduces the preliminaries to the theme. It has written to be a reference, then several proofs have been omitted. The content of the following chapters is primarily:

Construction of the spaces;

Riemannian Metric;

Structure of homogeneous and symmetric spaces;

Classiﬁcation and stabilizers of totally geodesic submanifolds;

We highlight the different topics of each chapter. The matrix model in Section 1.5 is the motivation to the main model of the octonionic projective plane. Chapter 2 starts with the configuration Jordan algebra and the exceptional group F4. In particular, Section 2.7 intends to detail a complete argumentation about the non-existence of higher-dimensional projective spaces over octonions. Chapter 3 deals with classical hyperbolic spaces. It contains the proofs of duality and rank. In Chapter 4, the configuration algebra of the octonionic hyperbolic plane is similar to that of the projective case, so several results have the same proof. We spend some work to describe and give characterizations of this space. Further, the original proofs of classification of totally geodesic submanifolds are given in the last two chapters and applied to the compact cases by duality. We also add an appendix in which we obtain the automorphism groups of Jordan subalgebras related to the classical planes. Although the constructions are not simple due to the non-associativity, they follow the same lines or motivations of the classical spaces. Finally, we note that the spaces studied here, together with euclidean spheres, form the symmetric spaces of rank one. The pictures were made in the free license software Ipe, version 7.2.23*.

1 Introdu¸c˜ao

Os númerosreais, númeroscomplexos, quatérniose octôniosformam as quatro álgebras de divisãonormadas. Enquanto os quatérniosnãosãocomutativos, os octôniosnãosão nem mesmo associativos. Apesar dessa peculiaridade, esses númerosestãorelacionados a estruturas excepcionais encontradas por E. Cartan e W. Killing no final do séculoXIX e possuem um crescente interesse em F´ısica. Nesse trabalho, estudamos os planos projetivo e hiperbólicooctoniônicos.Esses obje- tos tambémsãochamados de planos projetivo e hiperbólicode Cayley. Alémde conterem toda a geometria dos planos reais, complexos e quaterniônicos, esses espa¸cosrepresen- tam as rela¸cõesacima mencionadas. Entretanto, a constru¸cãousando projetiviza¸cãode espa¸cosvetoriais nãose aplica devido a nãoassociatividade, de modo que novas ferra- mentas sãointroduzidas para substitu´ıremo papel de uma estrutura vetorial e descrever os espa¸cosoctoniônicos. O cap´ıtulo0 introduz as preliminares ao tema. Escrito para ser uma referência,a maioria das provas ali contidas foram omitidas. O conteúdodos cap´ıtulosseguintes é essencialmente: Constru¸cãodos espa¸cos; MétricaRiemanniana; Estrutura de espa¸coshomogêneose simétricos; Classifica¸cãoe estabilizadores das subvariedades totalmente geodésicas; Destacamos os diferentes tópicosde cada cap´ıtulo.O modelo matricial na Se¸cão1.5 é a motiva¸cãopara o principal modelo do plano projetivo octoniônico.O cap´ıtulo2 inicia- se com a álgebrade Jordan de configura¸cãoe o grupo excepcional F4. Em particular, a se¸cão2.7 pretende detalhar uma completa argumenta¸cão sobre a nãoexistência de espa¸cos projetivos de dimensãomaior do que trêssobre os octônios. O cap´ıtulo 4 lida com os espa¸coshiperbólicosclássicos. Ele contém as provas de dualidade e posto. No cap´ıtulo4, a álgebrade configura¸cãoésimilar a do caso projetivo, entãováriosresultados possuem a mesma prova. Algumas descri¸cões e caracteriza¸cõessão dadas sobre este espa¸co.Alémdisso, as provas originais da classifica¸cãode subvariedades totalmente geodésicassãofeitas nos últimosdois cap´ıtulos e entãoaplicadas ao caso compacto por dualidade. Tambémadicionamos um apêndice no qual obtemos os grupos de automorfismos das subálgebrasde Jordan relacionadas aos planos clássicos. Apesar das constru¸cõesnãoserem simples devido a nãoassociatividade, elas seguem as mesmas linhas e motiva¸cõesdos espa¸cos clássicos.Finalmente, notamos que os espa¸cos tratados aqui, juntos com as esferas euclideanas, formam os espa¸cossimétricosde posto um. As figuras foram feitas no software de licen¸calivre Ipe, versão7.2.23*. *©1993–2020 Otfried Cheong.

2 Index of Notations

R Real numbers C Complex numbers K Quaternions O Octonions FR, C, or K F∗ F − {0}. Re(z) Real part of z ∈ F, O Im(z) Imaginary part of z ∈ F, O d Real dimension of the algebra F Mn(A) Square matrices of order n with entries in A Hn(A) Hermitian matrices of order n with entries in A diag(λ1, . . . , λn) Diagonal matrix of order n with entries λi ∈ R In Identity matrix of order n I1,n Diagonal matrix diag(−1, 1,..., 1) H Diagonal matrix I1,2 tr(A) Trace of a quadratic matrix A det(A) Determinant of a quadratic matrix A Sn Euclidean sphere in Rn+1 n FP Projective space over F of F-dimension n 2 OP Octonionic projective plane n Hyperbolic space over of -dimension n HF F F 2 Octonionic hyperbolic plane HO Iso(M) Isometry group of a Riemannian manifold M G0 Connected component of the identity of a Lie group G Id Identity map

3 Chapter 0

Preliminaries

This chapter is a concise introduction to concepts, results, and objects related to projective and hyperbolic spaces. We omit most of the proofs. The references are given at the beginning of each section.

0.1 Division Algebras

We introduce the quaternions and octonions from the formal definition of algebra. Although the abstraction, this allows us to derive some properties and characterizations succinctly. The relation with rotations in 3, 4, 7, and 8 dimensions is briefly described. The section ends with important results of classification. The main references are [51, Chapters 3 and 4], [7, Section 2.2], and [26, Chapter 6].

Deﬁnition 0.1.

1. An algebra A is a real vector space equipped with a bilinear map m: A × A −→ A and a nonzero element 1 ∈ A such that m(a, 1) = m(1, a) = 1 for all a ∈ A. The map m is called multiplication, 1 is the unit, and we abbreviate m(a, b) = ab.

2. An algebra A is a division algebra if given a, b ∈ A with ab = 0, then a = 0 or b = 0.

3. A normed division algebra is an algebra A that is also a normed vector space satisfying kabk = kak.kbk.

4. An algebra A has two-sided multiplicative inverses if for any nonzero a ∈ A there is an element a−1 ∈ A such that aa−1 = a−1a = 1.

5. An algebra A is associative if a(bc) = (ab)c for all a, b, c ∈ A.

6. An algebra A is alternative if (aa)b = a(ab) and (ba)a = b(aa) for all a, b ∈ A.

7. The automorphism group of an algebra A is deﬁned as

Aut(A) = {g ∈ GL(A): g(ab) = g(a)g(b) ∀ a, b ∈ A}.

Note that the conditions of item 3 are enough to A be a division algebra, and this justiﬁes the nomenclature. The next proposition characterizes a division algebra.

4 CHAPTER 0. PRELIMINARIES 5

Proposition 0.1. An algebra A is a division algebra if and only if for all a, b ∈ A, with a 6= 0, there exist unique x, y ∈ A such that ax = b and ya = b. In particular, every nonzero element has left and right inverses.

Proof. Let A be a division algebra and a, b ∈ A with a 6= 0. If there exist x, x0, y, y0 ∈ A satisfying these equations, the conditions imply x = x0 and y = y0, which proves uniqueness. For existence, let Ra and La be the right and left multiplications, respectively. It holds that Ker(Ra) = Ker(La) = 0, so the maps are invertibles and this guarantees the existence of x and y. Conversely, let a, b ∈ A with ab = 0. If a 6= 0, by the condition of uniqueness, b = 0 and similarly for b 6= 0.

As a consequence, an associative algebra has multiplicative inverses if and only if it is a division algebra. An important result to work with octonions is the following.

Theorem 0.1 (Artin - [26], Theorem 6.37). An algebra is alternative if and only if the subalgebra generated by any two elements is associative.

Deﬁnition 0.2.

1. An algebra A is an ∗-algebra if there is a conjugation map ∗: A −→ A such that (a∗)∗ = a and (ab)∗ = b∗a∗.

2. An ∗-algebra A is real if a = a∗ for all a ∈ A.

3. An ∗-algebra A is nicely normed if a + a∗ ∈ R and a∗a = aa∗ > 0 for all a ∈ A.

Theorem 0.2. Let A be a nicely normed ∗-algebra. Then:

1. A has multiplicative inverses.

2. If A is alternative, then it is a normed division algebra. Proof. For a nonzero element a the multiplicative inverse is given by a−1 = a∗/aa∗, which shows item 1. For item 2, deﬁne |x|2 = xx∗ for all x ∈ A. For a, b ∈ A, consider the subalgebra ∗ ∗ B = h1, a − a , b − b i, that is, the algebra generated by a − a∗ and b − b∗. Since a + a∗, b + b∗ ∈ R ⊂ B, it follows that a, b, a∗, b∗ ∈ B. By Artin’s theorem, B is associative, then

|ab|2 = (ab)(ab)∗ = (ab)(b∗a∗) = a(bb∗)a = |a|2|b|2.

This implies that A is normed. Now, given the equations ax = b and ya = b, with a 6= 0, alternativity implies x = a−1b and y = ba−1, and these solutions are unique. Therefore, the result follows from Proposition 0.1.

A systematic way to obtain new algebras from an ∗-algebra is by the Cayley-Dickson construction. Indeed, we shall study algebras obtained in this way.

Deﬁnition 0.3. Let A be an ∗-algebra. We deﬁne a new ∗-algebra A0 as the vector space A0 = A ⊕ A with the multiplication given by

(a, b)(c, d) = (ac − d∗b, da + bc∗). 6 0.1. DIVISION ALGEBRAS

The next proposition shows the eﬀect of repeatedly applying the Cayley-Dickson construction.

Proposition 0.2 ([26], Corollary 6.27). Let A be an ∗-algebra. 1. A0 is a ∗-algebra never real. 2. A is real if and only if A0 is commutative. 3. A is commutative and associative if and only if A0 is associative. 4. A is associative and nicely normed if and only A0 is alternative. 5. A is nicely normed if and only if A0 is nicely normed. The complex numbers C form the algebra obtained from R by the Cayley-Dickson construction. A basis is given by {1, i}, where i2 = −1. For z = a + bi, the conjugation is given by z = a − bi and the norm by |z|2 = a2 + b2. The real numbers a and b are the real part Re(z) and the imaginary part Im(z), respectively. The automorphisms of these algebras are given by the next theorem. Theorem 0.3 ([34], Propositions 2.9 and 2.10).

1. The only automorphism of R is the identity, then Aut(R) = 1. 2. The automorphism group of C consists of the identity and the complex conjugation, then Aut(C) ' Z2.

0.1.1 Quaternions

The quaternions K form the 4-dimensional algebra obtained from C by the Cayley- Dickson construction. A canonical basis is {1, i, j, k} which satisﬁes 1 is the identity; i2 = j2 = k2 = −1; ij = k, ji = −k.

k j

Figure 1: Quaternionic Multiplication

By Proposition 0.2, it follows that K is a nicely normed associative algebra, but it is not commutative. A mnemonic for the multiplication table is Figure 1, where the counter- clockwise direction gives the additive symmetric points. For q = q0 + q1i + q2j + q3k ∈ K, we have: CHAPTER 0. PRELIMINARIES 7

The conjugate q = q0 − q1i − q2j − q3k; the norm qq = qq = |q|2; the real and imaginary parts Re(q) = 1/2(q+q) and Im(q) = 1/2(q−q), respectively.

Indeed, a long and direct calculation also shows that for any p, q ∈ K, pq = qp. The quaternions were discovered by W. R.Hamilton in 1843 after a frustrating attempt to construct a 3-dimensional algebra.. The ﬁrst application arises from Eletromagnetism. In fact, the usual scalar and cross product in R3 can be described by the quaternionic multiplication: for v, w ∈ R3 ' Im(K), we have v.w = Re(vw) and v × w = Im(vw). (0.1)

Moreover, quaternions also describe rotations in R3 as follows. Fix q ∈ K with |q| = 1. For v ∈ Im(K), the properties above show that Re(qvq) = 0. Thus we get a well-deﬁned map

cq : Im(K) −→ Im(K) (0.2) v 7−→ qvq, that will be called a conjugation map. In particular, if q = eiα for some real α and v = a0i + a1j + a2k, then

−2iα −2iα qvq = a0i + a1e j + a2e k, which corresponds to a rotation by 2α in the plane jk. Similarly, ejα and ekα yields rotations in the planes ik and ij. But any rotation in this space is a composition of such operations, which still a conjugation, thus

iα jα kα SO(3) = hcq : q = e , e , or e , α ∈ Ri, 3 where SO(3) denotes the group of rotations in R . For a unit vector v1 ∈ Im(K), there are unit and orthogonal vectors v2, v3 ∈ Im(K) such that {1, v1, v2, v3} is a new basis for v1α K. The same computations above, show that the map cq, with q = e , corresponds to a rotation by 2α in the plane v2v3. On the other hand, any unit quaternion q can be written as evθ for some real θ and a unit vector v ∈ Im(K). Therefore

SO(3) = {cq : q ∈ K and |q| = 1}. (0.3) A simple rotation in R3 is a rotation about a line. So, we derive that the rotations in this space are simple and the well known Euler’s Theorem, which states that the product of iθ1 iθ2 iα any two simple rotations is simple. Now, let w = r1e + r2e j ∈ K and q = e . Then

i(θ1+2α) i(θ2+2α) rq(w) := qwq = r1e + r2e j, which corresponds to a rotation of w by 2α in the plane 1i. Similarly, ejα and ekα yields rotations in the planes 1k and 1j. Since SO(4) is generated by such rotations, we have

iα jα kα SO(4) = h{cq : q ∈ K and |q| = 1} ∪ {rq : q = e , e , or e , α ∈ R}i. (0.4) Theorem 0.4 ([34], Chapter 8). The automorphism group of K consists of conjugation maps. Thus Aut(K) ' SO(3). 8 0.1. DIVISION ALGEBRAS

0.1.2 Octonions

The octonions O form the 8-dimensional algebra obtained from K by the Cayley- Dickson construction. A basis is given by

{1, e1 = i, e2 = j, e3 = k, e4 = l, e5 = il, e6 = jl, e7 = kl}, which satisﬁes:

1 is the identity.

2 er = −1 for all r.

eres = −eser for r 6= s.

er(eset) = −(eres)et for all r, s, t.

(erer)ej = er(eres) for all r, s.

i j l

jl k il Figure 2: The Fano plane and the octonionic multiplication

This deﬁnes the full multiplication table. A mnemonic is the Fano plane (see Figure 2). The counter-clockwise direction gives the additive symmetric points. The vertices are understood to continue at the beginning of the line that contains them. By Proposition 0.2, O is a nicely normed alternative algebra, but not associative. For example: (kl)j = il 6= k(lj) = −k(jl) = −il, (il)(jl) = −k 6= (i(lj)) l = − (i(jl)) l = (kl)l = k.

Despite this fact, Artin’s Theorem allows us to realize some calculations with octonions. To measure the non-associativity of any three octonions, there is the associator:

[x, y, z] := x(yz) − (xy)z.

Since O is alternative, the associator is zero when two entries are equal. In particular, [x + y, x + y, z] = [x + z, y, x + z] = [x, y + z, y + z] = 0, CHAPTER 0. PRELIMINARIES 9 which implies the equalities [x, y, z] = −[y, x, z] = −[x, z, y] = −[z, y, x]. (0.5) P7 For an octonion x = x0 + r=1 xrer, we have: P7 The conjugate x = x0 − i=1 xiei; P7 2 the norm |x| = xx = xx = r=0 xr; the real and imaginary parts Re(x) = 1/2(x + x) and Im(x) = 1/2(x − x), respectively. The conditions over the conjugation can be recovered by a direct computation as follows. For any p, q ∈ K = h1, i, j, ki, we have p(ql) = −(ql)p and (pl)q = −q(pl).

This implies, for any x, y ∈ O, xy = yx. The octonions were first introduced by John Graves in 1843 in an exchange of letters with Hamilton after his discovery of the quaternions. But Arthur Cayley was the first who published a work about octonions. This also justifies the usual name Cayley Numbers. For vectors x, y ∈ R7 ' Im(O), the usual dot product can be described as hx, yi = Re(xy) = Re(yx). (0.6)

A notion of the cross product is also possible to O [26, Deﬁnition 6.46], which extends the concept to the 7-dimensional space. There are the following properties. Proposition 0.3. Let x, y, z ∈ O. Then 1. [x, y, z] ∈ Im(O). 2. Re(xy) = Re(yx). 3. Re (x(yz)) = Re (y(zx)) = Re (z(xy)). Proof. 1. Considering that x + x ∈ R for all x ∈ O and (0.5), we obtain that −[z, y, x] = [z, y, x] = [y, z, x] = −[x, y, z]. Since [x, y, z] = −[z, y, x], the result follows.

2. We may suppose x, y∈ / C. Let x, y ∈ Im(O) and recall the dot product as in (0.6). Then Re(xy) = −hx, yi = −hy, xi = Re(yx).

Now, let x = x0 + Im(x), y = y0 + Im(y) ∈ O. Then

Re(xy) = x0y0 + Re(Im(x)Im(y))

= x0y0 + Re(Im(y)Im(x)) = Re(yx), as stated. 10 0.1. DIVISION ALGEBRAS

3. By item 1, the expression Re(xyz) is well-deﬁned. Thus, the equalities follow from item 2.

Theorem 0.5 (Moufang Identities). Let x, y, z ∈ O. Then the following equations holds: 1. (xyx) z = x (y (xz)). 2. z (xyx) = ((zx) y). 3. (xy)(zx) = x (yz) x. An interesting fact is that Ruth Moufang proved this identities in the context of projective geometry (see Section 2.7). As for quaternions, octonions can be used to describe rotations in 7 and 8 dimensions. 3 Let p0 ∈ R . The rotation of this point in the xy plane by an angle of 2α can be characterized in the following way. Reflect it about the axis x to obtain a point p1. Choose a line in the xy plane making an angle of 2α with this axis and passing through the origin. Reflect p1 about this line to obtain p2. If the rotation has coordinates (x0, y0, z0), then p2 = (x0, y0, −z0). This construction is the same for the n-dimensional space, where the effect on the remainder coordinates is the same. Let x ∈ Im(O) ' R7. We apply the construction above to obtain a rotation in the ij plane. A calculation shows that reflect in i axis, is conjugate by i. The desired line is of the form w = i cos α + j sin α. Therefore, x is mapped to

ru ◦ ri(x) = (cos αi + sin αj)(ixi)(cos αi + sin αj).

Note that this expression is well deﬁned, since O is alternative, although it can not be simpliﬁed. The construction is the same if we replace i, j by orthogonal unit imaginary octonions u and v. Thus

2 2 SO(7) = h{rw ◦ ru : w = cos αu+sin αv, u, v ∈ Im(O), u = v = −1, hu, vi = 0, α ∈ Ri. As in the quaternionic case, this is extended to rotations in the 8-dimensional space. Let iθ1 iθ2 iφ1 iφ2 x = r1e + r2e j + s1e l + s2 e j l ∈ O. Then lα lα i(θ1+2α) iθ2 i(φ1+2α) iφ2 e xe = r1e + r2e j + s1e l + s2 e j l, which corresponds to a rotation by 2α in the plane 1l. Similarly to rotations in the planes 1i, 1j, and 1k. Hence

iα jα kα lα SO(8) = hSO(7) ∪ {rq : q = e , e , e or e , α ∈ R}i.

The ﬁrst connection between O and exceptional structures is the following remarkable result ﬁrst stated by E. Cartan. Theorem 0.6. The automorphism group of the octonions is the real compact exceptional Lie group G2: Aut(O) = G2 . CHAPTER 0. PRELIMINARIES 11

0.1.3 Further Results

The Cayley-Dickson contruction applied to O gives the sedenions O0. This algebra is nicely normed, in particular, it has multiplicative inverses. However, one can show the existence of zero divisors in this set. Since octonions are not associative, item 4 of Proposition 0.2 shows that O0 is not alternative. Indeed, if a, b ∈ O0 are nonzero elements with ab = 0, then 0 = a−1(ab) 6= (a−1a)b = b. The next theorems exemplify the importance of the algebras treated until here.

Theorem 0.7 (Hurwitz - [26], Theorem 6.37). The algebras R, C, K, O are, up to isomorphism, the only normed division algebras.

Theorem 0.8 (Zorn - [7], Theorem 2). The algebras R, C, K, O are, up to isomorphism, the only alternative division algebras.

Let A be a division algebra. Since the right and left multiplications are invertible, every nonzero element has left and right inverses, although they are not necessarily equal.

Example 0.1. We can modify the construction of the quaternions by setting i2 = −1+j for some real number 6= 0 and leaving the rest of the multiplication table unchanged. Then i(−i − k) = 1 6= (−i − k)i = 1 − 2j. That is, this algebra does not have a two-sided inverse. In particular, it is a division algebra which can not be isomorphic to K. However, the dimension of division algebras is restricted as shows the next theorem.

Theorem 0.9 (Bott-Milnor-Kervaire - [7], Theorem 3). A division algebra has dimension 1,2,4, or 8.

In some sense, this last theorem tells us that the usual cross product only exists in three and seven dimensions.

0.2 Lie groups and Lie Algebras

This section is devoted to review some basic facts about Lie groups and Lie algebras in order to introduce some theorems used throughout the work. The main references are [1, Chapters 1 and 2], [2, Chapters 1 and 8], and [26, Chapter 1].

0.2.1 Basic Theory We establish some deﬁnitions and terminology in this subsection.

Deﬁnition 0.4.

1. A smooth manifold G is a Lie group if it is a group and the product and inversion operations are smooth functions.

2. A Lie subgroup of a Lie group G is an abstract group and an immersed submanifold of G. 12 0.2. LIE GROUPS AND LIE ALGEBRAS

3. A closed subgroup of a Lie group G is an abstract subgroup and a closed subset of G.

An algebraic subgroup of a Lie group does not need to have the induced manifold structure, however we have the following result:

Theorem 0.10 ([1], Theorem 1.4). If H is a closed subgroup of a Lie group G, then H is a submanifold and hence a Lie subgroup of G. In particular, it has the induced topology.

Recall that an abstract Lie algebra g is a real or complex vector space equipped with a bracket operation [., .] satisfying anti-commutativity, bilinearity and the Jacobi identity. The tangent space at identity e of a Lie group G equipped with the bracket induced from bracket operation on vector ﬁelds is the Lie algebra of G.

Deﬁnition 0.5. Let g be an abstract Lie algebra and h ⊂ g a vector subspace.

1. h is a Lie subalgebra if for all X,Y ∈ h, [X,Y ] ∈ h.

2. h is an ideal in g if for all X ∈ h and A ∈ g, [A, X] ∈ h.

The relations between Lie algebras and Lie groups are expressed, in a convenient form, by the next result.

Theorem 0.11 ([1], Theorem 1.14).

1. For any abstract Lie algebra g there exists a Lie group G (not necessarily unique) whose Lie algebra is g.

2. If G is a Lie group with Lie algebra g, Lie algebras of Lie subgroups of G are Lie subalgebras of g. Conversely, for each subalgebra h of g there exists a unique connected Lie subgroup of G with Lie algebra h. Furthermore, normal subgroups correspond to ideals in g.

3. Let G and H be Lie groups with isomorphic Lie algebras g and h, respectively. Then G and H are locally isomorphic. If in addition G and H are simply connected, they are isomorphic.

Items 1 and 3 of Theorem 0.11 show that for each Lie algebra there is only one class of isomorphic simply connected Lie groups. Moreover, the Lie algebra of a Lie groups is determined by the identity component of the group.

0.2.2 Unitary Groups Matrix Lie groups represent one of the ﬁrst and most important classes of such spaces. The groups that preserve Hermitian forms in Fn will be of special interest. The real (complex) general linear group consists of the invertible real (complex) matrices:

GL(n; F) = {A ∈ Mn(F): detFA 6= 0}, where F = R, C. For the real case, the notation is GL(n). The quaternionic general linear group GL(n; K) is deﬁned in the same manner, but not in terms of a determinant form, which has not a useful notion in Mn(K). CHAPTER 0. PRELIMINARIES 13

Compact Unitary Groups

n+1 Let z = (z0, . . . , zn), w = (w0, . . . , wn) ∈ F , the Hermitian form is deﬁned as

hz, wi = z0w0 + ··· + znwn. (0.7)

For F = R this is the usual dot product. The unitary group U(n; F) consists of the matrices in GL(n; F) which preserve this form, that is, ∗ U(n; F) = {A ∈ GL(n; F): AA = In}. (0.8) In the common notation and nomenclature,

U(n; R) = O(n), the real orthogonal group; U(n; C) = U(n), the complex unitary group; U(n; K) = Sp(n), the quaternionic unitary group. The group Sp(n) is also called the compact symplectic group. These groups are related by the following inclusions as closed subgroups: Sp(n) ⊂ U(2n) ⊂ O(4n). Hence, the compactness of O(n) implies the others. For F = R, C the special unitary group SU(n; F) is the subgroup of U(n; F) of matrices with determinant 1: SU(n; R) = SO(n), the real special orthogonal group; SU(n; C) = SU(n), the special unitary group. Let Z(n; F) the center of U(n; F). The projective groups are deﬁned as the quotients

PU(n; F) = U(n; F)/ Z(n; F).

For F = R, Z(n; R) = {In, −In}. If n is odd, O(n) = SO(n) × O(1), and we obtain that PO(n) = PSO(n) = SO(n).

If n is even, it does not hold. In both cases, the projective groups are simple, because the Lie algebras o(n) and so(n) are equal and the centers are ﬁnite. For F = C, the map ψ : SU(n) × U(1) −→ U(n) (0.9) (A, eiφ) 7−→ Aeiφ is a surjection with the kernel consisting of the group of diagonal matrices with entries being the roots of unity, which is isomorphic to Zn. Thus

U(n) = (SU(n) × U(1)) /Zn.

Moreover, Z(n; C) = U(1) and the center of SU(n) is isomorphic to Zn. It follows that the inclusion map is an isomorphism:

PU(n) = PSU(n) = SU(n)/Zn.

Note that this group is also simple. For F = K, Z(n; K) = {In, −In}' Z2. Therefore,

PSp(n) = Sp(n)/Z2 is also simple. 14 0.2. LIE GROUPS AND LIE ALGEBRAS

Non-compact Unitary Groups

n+1 Let z = (z0, . . . , zn), w = (w0, . . . , wn) ∈ F , the Hermitian form of signature (1, n) is deﬁned as Φ(z, w) = −z0w0 + ··· + znwn. (0.10) The indeﬁnite unitary group U(1, n; F) consists of the matrices in GL(n; F) which preserve this form, that is,

∗ U(1, n; F) = {A ∈ GL(n; F): AI1,nA = I1,n}. (0.11) In the common notation and nomenclature,

U(1, n; R) = O(1, n), the real indeﬁnite orthogonal group;

U(1, n; C) = U(1, n), the indeﬁnite unitary group;

U(1, n; K) = Sp(1, n), the quaternionic indeﬁnite unitary group.

For F = R, C the indeﬁnite special unitary group SU(1, n; F) is the subgroup of U(1, n; F) of matrices with determinant 1. In details,

SU(1, n; R) = SO(1, n), the real indeﬁnite special orthogonal group;

SU(1, n; C) = SU(1, n), the indeﬁnite special unitary group. Similarly, the projective indeﬁnite groups are given by

PU(1, n; F) = U(1, n; F)/ Z(1, n; F).

The centers Z(n + 1; F) and Z(1, n; F) are equal. For F = C, the map (0.9) also provides the isomorphism U(1, n) = (SU(1, n) × U(1))/Zn. Thus

PU(1, n) = PSU(1, n) = SU(1, n)/Zn, PSp(1, n) = Sp(1, n)/Z2. All of these groups are simple by similar reasons.

0.2.3 Killing Form The Killing form, named after W. Killing and introduced by E. Cartan, is an inner product on g related to classification of semisimple Lie algebras and the theory of symmetric spaces (see Definition 0.19). First, some definitions are given.

Definition 0.6. A map φ : G → G is an automorphism if it is a diffeomorphism and −1 a group isomorphism. The automorphism defined by Ig(h) = ghg is called an inner automorphism.

Deﬁnition 0.7.

1. The adjoint representation of G is the homomorphism

Ad : G −→ Aut(g)

g 7−→ (dIg)e CHAPTER 0. PRELIMINARIES 15

2. The adjoint representation of g is the homomorphism

ad : g −→ End(g)

X 7−→ (d Ad)e(X)

Theorem 0.12 ([1], Theorem 2.8). The adjoint representation of a Lie algebra g of a Lie group G satisﬁes ad(X)Y = [X,Y ] (0.12) for all X,Y ∈ g.

Deﬁnition 0.8. Suppose that g is a complex Lie algebra, then the adjoint representation is deﬁned by equation (0.12).

Deﬁnition 0.9. The Killing form of a Lie algebra g is the function

B : g × g −→ F (0.13) (X,Y ) 7−→ tr(ad(X) ◦ ad(Y ))

Example 0.2.

1. For g = so(n), B(X,Y ) = (n − 2)tr(XY ).

2. For g = su(n), B(X,Y ) = 2ntr(XY ).

3. For g = u(n), B(X,Y ) = 2ntr(XY ) − 2tr(X)tr(Y ).

4. For g = sp(n), B(X,Y ) = 2(n + 1)tr(XY ).

Deﬁnition 0.10. Let B be the Killing form of a Lie algebra g.

1. If B is non-degenerate, the Lie algebra is called semisimple.

2. If g is semisimple and the unique ideals are g and {0}, then g is called simple.

3. If B is negative deﬁnite, g is said to be of compact type.

4. The Killing form of a Lie group G is the Killing form of its Lie algebra g.

5. A Lie group G is semisimple if its Lie algebra is semisimple.

The next theorem justiﬁes the nomenclature in Deﬁnition 0.10.

Theorem 0.13 ([1], Theorems 2.13 and 2.14).

1. If G is a compact semisimple Lie group, then its Killing form is negative deﬁnite.

2. If G is a connected Lie group and its Killing form is negative deﬁnite, then G is compact and semisimple.

Example 0.3. The Lie algebras in Example 0.2 are of compact type. 16 0.2. LIE GROUPS AND LIE ALGEBRAS

1. A matrix (xij) ∈ so(n) satisﬁes ( xij = 0, if i = j

xij = −xji, if i 6= j

See the proof of Theorem 3.3. Then, for X 6= 0,

X X 2 B(X,X) = (n − 2)tr(XX) = (n − 2) xklxlk = (n − 2) −xkl < 0. k,l k,l

2. A matrix (xij) ∈ u(n) satisﬁes ( Re(xij) = 0, if i = j

xij = −xji, if i 6= j

See the proof of Theorem 3.6. Then, for X 6= 0,

X 2 B(X,X) = 2ntr(XX) = −2n |xkl| < 0. k,l

Since su(n) and sp(n) are contained in u(2n), the statement follows.

0.2.4 Classification of Simple Lie Algebras Complex simple Lie algebras were classified by Cartan and Killing in the end of 19th century. Since a Lie algebra is semisimple if and only if it is a direct sum of simple ideals [1, Proposition 2.18], complex semisimple Lie algebras are also classified. A brief description of the subject is given and the results are summarized in the table 1.

Deﬁnition 0.11.

1. If g is a real Lie algebra, its complexiﬁcation gC is the complex Lie algebra consisting of the vector space {X + iY : X,Y ∈ g} equipped with the bracket

[X1 + iY1,X2 + iY2] = ([X1,Y1] − [Y1,Y2]) + i ([Y1,X2] + [X1,Y2]) . (0.14)

2. A real Lie algebra h is a real form of a complex Lie algebra g if hC ' g.

The real forms of complex Lie algebras are not unique. However, a Theorem of H. Weyl states that a compact real form is unique up to conjugation by an inner automorphism. E. Dinkyn solved the classification problem to Lie algebras over general algebraic closed fields of characteristic zero and gave a modern treatment to the subject. Each Lie algebra is associated with a graph, called Dinkyn diagram, which reduces the question to a combinatorial problem. The second column of the table below shows all the possibilities to a complex simple Lie algebra. By Weyl’s result, real simple Lie algebras of compact type are also classified. On the other hand, the complexification of a real simple Lie algebra can be semisimple or simple, so the real case does not follow from the complex. The last five cases are the exceptional. Their existence has never been suspected until Killing’s work. As we have seen, the exceptional group G2 is the automorphism group of O. In the middle of the 20th century, the group F4 was related to octonionic projective CHAPTER 0. PRELIMINARIES 17

Family Complex Algebra Complex Group Compact Form Real Group Real Dimension An(n ≥ 1) sl(n + 1; C) SL(n + 1; C) su(n + 1) SU(n + 1) n(n + 2) Bn(n ≥ 2) so(2n + 1; C) SO(2n + 1; C) so(2n + 1) Spin(2n + 1) n(2n + 1) Cn(n ≥ 3) sp(n; C) Sp(n; C) sp(n) Sp(n) n(2n + 1) Dn(n ≥ 4) so(2n; C) SO(n; C) so(n) Spin(n) n(2n − 1) C C G2 g2 G2 g2 G2 14 C C F4 f2 F4 f4 F4 52 C C E6 e6 E6 e6 E6 78 C C E7 e7 E7 e7 E7 133 C C E8 e8 E8 e8 E8 248 Table 1: Classiﬁcation of Complex Simple Lie Algebras

geometry. In a series of papers, Freudenthal described the other exceptional Lie groups also in these terms [12]. The exceptional Lie algebras can be also constructed using the magic square, a technic developed independently by Freudenthal and Tits [7, Section 4.3]. Let g be a real compact semisimple Lie algebra. By Theorem 0.13, the associated simply connected Lie group G is compact and semisimple. Conversely, by the same theorem, if G is a compact and simply connected Lie group, then its Lie algebra is compact and semisimple. Since a simply connected group is determined by its Lie algebra and the real compact semisimple Lie algebras are classified, the classification of compact simply connected Lie groups is derived: If G is simple, it is isomorphic to one of the groups in the fifth column, otherwise, G is isomorphic to a direct product of such simple groups [1, Theorem 2.17].

0.3 Spin Groups

Spin groups are related to Cliﬀord algebras. They are introduced following [32, chapter 10] and a characterization in [33, Propositions 13.5-7].

0.3.1 Clifford Algebras Definition 0.12. Given a real vector space V of finite dimension n equipped with the Euclidean norm k.k and a orthonormal basis {e1, . . . , en}, the associative algebra freely generated by V modulo the relations

2 ei = −1 and eiej = −ejei for i 6= j (0.15) is called the Cliﬀord Algebra and is denoted by Cl(n).

In particular, v2 = −kvk2 ∀ v ∈ V. A basis for Cl(n) is

{ei1 . . . eir : i1 < ··· < ir , 0 ≤ r ≤ k}, where r = 0 denotes the element 1 of the basis. If V has dimension n, Cl(n) has dimension 2n. The vector space V is naturally embedded in Cl(n) and this inclusion is denoted by i. 18 0.3. SPIN GROUPS

Example 0.4 ([33], Proposition 13.22). Consider i, j, k the imaginary units in K. The following identiﬁcations e1 7→ i, e2 7→ j, e3 7→ k, provide the isomorphisms

Cl(0) ' R, Cl(1) ' C, Cl(2) ' K.

Since Cliﬀord Algebras are associative, the case n = 3 is not isomorphic to O as an algebra, even though they are identiﬁed as sets. For the next cases identify

i 0 j 0 k 0 e 7→ , e 7→ , e 7→ . 1 0 −i 2 0 −j 3 0 −k

This provides the isomorphisms

Cl(3) ' K ⊕ K and Cl(4) ' M2(K).

For the next cases see [7, Section 2.3]. The following table summarize the cases for n ≤ 8. n Cl(n) 0 R 1 C 2 K 3 K ⊕ K 4 M2(K) 5 M4(C) 6 M8(R) 7 M8(R) ⊕ M8(R) 8 M16(R) For n > 8, an interesting fenomenous occurs:

Cl(n + 1) ' Cl(n) ⊗ M16(R).

It is related to Bott periodicity and provides a proof of Hurwitz’s theorem.

0.3.2 Spin Groups Our main interest is about Spin groups. First, considerer the group of invertible elements Cl∗(n) = {v ∈ Cl(n): v 6= 0}.

Deﬁnition 0.13.

∗ Pin(n) = {a ∈ Cl (n): a = v1 . . . vr, vi ∈ V and kvik = 1} is the Pin group and

∗ Spin(n) = {a ∈ Cl (n): a = v1 . . . v2r, vi ∈ V and kvik = 1} is the Spin group. CHAPTER 0. PRELIMINARIES 19

−1 If v 6= 0, deﬁne the adjoint action Adv : V → V by Adv(x) = v.x.v . Recall that

v−1 = −v/kvk2 and v.x + x.v = −2hx, vi.

Then hx, vi Ad (x) = −x + 2 = −R (x), v kvk2 v where Rv denotes the reflection along of the line v. The map w 7→ −w is an isometry and it can be extended to an isomorphism w 7→ w˜. This map is called the canonical automorphism. To avoid the minus sign, define the ˜ −1 ˜ ˜ ˜ twisted adjoint action Adv(x) =vxv ˜ . Note that Adv ◦ Adw = Advw. If a ∈ Pin(n), ˜ then Adv ∈ O(n), because it is a composition of reflections. Since the determinant of a reflection is −1, the elements of Spin(n) are mapped to SO(n).

Theorem 0.14 ([26], Theorem 4.23). Every orthogonal transformation A ∈ O(n) can be expressed as the product of a ﬁnite number of reﬂections

A = Ru1 ◦ ... ◦ Rur along non-null lines, so that kujk= 6 0 for all j, 1 ≤ j ≤ r. Moreover, r ≤ n.

Hence the map Ad˜ restricted to Pin(n) and Spin(n) is surjective.

Proposition 0.4 ([26], Theorem 10.11). The following sequences are exact:

Ad˜ 1 Z2 Pin(n) O(n) I

Ad˜ 1 Z2 Spin(n) SO(n) I

In addition, the above maps are continuous and if n ≥ 2, Spin(n) is connected and if n ≥ 3 it is simply connected. Therefore, in the latter case, Spin(n) is the universal covering of SO(n) and, in particular, a double covering [30, p.16]. For n ≤ 6, spin groups are classical Lie groups. For n = 7, 8, 9 octonions provide useful models.

Example 0.5 (Spin(1) and Spin(2)). The double covering of the trivial group SO(1) is 1 1 O(1) ' Z2. The group SO(2) ' S has only one connected double covering, which is S itself. Therefore, Spin(1) ' O(1) and Spin(2) ' SO(2).

Example 0.6 (Spin(3) and Spin(4)). By (0.3) each g ∈ SO(3) corresponds to a conjugation by a unit quaternion. The map

ψ : Sp(1) −→ SO(3)

p 7−→ gp is a continuous and surjective homomorphism . If h ∈ Ker(ψ), h commutes with i, j and k ∈ K and has norm 1, then Ker(ψ) ' Z2. Thus Sp(1)/Z2 ' SO(3). Since Sp(1) is simply connected, Spin(3) ' Sp(1). 20 0.3. SPIN GROUPS

Deﬁne the map

Ψ : Sp(1) × Sp(1) −→ SO(4)

(q1, q2) 7−→ A,

4 −1 where A is a linear transformation of R ' K deﬁned by A(v) = q1.v.q2 , where v ∈ K. −1 The map Ψ is a homomorphism of groups, and if (q1, q2) ∈ Ker(Ψ), then q1.v.q2 = v −1 for all v ∈ K. This implies q1, q2 ∈ R, so Ker(Ψ) ' Z2. Note that Ψ(q, q ) = rq and −1 Ψ(q, q ) = cq, and these maps generate SO(4), by (0.4). Thus the map is surjective. Since Sp(1) × Sp(1) is simply connected, it follows that

Sp(1) × Sp(1) ' Spin(4).

For the cases n = 5 and n = 6, it holds that Spin(4) ' Sp(2) and Spin(6) ' SU(4) [32, p.141]. The cases n = 8 and n = 9 will be of special interest in this work. For the latter cases we based on [26, chapter 14] and [31].

0.3.3 The Group Spin(8)

Identify the 8-dimensional real vector space V (8) ' O with the subspace of EndR(O ⊕ O) 0 Lu V (8) = : u ∈ O , −Lu 0 0 Lu where Lu denotes the left multiplication. In addition, if Au = , deﬁne kAuk = −Lu 0 2 |u|. It holds that Au.Au = −|u| I2. Therefore, using the Fundamental Lemma of Cliﬀord

Algebras [26, Lemma 9.7], one can show that Cl(8) ' EndR(O ⊕ O). Thus, Spin(8) is generated by the 7-sphere Lu 0 2 : u ∈ O, |u| = 1 . (0.16) 0 Lu

8 2 The left multiplication Lu is a linear transformation of R ' O. Since |u| = 1, we must have Spin(8) ⊂ SO(8) × SO(8).

0.3.4 The Group Spin(9)

Identify the 9-dimensional real vector space V (9) ' R ⊕ O with the subspace of End (C ⊗ O2) C r Lu V (9) = i : r ∈ R, u ∈ O . Lu −r r Lu 2 2 If Ar,u = i , deﬁne kAr,uk = |u|. It holds that Ar,u.Ar,u = −(r + |u| )I2. Lu −r 2 Therefore, using the fundamental lemma, one can show that Cl(9) ' EndC(C ⊗ O ). Thus, Spin(9) is generated by the 8-sphere r Lu 2 2 : r ∈ R, u ∈ O and r + |u| = 1 . (0.17) Lu −r CHAPTER 0. PRELIMINARIES 21

See also [26, Lemma 14.7]. Note that A(r, u).A(s, v) is of the form

z L w , −Lw z with z, w ∈ O. Solving a linear equation in O, this shows that Spin(8) ⊂ Spin(9). 16 Moreover, each generator Ar,u is a linear transformation in O ⊕ O ' R which preserves the quadratic form, then Spin(9) ⊂ O(16). Note that det(A1,0) = 1, hence Spin(9) ∩ SO(16) 6= ∅. Since O(16) has two connected components and Spin(9) is connected, we must have Spin(9) ⊂ SO(16). The case n = 7 is treated in a similar way considering Im(O)[26, p.286].

0.4 Principal Bundles

We introduce principal fiber bundles. This concepts will be used to describe the tangent spaces of projective and hyperbolic spaces. The references are [2, Section 13.2] and [54, Sections 1.5 and 2.1, vol. I]. Definition 0.14. Let M be a manifold and G a Lie group. A principal fiber bundle with base M and fiber G is a manifold E such that: 1. G acts freely on E on the right, that is, (p, g) ∈ E × G is mapped to p.g ∈ E.

2. M is the quotient space E/G, and the canonical projection π : E −→ M is diﬀer- entiable.

3. E is locally trivial, that is, every point p ∈ M has a neighborhood U such that π−1(U) is isomorphic with U × G in the sense that there is a diffeomorphism ψ : π−1(U) −→ U × G such that ψ(x) = (π(u), φ(u)), where φ is a mapping of π−1(U) into G satisfying φ(xg) = (φ(x))g for all x ∈ π−1(U) and g ∈ G. For each q ∈ E there exists a neighborhood V of q diffeomorphic to U × G, where U is a neighborhood of M. It follows that each tangent space TqE of E contains a subspace isomorphic to the tangent space to the fiber. Moreover, the local condition p(φ(q)) = π(q) implies that this tangent space can be characterized as the kernel of (dπ)q. Its notation is VqE. Note also that VqE is isomorphic to the Lie algebra of G. While such spaces are intrinsic to every TqE, their complements are not necessarily. A connection is a suitable assingment of such complements on E. Definition 0.15. A connection in a principal bundle (E, M, F, π) is a smoothly-varying assignment to each point q ∈ E of a subspace HqE of TqE such that:

1. TqE = VqE ⊕ HqE.

2. (dφg)(HqE) = Hp.gE, for all g ∈ G, q ∈ E.

The space HqE is called the horizontal space at q and VqE the vertical space at q. Note that dπq (HqE) ⊂ Tπ(q)M and both spaces have the same dimension, therefore

dπq (HqE) = Tπ(q)M. (0.18)

Indeed, this condition is equivalent to condition 1 in Deﬁnition 0.15. 22 0.5. SYMMETRIC SPACES

0.5 Symmetric Spaces

Symmetric spaces form a special class of Riemannian manifolds with strong geometric properties. Since projective and hyperbolic planes lie in this class, important properties about these objects are obtained from the theory. In this section, M will always denote a Riemannian manifold with a ﬁxed Riemannian metric g.

0.5.1 Introduction We start with basic deﬁnitions and properties. Later, symmetric spaces will be recovered from other structures.

Deﬁnition 0.16 ([4], Deﬁnition 2.1). A Riemannian manifold M is a symmetric space if for each p ∈ M, there exists a isometry jp such that:

1. jp(p) = p.

2. djp = −IdTpM . jp is called involution or the symmetry at p. The next lemma justiﬁes the nomenclature.

Lemma 0.1 ([4], Lemmas 2.2 and 2.3). Let M be a Riemannian manifold, jp the symmetry at a point p, γ :[−1, 1] −→ M a geodesic such that γ(0) = p and γ(t0) = q. Then:

1. jp(γ(t)) = γ(−t) ∀ t ∈ [−1, 1].

2. jq ◦ jp(γ(t)) = γ(t + 2t0) ∀ t ∈ [−1, 1]. Proof.

1. Since jp is an isometry, jp(γ(t)) is also a geodesic. Moreover, jp(γ(0)) = p and

j˙p(γ(0)) = (jp)p.γ˙ (0)

=γ ˙ (0) ∈ TpM.

The geodesic γ(−t) satisﬁes the same equalities for t = 0, then the result follows from the theorem of existence and uniqueness.

2. The geodesicγ ˜(t) = γ(t + t0) satisﬁesγ ˜(0) = q. By item 1, it follows that

jq(jp) = jq(γ(−t))

= jq(˜γ(−t − t0)

=γ ˜(t + t0)

= γ(t + 2t0)

The property in item 1 is called Geodesic Reﬂection.

Example 0.7. CHAPTER 0. PRELIMINARIES 23

1. The euclidean space Rn endowed with usual metric is symmetric. The symmetry at n p ∈ R is given by jp(x) = 2p − x.

2. The sphere Sn endowed with the induced metric from Rn is symmetric. Let p = n (1, ..., 0) ∈ S and deﬁne jp(x1, ..., xn) = (x1, −x2, ..., −xn). For any other point q ∈ Sn, there exists an isometry A ∈ O(n + 1) ' Iso(Sn) such that A(p) = q. Thus, −1 jp ◦ A is the symmetry at x. Theorem 0.15 ([4], Theorem 2.11). Let M be a symmetric space. The identity component Iso0(M) of the isometry group Iso(M) acts transitively on M. In addition, Iso(M) has a Lie group structure and the action is smooth, hence M is a homogeneous space [4, Theorems 2.14 and 2.16]. It follows that M is a complete 2 Riemannian manifold and jp = Id.

0.5.2 Symmetric Pairs Symmetric pairs are one of the structures that characterize symmetric spaces. This relation and its deﬁnition are introduced here.

Deﬁnition 0.17. If G is a connected Lie group, K a closed subgroup of G, σ an involutive automorphism, and Gσ is the subgroup of elements ﬁxed by σ, then the pair (G, K, σ) is called a symmetric pair if:

1. (Gσ)0 ⊆ K ⊆ Gσ. 2. Ad(K) is a compact subset of GL(g).

If θ = (dσ)e and E1(θ) and E−1(θ) are the eigenspaces related to θ, then g = E1(θ)⊕E−1(θ) is the Cartan decomposition of g.

The relation between symmetric spaces and symmetric pairs is given by the next theorem.

Theorem 0.16. Let M be a symmetric space such that M ' G/K, where G = Iso0(M) and K the isotropy group at p for some p ∈ M. The involutive automorphism given by σp(g) = jp ◦ g ◦ jp, g ∈ G, satisﬁes

(Gσp )o ⊆ K ⊆ Gσp .

In particular, (G, K, σp) is a symmetric pair. Conversely, if (G, K, σ) is a symmetric pair, there exists a G-invariant metric on M ' G/K such that M is a symmetric space equipped with this metric. Furthermore, if π is the canonical projection and jo the symmetry at o = eK, then

π ◦ jo = σ ◦ π.

The next theorem shows the importance of actions of semisimple groups.

Theorem 0.17 ([3], Theorem V.4.1). Let (G, K, σ) be a symmetric pair. Suppose that G is a connected semisimple group and acts eﬀectively on the coset space M = G/K. Let g be any G-invariant Riemannian metric on M. Then G = Iso0(M) as Lie groups. 24 0.5. SYMMETRIC SPACES

0.5.3 Orthogonal Symmetric Lie Algebras Deﬁnition 0.18 ([5], Deﬁnitions 2.53 and 2.54). 1. A Lie subalgebra h ⊂ g is said to be compactly embedded if ad(h) is the Lie algebra of some compact subgroup of GL(g).

2. An orthogonal symmetric Lie algebra is a pair (g, θ), where g is a Lie algebra and θ 6= Id is an involutive automorphism such that E1(θ) is compactly embedded in g. An orthogonal symmetric Lie algebra is eﬀective if Z(g) ∩ E1(θ). In addition to the compact type, there are two distinct classes of orthogonal symmetric Lie algebras. Deﬁnition 0.19. Let (g, θ) be an orthogonal symmetric Lie algebra.

1. If B is non-degenerate and B restricted to E−1(θ) × E−1(θ) is positive deﬁnite, then g is said to be of non-compact type.

2. If E−1(θ) is an abelian ideal in g, then (g, θ) is said to be of euclidean type. Definition 0.20. 1. A symmetric pair is effective if the associated Lie algebra is effective.

2. A symmetric space is effective if the associated symmetric pair is effective. Symmetric pairs associated to symmetric spaces are effective [5, Lemma 2.66]. This fact closes the relation between symmetric spaces, symmetric pairs and orthogonal symmetric Lie algebras. Another remarkable fact is the following: Theorem 0.18 ([6], Theorem 35). Let M be a symmetric space. 1. If M is of compact type, the sectional curvature is non-negative and M is compact.

2. If M is of non-compact type, the sectional curvature is non-positive, M is simply connected and not compact.

3. If M is of euclidean type, the sectional curvature is zero.

0.5.4 Lie Triple Systems and Totally Geodesic Submanifolds Totally geodesic submanifolds of a symmetric space can be studied through of special subsets of the Lie algebra associated to the Lie group. Deﬁnition 0.21. Let n be a subset of a Lie algebra g. If [n, [n, n]] ⊆ n, then n is called a Lie triple system.

Theorem 0.19 ([5], Theorem 2.47). Let M be a symmetric space, G = Iso0(M), o = eK, K the isotropy group at o, and g = E1(θ) ⊕ E−1(θ) the Cartan decomposition.

1. If n ⊂ E1(θ) is a Lie triple system, then

N := Expo (deπ(n)) ⊆ M is a totally geodesic submanifold of M. CHAPTER 0. PRELIMINARIES 25

2. If N is a totally geodesic submanifold of M such that o ∈ M, then

−1 n := ((dπ)e) (ToN)

is a Lie triple system. Since the action is transitive and it is possible take a copy of N containing o, the condition over N in item 2 is not essential. Related to spaces of non-compact type, the next result will be of interest in the classiﬁcation of totally geodesic submanifolds. Theorem 0.20 (Mostow-Karpelevich - [50], Theorem 1.1). Let M be a Riemannian symetric space of non-compact type. Then any connected and semisimple subgroup G ⊂ Iso(M) has a totally geodesic orbit G.p ⊂ M. Remark 0.1 (Symmetric Subspaces). Totally geodesic submanifolds also can be characterized by symmetric pairs as follows. Given a symmetric pair (G, K, σ), a symmetric subspace consists of a connected Lie subgroup G0 ⊂ G invariant by σ, the intersection 0 0 0 0 0 0 K = G ∩ K, and the restriction σ = σ|G. In this case, M = G /K is a totaly geodesic submanifold of M = G/K. Conversely, given a totally geodesic submanifold M 0, let G0 the largest connected subgroup of G leaving M 0 invariant and K0 = G0 ∩ K, then (G0,K0, σ0) is a symmetric subspace. For details, see [54, Theorems 4.1 and 4.2].

0.5.5 Duality and Rank The duality theory establishes the correspondence between symmetric spaces of compact and non-compact type. In particular, projective and hyperbolic planes are closely related. As consequence, curvature, compactness, and totally geodesic submanifolds, for example, are correspondents in these spaces. ∗ If g = E1(θ) ⊕ E−1(θ) is an orthogonal symmetric Lie algebra, g = E1(θ) ⊕ iE−1(θ) is a R-subalgebra of gC. If ξ denotes the complex conjugation ξ(X + iY ) = X − iY in gC, ξ preserves g∗ and its restriction (which will also be denoted by ξ) to this subalgebra is an involutive automorphism. Furthermore, (g∗, ξ) is also an orthogonal symmetric Lie algebra [5, Lemma 2.86]. Deﬁnition 0.22. Let (g, θ) be an orthogonal symmetric Lie algebra, (g∗, ξ) is the dual orthogonal symmetric Lie algebra. Theorem 0.21 ([5], Proposition 2.88). 1. If (g, θ) is an orthogonal symmetric Lie algebra of non-compact (respectively compact) type, then the dual (g∗, ξ) is an orthogonal symmetric Lie algebra of compact (respectively non-compact) type.

2. If (g, θ) and (h, ω) are orthogonal symmetric Lie algebras, then (g, θ) ' (h, ω) if and only if (g∗, ξ) ' (h∗, ξ). Proof. For the next items, consider the isomorphism of real vector spaces

Ψ: E1(θ) ⊕ E−1(θ) −→ E1(θ) ⊕ iE−1(θ) (0.19)

(X1 + X2) 7−→ X1 + iX2. 26 0.5. SYMMETRIC SPACES

1. Given an isomorphism φ from (g, θ) to (h, ω), an isomorphism φ∗ from (g∗, ξ) to (h∗, ξ) is obtained by equation

∗ φ (X1 + iX2) = Ψ(φ(X1 + X2)),

and vice-versa. The statement follows from the fact that Ψ preserves the bracket:

Ψ[X1 + X2,Y1 + Y2] = [Ψ(X1 + X2), Ψ(Y1 + Y2))]. (0.20)

2. If Z1,Z2 ∈ E−1(θ), then

adg∗ (iZ1)adg∗ (iZ2)(X + iY ) = [iZ1, [iZ2,X + iY ]]

= [iZ1, [−Z2,Y ] + i[Z2,X]]

= −[Z1, [Z2,X]] − i[Z1, [−Z2,Y ]]

= −Ψ([Z1, [Z2,X]] + [Z1, [Z2,Y ]])

= −Ψ([Z1, [Z2,X + Y ]])

= −Ψ(adg(Z1)adg(Z2)(X + iY )),

where we have applied Deﬁnition 0.8 and (0.14). This implies that Bg∗ (iZ1, iZ2) = −Bg(Z1,Z2) and the result follows.

Definition 0.23. Suppose that (g∗, ξ) is the dual of the orthogonal symmetric Lie algebra ∗ ∗ (g, θ) and (G, K, σ1) and (G ,K , σ2) are the associated symmetric pairs. The associated symmetric spaces G/K and G∗/K∗ are called duals. We introduce now the notion of rank for symmetric spaces. Definition 0.24. The rank of a symmetric space is the maximal dimension of a flat totally geodesic submanifold, that is, the maximal dimension of a totally geodesic submanifold of curvature zero. Lemma 0.2 ([5], Lemma 3.8). Let (G, K, σ) be a symmetric pair of non-compact type, M = G/K, o ∈ M the origin, g = E1(θ) ⊕ E−1(θ). There is a one-to-one correspondence between flat totally geodesic submanifolds of M through o ∈ M and abelian subspaces of g contained in E−1(θ). Proof. By Theorem 0.19 there is a one-to-one correspondence between Lie triple systems and totally geodesic submanifolds. On the other hand, the G-invariant metric on G/K at o can be recovered as the restriction −Bg|p×p, where p = E1(θ). For an orthonormal frame (X1,X2) ∈ p the sectional curvature is

Bg([X1,X2], [X1,X2]).

Since Bg is negative deﬁnite on this restriction, the above equation vanishes if and only if [X1,X2] vanishes. Since abelian subspaces of p are trivially Lie triple systems, this proves the statement. Theorem 0.22 ([5], Theorem 3.12). Let (G, K, σ) be a symmetric pair of non-compact type with associated Cartan decomposition g = l ⊕ p. Further, let a, a0 ⊂ p be maximal 0 abelian subspaces. Then there is k ∈ K such that AdG(k)(a ) = a. CHAPTER 0. PRELIMINARIES 27

In particular, the dimension of such a space is invariant.

Remark 0.2. Let (M, g) and (M ∗, g∗) be dual symmetric spaces. Each totally geodesic submanifold N ⊂ M is associated to a subalgebra h ⊂ g and to a Lie triple system n ⊂ p, where g = l⊕p is the Cartan decomposition. The image of n under Ψ: g → g∗ is also a Lie triple system, because the bracket is preserved by (0.20). The associated totally geodesic subspace has associated Lie algebra h∗. We conclude that totally geodesic submanifolds are in correspondence, so the rank is an invariant of the pair.

We ﬁnish with Cartan’s classiﬁcation of symmetric spaces of rank one. Except for the sphere, they are the spaces studied in this work.

Theorem 0.23 ([19], Theorem 8.12.2). Let M be a symmetric space of rank one. Then M is isometric to one of the following spaces:

1. The euclidean sphere Sn ' O(n + 1)/ O(n). n 2. The F-projective space FP ' U(n + 1; F)/ U(n; F) × U(1; F). 2 3. The octonionic projective plane OP ' F4 / Spin(9). 4. The -hyperbolic space n ' U(1, n; )/ U(n; ) × U(1; ). F HF F F F 5. The octonionic hyperbolic plane 2 ' F* / Spin(9). HO 4 Chapter 1

F-Projective Spaces

In this chapter, we study the projective spaces over R, C, and K. They are also called the classical projective spaces and are in duality with the objects of chapter 3. In addition to construction, structure of compact symmetric space, and totally geodesic submanifolds, we also present some deﬁnitions and the fundamental theorem of projective geometry, this latter applied to the spaces in question. The main references are [29, Chapter 3] and [7].

1.1 Introduction

Projective geometry has its origins in the study of perspective by Renaissance painters. The concepts of points and lines are extended to a new space. In such spaces, there are no parallel lines, which contradicts the ﬁfth Euclid’s postulate. We introduce the basic deﬁnitions following [7, Section 3] and [39, Chapters 10 and 11].

Deﬁnition 1.1. A projective plane P is a triple consisting of a set of points, a set of lines, and a relation whereby a point lies on a line, also called incidence relation, satisfying the following axioms:

1. For any two distinct points, there is a unique line on which they both lie.

2. For any two distinct lines, there is a unique point which lies on both of them.

3. There exist four points, no three of which lie on the same line.

4. There exist four lines, no three of which have the same point lying on them.

Projective geometry is extendable to higher dimensions under the concept of projective space.

Deﬁnition 1.2. A projective space P is a triple consisting of a set of points, a set of lines, and a relation whereby a point lies on a line, also called incidence relation, satisfying the following axioms:

1. For any two distinct points P, Q, there is a unique line PQ on which they both lie.

2. For any line, there are at least three points lying on this line.

3. If A, B, C, D, are distinct points and there is a point lying on both AB and CD, then there is a point lying on both AC and BD.

28 CHAPTER 1. F-PROJECTIVE SPACES 29

Deﬁnition 1.3. Let P be a projective space and S = {A1,...,An} a ﬁnite set of points. The span of S is the smallest set T of points containing S and such that if A and B lie in T , the line AB is also contained in this set.

A subspace is a projective space. Moreover, if two finite sets span the subspace T , they have the same cardinality m. Thus, the dimension of T is defined to be m − 1. This definition establishes points, lines, and planes as 0,1, and 2-dimensional projective subspaces, respectively. Let Rn be the euclidean space. A k-subspace, with k ≤ n, is a set of the form

{α1x1 + ··· + αkxk : αi ∈ R}, for some fixed and linearly independent vectors x1, . . . , xk. Lines and planes are 0, 1- subspaces, respectively. Subspaces of the complex vector space Cn are defined in the same way. In this case, a k- complex vector subspace is identified with a 2k-real subspace. The lack of commutativity in K leads us to consider a right vector space structure in the space Kn+1. That is, the action of K∗ on Kn+1 is given by a right multiplication:

v.λ = (v1, . . . , vn)λ = (v1λ, . . . , vnλ), for v ∈ Kn+1 and λ ∈ K. Naturally, the right and left vector spaces are isomorphic. The deﬁnition of subspaces is extended, and a k-quaternionic subspace is identiﬁed with a 4k-real subspace. We conclude with the following facts for the vector space Fn+1:

1. Any two distinct lines in Fn determine uniquely a plane in this space.

2. Any two distinct planes in Fn passing though the origin intersect in a unique line also containing the origin.

3. Any plane in Fn contains at least three lines.

4. There exist four lines through origin, no three of which lie on the same plane.

5. There exist four planes, no three of which have the same line lying on them.

Let Fn+1 equipped with the canonical F-right vector space structure. Two non-zero vectors z, w ∈ Fn+1 are called equivalent if there exists λ ∈ F∗ such that z = wλ and we denote z ∼ w. This relation is reﬂexive and symmetric. Let z ∼ w ∼ v. Then z = wλ1, ∗ w = vλ2, for some λ1, λ2 ∈ F , and

z = (vλ1)λ2 = v(λ1λ2), (1.1) where we used the associativity of F. Therefore, ∼ is an equivalence relation.

n Deﬁnition 1.4. The F-projective space FP is deﬁned by

n n+1 FP = F − {0}/ ∼, where ∼ denotes the equivalence relation above. 30 1.1. INTRODUCTION

n The canonical map p : Fn+1 → FP sends a non-zero vector z ∈ Fn+1 to its class [z]. n A line in FP is a set of the form

n {[z] ∈ FP : z = v1λ1 + v2λ2}, (1.2)

n+1 where λi ∈ F and v1, v2 ∈ F are linearly independent and ﬁxed vectors. Thus, lines n and planes in Fn+1 corresponds to points and lines in FP , respectively. Comparing to n n 1.1, it follows that FP is a projective space. In general, k-subspaces of FP correspond to n k + 1-subspaces of Fn+1 under the projectivization map p. In particular, lines in FP are 1 isomorphic to FP . These spaces can be also described as quotients of spheres as follows. Each line in Rn+1 through the origin intersects the sphere Sn in two antipodal points. n Consider the action of Z2 = {Id, −Id} on the sphere. Thus, the points on S the corre- n spond to points in RP up to an action of Z2:

n n RP = S /Z2. (1.3)

Each complex line in Cn+1 passing through the origin intersects the sphere S2n+1 ⊂ Cn+1 ' R2n+2 in a circle S1. The action of the group U(1) on the sphere is deﬁned by

z.λ = (z0λ, . . . , znλ), where z ∈ S2n+1 and λ ∈ U(1). The point zλ lies on the same complex line that z, thus n in the same circle. It follows that the points in CP are in correspondence to points of S2n+1 up to an action of U(1):

n 2n+1 CP = S / U(1). (1.4)

Each quaternionic line passing through the origin in Kn+1 intersects the sphere S4n+3 ⊂ Kn+1 ' R4n+4 in a sphere S3. The action of the group Sp(1) on the sphere is similarly deﬁned. For λ ∈ Sp(1), the point zλ lies on the same quaternionic line that z, thus in n the same sphere S3. It follows that the points in KP are in correspondence to points on S4n+3 up to an action of Sp(1). Then

n 4n+3 KP = S / Sp(1). (1.5)

More succinctly, n dn+d−1 d−1 FP = S /S . n This identiﬁcation endows FP with the quotient topology and shows that these spaces are compact. The restriction of the map p to the spheres is denoted by π. The manifold structure is given by the following theorem.

n Theorem 1.1. The F-projective space FP is a dn-real-dimensional manifold.

n Proof. For every [z0, . . . , zn] ∈ FP at least one of the zi is non-zero. Let

Ui = {[z0, . . . , zn]: zi 6= 0}.

n The sets Ui, for i = 0, ..., n, are open in the quotient topology and cover FP . The maps

n dn ϕi : Ui −→ F ' R , CHAPTER 1. F-PROJECTIVE SPACES 31 given by z0 zi−1 zi+1 zn ϕi[z0, . . . , zn] = ,..., , ,..., zi zi zi zi −1 are smooths and has inverse ϕi : ϕi(Ui) −→ Ui which operates as

−1 ϕi (z0, . . . , zn) = [z0, . . . , zi−1, 1, zi+1, . . . , zn].

Thus ϕi is a diﬀeomorphism. For i < j, the transition functions

−1 ϕi ◦ ϕj : ϕj(Ui ∩ Uj) −→ Uj ∩ Ui −→ ϕi(Ui) are given by

−1 ϕi(ϕj (z0, . . . , zn)) = ϕi[z0, . . . , zj−1, 1, zj+1, . . . , zn] z z z z z z = 0 ,..., i−1 , 1, i+1 ,..., j−1 , 1, j+1 ,..., n zi zi zi zj zi zi −1 = ϕj ◦ ϕi .

n Therefore, the atlas {Ui, ϕ} gives a smooth real manifold structure for FP .

The charts above are called homogeneous coordinates. Let [z] ∈ Ui, we may suppose the i-th entry equals to 1. These coordinates are called non-homogeneous. If we restrict m n z to F0 ⊂ F, the spaces F0P , for m ≤ n, are obtained as submanifolds of FP . Moreover, 1 the projective line FP is diffeomorphic to the sphere Sd (see Theorem 2.1). n n Theorem 1.2. The projective spaces CP and KP are simply connected. S Proof. Set F = C, K and consider the coordinate charts (Ui, φ). Then i Ui is a cover to n FP . Moreover, Ui ∩ Uj = {[x0, ..., 1, ..., xj, ..., xn]: xj 6= 0} is diffeomorphic to Fn−1 × F − {0}, which is a path-connected space. In adittion, the n open sets Ui are simply connected, because they are diffeomorphic to F . The conclusion follows from van Kampen theorem.

Remark 1.1. Since R − {0} is not a path-connected set, the conclusion does not hold n n for RP . Indeed, the projection map π : Sn −→ RP deﬁnes Sn as the universal covering n of RP .

1.2 Collineation Groups

This section contains the deﬁnition of collineation maps and the statement of the fundamental theorem of projective geometry. Also, we apply this result to the projective n spaces FP .

Deﬁnition 1.5. A division ring is a ring (D, +,.) with the property that every non-zero element has a unique nonzero product inverse.

In this deﬁnition, a division ring is assumed to satisfy the associativity of the multiplication, but we will call alternative division ring when the associativity is replaced by alternativity. 32 1.2. COLLINEATION GROUPS

Deﬁnition 1.6.

1. A bijective function φ between projective spaces P1 and P2 is a collineation map if preserves collinearity, that is, for all triple of collinear points A, B, and C, φ(A), φ(B), and φ(C) are also collinear.

2. Let D be a division ring. A semilinear transformation of the right vector space Dn is a map T : Dn → Dn such that:

T (αv + βw) = T (v)σ(α) + T (w)β,

for all α, β ∈ A, v, w ∈ Dn, and σ a ﬁxed automorphism of D.

An automorphism σ of a division ring D induces the following map on Dn: X X viαi 7−→ viσ(αi). i i

The space Dn has a right-vector space structure and similarly we may construct the pro- n jective space DP . In this case, semilinear transformations induce a similar transformation n on DP which are collineations. The next theorem, also called fundamental theorem of projective geometry, shows the converse.

n Theorem 1.3 ([8], Theorem 3.1). Let D be a division ring and DP the associated pro- n n jective space. Every collineation map φ : DP → DP is induced by a semilinear automorphism of Dn.

The algebras F = R, C, and K are division rings. Then, the previous theorem can be applied for the projective spaces treated here.

n Theorem 1.4. The collineation group of the projective space FP is

1. The projective real linear group PGL(n + 1), if F = R.

2. The group generated by the projective complex linear group PGL(n + 1, C) and the collineation induced by the conjugation ξ on C, if F = C.

3. The projective quaternionic linear group PGL(n + 1, K), if F = K. Proof.

1. Recall that the identity is the unique automorphism of R. Thus, semilinear automorphisms are induced only from invertible linear transformations. Denote by H n the subgroup of GL(n + 1) which leaves RP pointwise ﬁxed. If A ∈ H,

A(v) = vλ,

for all non-zero vector v ∈ Rn+1 and some ﬁxed real λ. Then H is the subgroup of diagonal matrices: H = {λ.Id : λ ∈ R}. (1.6) The group H is the center of GL(n + 1) and the result follows. CHAPTER 1. F-PROJECTIVE SPACES 33

n 2. Similarly, the subgroup H of PGL(n + 1, C) leaving CP pointwise ﬁxed is

H = {λ.Id : λ ∈ C}.

The group H is the center of GL(n + 1, C) and the result follows.

n 3. If A ∈ H, the subgroup H of GL(n + 1, K) leaving KP pointwise ﬁxed, then A is a diagonal matrix λ.Id for some λ ∈ K and

λvi = viλ,

for all vi ∈ K. This implies that λ ∈ R and H is the same group in (1.6). On the other hand, an automorphism of K is of the form p 7→ qpq−1 for some ﬁxed q ∈ K∗. n The induced action on KP is

−1 −1 [v0, . . . , vn] 7−→ [qv0q , . . . , qvnq ] = [qv0, . . . , qvn],

which is a linear tranformation. Thefore, semilinear automorphisms are induced only by invertible linear transformations and the result follows.

1.3 Riemannian Metric

n In order to introduce a metric for FP , we describe its tangent space at a point. Although the description is in an abstract way, this metric is obtained taking the real part of a restriction of the usual bilinear form. We also derive the distance formula. First, we shall describe the structure of principal bundle (see Subsection 0.4). The n projection map π : Snd+d−1 −→ FP is a local diffeomorphism by the inverse function theorem. In particular, there exists a neighborhood V ⊂ Snd+d−1 such that the restriction of π to V is injective. Let U = π(V ). It follows that π−1(U) = V × Sd−1, which n is diffeomorphic to U × Sd−1. We conclude that (Snd+d−1, FP , Sd−1, π) is a principal fiber bundle. Derivating the equation hz, zi, the tangent bundle of the sphere Snd+d−1 is identified with

nd+d−1 nd+d−1 n+1 T S = (z, v): z ∈ S , v ∈ F , Rehz, vi = 0 .

Indeed, z and v can be seen as vectors of Rnd+d and Rehz, vi turns out to be the usual scalar product. Considering the general form of the Lie algebra u(1; F) (see Example 0.3), the tangent space to the ﬁber at z is identiﬁed with

nd+d−1 VzS = {(z, zλ): λ ∈ F, Reλ = 0} . (1.7)

Consider also the F-subspace

nd+d−1 n+1 HzS = (z, v): v ∈ F , hz, vi = 0 . (1.8)

Hence, we obtain an orthogonal decomposition

nd+d−1 nd+d−1 nd+d−1 TzS = VzS ⊕ HzS . 34 1.3. RIEMANNIAN METRIC

The action of Sd−1 on T Snd+d−1 is given by ((z, v), λ) 7−→ (zλ, vλ), (1.9) which leaves the decomposition invariant, because hzλ, vλi = λhz, viλ. Therefore, this deﬁnes a connection on the bundle, and the spaces (1.7) and (1.8) are the vertical and n nd+d−1 horizontal spaces, respectively. It follows that Tπ(z)FP is isomophic to HzS . More speciﬁcally, by (0.18), this tangent space is isomorphic to the set of classes

d−1 [zλ, vλ]: hz, vi = 0, λ ∈ S . (1.10) Now we move on to the Riemannian metric.

n Definition 1.7. The metric at a point ζ ∈ FP is defined by 1 hz, zi hdz, zi ds2 = det , (1.11) hz, zi2 hz, dzi hdz, dzi where z ∈ Fn+1 is a lifting of ζ, that is, p(z) = ζ. n Proposition 1.1. The expression (1.11) defines a Riemannian metric for FP . Proof. First, note that hvλ, vλi = |λ|2hv, vi for all v ∈ Fn and λ ∈ F, therefore the equation independs on liftings. Let z = (z0, ..., zn) non-homogeneous on Ui and λ ∈ F coordinate functions. If w = zλ, then dw = dzλ + zdλ. We obtain that

hw, wi = |λ|2hz, zi hdw, wi = |λ|2hdz, zi + |z|2dλλ hw, dwi = |λ|2hz, dzi + |z|2λdλ hdw, dwi = |λ|2|dz|2 + 2Rehdzλ, zdλi + |z|2|dλ|2.

Substituting these equalities in (1.11) results in the same expression deﬁned at z. This implies that (1.11) depends only on the class [z] = [w] in a smooth way. Finally, since h., .i is positive deﬁnite on Fn the same holds for ds2, by Cauchy-Schwarz inequality. This proves the statement.

n For F = R and ζ ∈ RP , we may suppose a lifting z ∈ Sn. It follows that hz, dzi = 0, and the metric becomes the induced metric of Rn. For F = C or K, this metric is called m n the Fubini-Study metric. Endowed with this metric, the embeddings F0P ⊂ FP , where 0 3 m ≤ n and F ⊂ F, are Riemannian. For a lifting (1, z2, z3) ∈ F , the expression for the metric is

2 3 2 2 2 2 2 |dz2| + |dz3| + |z2| |dz3| + |z3| |dz2| − 2Re[(z2z3)(dz3dz2)] ds = 2 2 2 (1.12) (1 + |z2| + |z3| ) In particular, if n = 1 the metric becomes

|dw|2 ds2 = . (1.13) (1 + |w|2)2

1 1 The projective lines CP and KP turn out to be isometric to euclidean spheres with constant curvature 4 (see Proposition 2.2). CHAPTER 1. F-PROJECTIVE SPACES 35

n nd+d−1 Let gζ denote the associated inner product on the tangent space Tζ FP ' HzS . 2 dn+d−1 Then gζ (dζ, dζ) = ds . Let p(z) = ζ and z ∈ S . We obtain that hz, dzi = 0 and 1 g (v , v ) = |v + v |2 − |v |2 − |v |2 , ζ 1 2 2 1 2 1 2

nd+d−1 for v1, v2 ∈ HzS . The F-hermitian product is not commutative, but considering its real part, we obtain gζ (v1, v2) = Rehv1, v2i. (1.14) Although the next proposition uses results from the following sections, it is convenient to state and prove it here.

n Theorem 1.5. Let ρ(p, q) denote the distance in FP . If z, w ∈ Fn+1 − {0}, then |hz, wi| cos (ρ(p(z), p(w))) = . (1.15) (hz, zihw, wi)1/2

n Proof. The equation is invariant under the action of the isometry group of FP , which acts transitively, and independs on liftings. Recall the identiﬁcation with spheres. We may suppose

z = (1, 0,..., 0), w = (cos α, sin α, . . . , 0),

1 for some α ∈ [0, π/2]. Consider the arcwise of RP connecting these points given by γ(t) = (cos t, sin t, . . . , 0), for 0 ≤ t ≤ α.

With respect to (1.11), |γ0(t)|2 = 1 1 It will be seen in Theorem 1.9 that RP is, up to isometry, the geodesic connecting p(z) and p(w). In other words, the distance between any two points can be calculated by translating them to a real projective line. Thus Z α ρ(p(z), p(w)) = dt = α. 0 On the other hand, ! hz, wi cos−1 = α, (hz, zihw, wi)1/2 which proves the statement.

m n Corollary 1.1. For F0 ⊂ F and m ≤ n the Riemannian embedding F0P ⊂ FP is isometric.

2 Taking two points in the coordinate chart U1 of FP with standard liftings z = 3 (1, z2, z3), w = (1, w2, w3) ∈ F , the expression becomes

|1 + z2w2 + z3w3| cos (ρ(p(z), p(w))) = 2 2 2 2 1/2 . (1.16) [(1 + |z2| + |z3| )(1 + |w2| + |w3| )] 36 1.4. THE PROJECTIVE SPACES AS SYMMETRIC SPACES

1.4 The Projective Spaces as Symmetric Spaces

n In order to associate a symmetric pair, we give the description of FP as a homogeneous space. Whence, these structures are compatible and the connected isometry group is obtained. We consider a involution σ : U(n; F) −→ U(n; F) given by

σ(A) = I1,nAI1,n.

1.4.1 The Real Case

The group O(n + 1) acts on Rn+1 leaving the sphere Sn = {z ∈ Rn+1 : hz, zi = 1} invariant. Moreover, this restricted action is transitive. Recall the identiﬁcation (1.3). It n follows that the map π induces a transitive action of O(n + 1) on RP . In fact, there is the following result:

n Theorem 1.6. The real projective space RP is diﬀeomorphic to the coset space O(n + 1)/ O(1) × O(n).

Proof. See the proof of Theorem 1.8.

The group O(n) is a subgroup of SO(n + 1) considering the inclusion

det(A) 0 A 7−→ 1×n . 0n×1 A

Then the real projective space can be also described as the quotient SO(n + 1)/ O(n). Recall the map σ on SO(n + 1). Then σ(A) = A if and only if A ∈ O(n). Since this group is compact and σ is an involutive automorphism, it follows that

(SO(n + 1), O(n), σ) is a symmetric pair. For F = R the metric (1.11) is O(n+1)-invariant. Moreover, PO(n+1) n is a simple group which acts eﬀectively on RP (see item 1 of Theorem 1.4). On the other n hand, each isometry of Sn+1 induces an isometry on RP and the full isometry group of n the sphere is O(n + 1). Conversely, the isometries of RP are induced from Sn. Thus

n Iso(RP ) = PO(n + 1).

1.4.2 The Complex Case

The group U(n + 1) acts on Cn+1 leaving the sphere S2n+1 = {z ∈ Cn+1 : hz, zi = 1} invariant. Moreover, this restricted action is transitive. Recall the identiﬁcation (1.4). It n follows that the map π induces a transitive action of U(n + 1) on CP . In fact, there is the following result:

n Theorem 1.7. The complex projective space CP is diﬀeomorphic to the coset space U(n + 1)/ U(1) × U(n).

Proof. See the proof of Theorem 1.8. CHAPTER 1. F-PROJECTIVE SPACES 37

The group U(n) is a subgroup of SU(n + 1) considering the inclusion

det(A) 0 A 7−→ 1×n . 0n×1 A

Then the complex projective space can be also described as the quotient SU(n+1)/ U(n). Recall the map σ on SU(n + 1). Then σ(A) = A if and only if A ∈ U(n). Since this group is compact and σ is an involutive automorphism, it follows that

(SU(n + 1), U(n), σ) is a symmetric pair. For F = C the metric (1.11) is SU(n + 1)-invariant. Moreover, n PU(n + 1) is a simple group which acts eﬀectively on CP (see item 2 of Theorem 1.4). By Theorem 0.17, it follows that

n Iso0(CP ) = PSU(n + 1).

1.4.3 The Quaternionic Case

The group Sp(n + 1) acts on Kn+1 leaving the sphere S4n+3 = {z ∈ Kn+1 : hz, zi = 1} invariant. Moreover, this restricted action is transitive. Recall the identiﬁcation (1.5). It n follows that the map π induces a transitive action of Sp(n + 1) on KP .

n Theorem 1.8. The quaternionic projective space KP is diﬀeomorphic to the coset space Sp(n + 1)/ Sp(1) × Sp(n).

Proof. By the previous paragraph, it remains to show that the isotropy group is Sp(1) × n+1 n Sp(n). Let he0, . . . , eni be the standard basis in K and W = he1, . . . , eni ' K . Suppose n that g ∈ Sp(n + 1) ﬁxes the point 0 ∈ KP , then g(e0) = e0λ, for some λ ∈ K. Note that

hg(e0), g(ei)i = he0λ, g(ei)i = he0, eii = 0, ∀i = 1, . . . , n.

λ 0 Therefore, g leaves W invariant and has the form , for some A ∈ Sp(n). 0 A

By the same map σ, (Sp(n + 1), Sp(1) × Sp(n), σ) is a symmetric pair. For F = K the metric (1.11) is Sp(n + 1)-invariant. The group n PSp(n + 1) acts eﬀectivelly on KP and it is simple (see item 3 of Theorem 1.4). By Theorem 0.17, it follows that

n Iso0(KP ) = PSp(n + 1).

1.5 The Matrix Model

n In this section, we construct a diﬀerent model for the projective space FP . This description is contained succinctly in [26, p.289] and is extended to octonionic case in Section 2.3. 38 1.5. THE MATRIX MODEL

1.5.1 The Real Case

For each vector v ∈ Rn+1 consider the projection on v given by a matrix A. Then, for w ∈ Rn+1, hv, wi A(w) = · v. (1.17) kvk2

2 v.vt It follows that A = A. Moreover, A = kvk2 and tr(A) = 1. In particular, A ∈ Hn+1(R). 2 Conversely, suppose that A ∈ Hn+1(R) and A = A. Since A is symmetric, there exists a diagonal matrix D and an orthogonal matrix P such that A = PDP −1. The eigenvalues of A are 0 and 1. The matrix A represents a projection on the space

n+1 W = {w ∈ R : A(w) = w}.

n −1 n If {ei}i=1 is an orthonormal basis for W , then {P (ei)}i=1 is also. Therefore

X −1 −1 tr(A) = tr(D) = hD(P (ei)),P (ei)i = dim(W ). (1.18) i

If dim(W ) = 1, choose a unit vector w ∈ W , then A = w.wt. Any other choice of a unit vector gives the same matrix. Thus

n 2 RP = {A ∈ Hn+1(R): A = A and tr(A) = 1}. (1.19)

n+1 Now consider a k-subspace V of R generated by linearly independent vectors v1, . . . , vk. The projection A on W is the linear sum of projections Ai on the lines generated by each vi: k X A(w) = Ai(w), i=1 where w ∈ Rn+1. Note that the matrix A has trace equals to k and it is symmetric. Conversely, given a idempotent matrix A in Hn+1(R) with trace k, equation (1.18) shows that A is a projection on a k-subspace W . Choosing k linearly independent vectors on W , A is of the form t t A = v1v1 + ··· + vkvk. n The real grassmannian space Gn,k(R) is the set of k-subspaces of R . The same reasoning enable us to identify

2 Gn,k(R) = {A ∈ Hn+1(R): A = A and tr(A) = k}. (1.20)

1.5.2 The Complex and Quaternionic Cases The same identiﬁcation can be extended to the complex and quaternionic cases. The diﬀerences with the real case are the Hermitian form, dimensions, and the structure of right vector space. In this subsection, F = C, K. A projection on a line in Fn+1 is also given by hv, wi A(w) = · v. (1.21) kvk2

vvt with respect to the Hermitian form and has the form kvk2 . Note that the matrix A is Hermitian idempotent, and has real trace equals to 1. CHAPTER 1. F-PROJECTIVE SPACES 39

Conversely, the well-known spectral theorem shows that a matrix in Hn+1(C) is diagonalizable by a matrix P ∈ U(n + 1). The quaternionic eigenvalue problem shows that the same is valid for a matrix in Hn+1(K), which is diagonalizable by a matrix P ∈ Sp(n + 1) [55, Theorem 3.3]. Then tr(A) = dim(W ), where

n+1 W = {w ∈ F : A(w) = w}. Thus n 2 FP = {A ∈ Hn+1(F): A = A and tr(A) = 1}. (1.22) n The grassmannian space Gn,k(F) is the set of k-subspaces of F . The same reasoning enable us to identify

2 Gn,k(F) = {A ∈ Hn+1(F): A = A and tr(A) = k}. (1.23)

1.6 Totally Geodesic Submanifolds

n In this section, we show the classiﬁcation of totally geodesic submanifolds of FP . To obtain this result, we apply the classiﬁcation of such spaces in the non-compact case by duality. However, a result of the suitable inclusion of algebras is proved here.

n Lemma 1.1 ([18], Lemma 1). Let N be a connected submanifold of FP . Then N is a n totally geodesic submanifold of FP which is isometric to an r-sphere if and only if N is 1 a totally geodesic submanifold of FP . Proof. Projective lines are totally geodesic submanifolds isometric to spheres (this is proved in Theorem 1.9 using duality, see also Proposition 2.2). Since totally geodesic 1 submanifolds of FP are euclidean spheres, the suﬃciency is proved. Suppose that N n is totally geodesic in FP and is isometric to a sphere. Choose p ∈ N, and let −p the 1 1 antipodal point of p in N. Given q ∈ N − {p, −p}, there is a copy γq of RP ' S which contains p, −p, and q. If Lq is the projective line determined by p and q, then Lq is a copy 1 of FP and γq ⊂ Lq. If L is the projective line determined by p and −p, it follows that L ∩ Lq contains p and −p. By item 1 of Deﬁnition 1.2, L = Lq. Since q is an arbitrary n point, N ⊂ L. A geodesic of N is a geodesic of FP which is contained in L. Since L is 1 also totally geodesic, it is a geodesic of L. Hence N is totally geodesic in FP . n m Corollary 1.2. Let F0 ⊂ F. Then FP is not a totally geodesic submanifold of F0P for any m, n.

1 m 1 Proof. Otherwise, FP ⊂ F0P would be totally geodesic. Since FP is isometric to a sphere, considering dimensions and Lemma 1.1 this is a contradiction.

n The classiﬁcation of totally geodesic submanifolds in FP is the following. n Theorem 1.9. Let M be a totally geodesic submanifold of FP . Then M is isometric to 1 r 1. S1 = RP or, FP (1 ≤ r ≤ n), for F = R; or r 2. Sr (1 ≤ r ≤ 2) or, FP (1 ≤ r ≤ n), for F = R, C; or r 3. Sr (1 ≤ r ≤ 4) or, FP (1 ≤ r ≤ n), for F = R, C, K. Moreover, all of these inclusions exist. 40 1.6. TOTALLY GEODESIC SUBMANIFOLDS

Proof. The statement follows by duality and the one-to-one correspondence between Lie triple systems and totally geodesic submanifolds (Theorem 0.19, and Remark 0.2) from Theorem 3.12. According to the classiﬁcation of irreducible symmetric spaces of rank one, the dual of r ⊂ n in n is r or r (see Theorem 3.3). On the other hand, the HR HF FP RP S following dualities hold

1 1 HR ←→ RP ; 1 1 HC ←→ CP ; (1.24) 1 1 HK ←→ KP . Therefore, if r is not contained in a quaternionic hyperbolic line, its dual in n must HR FP be r. This implies that the real space 3 ⊂ 1 is associated to the Lie algebra RP HR HK so(1, 3) ⊂ sp(1, 1) (see Theorem 3.3). The correspondences are

r so so r HR ←→ (1, r) ←→ (r + 1) ←→ RP ; r su su r HC ←→ (1, r) ←→ (r + 1) ←→ CP ; (1.25) r sp sp r HK ←→ (1, r) ←→ (r + 1) ←→ KP . where r 6⊂ 1 . The special case is HR HK 3 1 so sp sp 3 1 HR ⊂ HK ←→ (1, 3) (⊂ (1, 1)) ←→ (1) ←→ S ⊂ KP .

m n Remark 1.2. The spaces F0P ⊂ FP can be obtained as totally geodesic submanifolds also considering Lie triple systems. For this purpose it is necessary to use the expression of the Lie bracket on the associated Lie algebras. Chapter 2

The Octonionic Projective Plane

The projective plane coordinated by the octonions was ﬁrst discovered by Ruth Mo- ufang [44]. Moufang’s construction derives from axiomatic projective geometry and hence in an abstract way. In the middle of the 20th century, mathematicians such as M. Hirsh, P. Jordan, A. Borel, and H. Freudenthal advanced in the discovery of characterizations of these spaces. The main construction is based on a special Jordan algebra. This matter is the content of Section 1.2, and it will be used in Section 1.3 for the projective structure of the Matrix model. The second connection between octonions and exceptional structures arises from the characterization of this projective plane as a homogeneous space, which is done in Section 1.5. The last section illustrates the close relation between projective planes over classical algebras and the octonionic plane. While this chapter closes the study of projective planes over the four normed division algebras, their results are in correspondence with Chapter 4 and will be used there. The main references are [7], [13], and [26, Chapter 14].

2.1 The Octonionic Projective Line

The lack of associativity motivates us to define the octonionic projective line by extend- ing the matrix model. As expected, this object is diffeomorphic to the compactification of O ' R8 and axioms 1.1 are immediatelly verified. We also describe an approach by equivalence sets of O2. A notion of a line in O2 can be recovered. This space is related to the Bott peridiocity, Lorentzian geometry, and Hopf fibrations [7, Section 3.1].

1 Deﬁnition 2.1. The octonionic projective line OP is the set of points

2 {X ∈ H2(O): X = X and tr(X) = 1} (2.1) taken projectively, that is, each matrix X is identiﬁed with the vector subspace consisting of its non-zero scalar multiples. The identity matrix I2 represents the unique line which all points lies on.

1 Remark 2.1. For convenience, OP is usually identiﬁed with the set 2.1. 1 If X ∈ OP , a calculation (see Theorem 2.6) shows that X is of the form

2 t |z1| z1z2 zz = 2 , (2.2) z2z1 |z2|

41 42 2.1. THE OCTONIONIC PROJECTIVE LINE

2 2 1 with z1, z2 ∈ O and |z1| +|z2| = 1. Another useful model to OP is the following. Deﬁne

U1 = 1 × O,U2 = O × 1,U = U1 ∪ U2. The pairs (1, y) and (x, 1) in U are called equivalent if and only if x, y 6= 0 and x = y−1. Denote by ∼ this relation and the class of (1, y) and (x, 1) by [1, y] and [x, 1], respectively. The map

1 ψ : U/ ∼ −→ OP z 7−→ zzt

1 provides an isomorphism from U/ ∼ to OP . 1 The notion that a point in OP represents a line passing through the origin in O2 can 2 −1 be recovered as follows. If z = (x, y) ∈ O is a non-zero element, set z1 = (1, x y), if −1 t t x 6= 0, and z2 = (y x, 1), if y 6= 0. In both cases zz is a scalar multiple of z1z1 and t z2z2 . For each α ∈ O: −1 2 −1 zα = 1, α(α(x y)) = |α| 1, (x y) , −1 2 −1 wα = α(α(y x)), 1 = |α| (y x), 1 , by Artin’s theorem. Hence

−1 t −1 t ψ (zαzα ) = ψ (z1z1 ) = [z1] −1 t −1 t ψ (wαwα ) = ψ (z2z2 ) = [z2].

Therefore, the line passing trough the origin and (x, y) in O2 is the set:

−1 {(α, α(x y)) : α ∈ O} , if x 6= 0, −1 {(α(y x), α): α ∈ O} , if y 6= 0.

1 Theorem 2.1. Let F = R, C, K, or O. Then the projective line FP is a d-dimensional diﬀerentiable manifold diﬀeomorphic to Sd. 1 Proof. The following two charts cover FP :

d ϕ1 : U1/ ∼ −→ F ' R [1, x] 7−→ x d ϕ2 : U2/ ∼ −→ F ' R [x, 1] 7−→ x

−1 d d The transition functions ϕi ◦ ϕj : R −→ R are:

−1 −1 −1 −1 ϕ1 ◦ ϕ2 (x) = ϕ1([1, x ]) = x = ϕ1 ◦ ϕ2 (x). The map

1 d d f : FP −→ R ∪ {∞} ' S [1, x] 7−→ y [0, 1] 7−→ ∞

1 is a diﬀeomorphism from FP to the sphere Sd. CHAPTER 2. THE OCTONIONIC PROJECTIVE PLANE 43

2.2 The Jordan Algebra H3(O) In 1949, P. Jordan stated without proof a presentation of the octonionic projective plane using idempotent matrices in a special algebra over O. Independently, Freudenthal systematized this approach introducing multilinear forms, which provides a projective structure to the matrix model [12, Sections 4.6 and 4.7]. Deﬁnition 2.2. A Jordan algebra J is an algebra with a bilinear Jordan product denoted by X ◦ Y satisfying: 1. X ◦ Y = Y ◦ X

2. (X2 ◦ Y ) ◦ X = X2 ◦ (Y ◦ X) for all X,Y ∈ J. Note that X2 = X ◦ X where X2 is the usual algebra product. Example 2.1.

1. The set Mn(F) is a Jordan algebra with the Jordan product given by 1 X ◦ Y = (XY + YX), 2 where XY is the usual matrix product. The associativity of F is essential to verify item 2 of Deﬁnition 2.2.

∗ 2. Let Hn(F) be the hermitian matrices with entries in F. The condition X = X implies aij = aji for all i, j and (XY )∗ = (XY )T = (Y X)T = X∗Y ∗ = XY.

Note that these calculations still valid if F = O and n = 3. If X,Y ∈ Hn(F): 1 (X ◦ Y )∗ = ((XY )∗ + (YX)∗) 2 1 = (Y ∗X∗ + X∗Y ∗) 2 = X ◦ Y.

Therefore, Hn(F) is a Jordan subalgebra of Mn(F).

3. In general, if A is an associative algebra, a Jordan algebra JA is obtained considering the Jordan product 1/2(XY + YX). Deﬁnition 2.3. Let J be a Jordan algebra.

2 2 1. J is formally real if for Xi ∈ J, X1 + ...Xn = 0 implies Xi = 0 for all i.

2. J is special if it can be described as a Jordan subalgebra JA of some associative algebra A. 44 2.2. THE JORDAN ALGEBRA H3(O)

3. J is exceptional if it is not special.

The algebra Mn(F) is formally real and Hn(F) is special. For the octonionic case there is the important Albert’s theorem:

Theorem 2.2 ([16], Principal Theorem). The set

∗ H3(O) = {X ∈ M3(O): X = X} equipped with the product X ◦ Y = 1/2(XY + YX) is an exceptional and formally real Jordan algebra of dimension 27.

Deﬁnition 2.4. In H3(O) are deﬁned: 1. A quadratic form by 1 Q(X) = tr(X2). (2.3) 2 The correspondent bilinear form, denoted by (., .), is

1 (X,Y ) = tr(X ◦ Y ). (2.4) 2

2. The Freudenthal product by 1 1 1 1 X ∗ Y = X ◦ Y − (Y,I )X − (X,I )Y − (X,Y )I + (X,I )(Y,I )I . (2.5) 2 3 2 3 2 3 2 3 3 3 Equivalently, 1 1 X ∗ Y = X ◦ Y − (Xtr(Y ) + Y tr(X)) + I (tr(X)tr(Y ) − tr(X ◦ Y )). (2.6) 4 4 3

3. A symmetric trilinear form by 1 (X,Y,Z) = (X,Y ∗ Z). 3

4. The determinant by 1 det(X) = (X,X,X). 3 Equivalently, 1 1 1 det(A) = tr(A)3 − (tr(A)2)(tr(A)) + tr(A3) (2.7) 6 2 3

The bilinear form 2.3 is related to the usual dot product in euclidean space as follows. Let     r1 x11 ··· x1n−1 s1 y11 ··· y1n−1  x r ··· x   y s ··· y   11 2 2n−2  11 2 2n−2 X =  . . .. .  and Y =  . . .. .  ∈ Hn(F),  . . . .   . . . .  x1n−1 ······ rn y1n−1 ······ sn CHAPTER 2. THE OCTONIONIC PROJECTIVE PLANE 45

n(d+1) where F = R, C, K or O (n ≤ 3). If dim(F) = d, then Hn(F) ' R and

n 1 1 X tr(X ◦ Y ) = (2r s + x y + y x ) 2 2 i i ii ii ii ii i=1 n 1 X = 2r s + 2Re(x y ) 2 i i ii ii i=1 = hX,Y i. (2.8)

In the last equality X and Y are seen as vectors in Rn(d+1). A useful relation envolving the above forms is given by the next theorem.

Theorem 2.3 ([26], Equation 14.97). A matrix A ∈ H3(O) satisﬁes

3 2 A − tr(A)A + σ(A)A − det(A)I3 = 0, (2.9)

1 2 2 where σ(A) = 2 (tr(A) − tr(A )).

Deﬁnition 2.5. A linear map g ∈ GL(H3(O)) is an automorphism of the Jordan Algebra H3(O) if g(A ◦ B) = g(A) ◦ g(B) for all A, B ∈ H3(O).

2 The condition g(A ) = g(A) ◦ g(A) over a linear map g ∈ GL(H3(O)) implies g ∈ F4 by polarization.

Theorem 2.4 ([25], Theorem). The compact simply connected real Lie group of type F4 is the automorphism group of the Jordan algebra H3(O) and it is denoted by F4.

The compactness of F4 can be veriﬁed as follows. Recovering the dot product in 27 R ' H3(O) by (2.8), 1 hg(X), g(Y )i = tr(g(X) ◦ g(Y )) 2 1 = tr(g(X ◦ Y )) 2 = hX,Y i, by Lemma 2.1. Then F4 ⊂ O(27) is a compact group.

Lemma 2.1 ([26], Lemma 14.96). Let g ∈ F4. Then

tr (g(A)) = tr(A) for all A ∈ H3(O). Consequently, the group F4 ﬁxes (., .), det, and the Freudenthal product.

Proof. Since g ∈ F4 is a bijection, g(I3) = I3. If A ∈ H3(O), the matrix tr(A) A0 = A − I 3 3 46 2.2. THE JORDAN ALGEBRA H3(O) has zero trace. Then tr(g(A0)) = tr(A0) if and only if tr(g(A)) = A. So, it is suﬃcient prove that g preserves matrices with zero trace. Consider

3 Λ1 = {A ∈ H3(O): A = aA + bI3, for some a, b ∈ R} and 2 Λ2 = {A ∈ H3(O): A = aA + bI3, for some a, b ∈ R}.

The sets Λi are invariants by the action of F4. If tr(A) = 0, by (2.3), A ∈ Λ1. If B = g(A), then 3 B = aB + cI3, (2.10) for some a, b ∈ R. Replacing B in (2.3) and subtracting from (2.10),

2 tr(B)B − (σ(B) + a)B + (det(B) − b)I3 = 0. (2.11)

Let Ω = {A ∈ H3(O): tr(A) = 0}.

If B/∈ Λ2, then tr(B) = 0 by (2.11). Thus g maps Ω \ Λ2 to Ω. Let A ∈ H3(O) such that A = diag(λ1, λ2, λ3) with distinct λi and zero trace. Therefore, A ∈ Ω \ Λ2 and this set is non-empty. Note that Ω is a hyperplane in R27, then it is open and connected. It follows that, Ω \ Λ2 = Ω and g maps Ω to itself.

We shall describe brieﬂy the following subgroups of F4 that will appear later. For details see [27, Sections 2.11 and 2.12].

T SO(3): For A ∈ H3(O) and g ∈ SO(3) deﬁne g(A) := gAg . Then g(A) ∈ H3(O) and

g(A)2 = (gAgT )(gAgT ) = gA2gT = g(A2),

which is an embedding of SO(3) in F4. SU(3): The Jordan algebra H3(O) can be decomposed as H3(C) ⊕ M3(C), because 3 0 O ' C ⊕ C . Given g ∈ SU(3) and A = A + M ∈ H3(C) ⊕ M3(C) one can show that the map g(A0 + M) := gA0g∗ + Mg∗ (2.12)

deﬁnes an action of SU(3) in H3(O) which preserves the Jordan product. For details see [27, Theorem 2.12.2].

3 Sp(3): The Jordan algebra H3(O) can be decomposed as H3(K) ⊕ K . Given g ∈ 0 3 Sp(3) A = A + a ∈ H3(K) ⊕ K one can show that the map

g(A0 + a) := gA0g∗ + ag∗ (2.13)

deﬁnes an action of Sp(3) in H3(O) which preserves the Jordan product. For details see [27, Theorem 2.11.2]. CHAPTER 2. THE OCTONIONIC PROJECTIVE PLANE 47

2.3 The Octonionic Projective Plane

Given the machinery developed in the previous section, we are in conditions to deﬁne the octonionic projective plane. Besides the matrix model, an approach by equivalence sets of O3 is studied. It provides an useful identiﬁcation that is similar to the other projective planes. 2 The cases FP leads us to consider the set

2 {X ∈ H3(O): X = X and tr(X) = 1} (2.14) to construct the projective plane over O. The next deﬁnition puts the projectivization of (2.14) in suitable terms.

2 Deﬁnition 2.6. The octonionic projective plane OP is the set of points

{X ∈ H3(O): X 6= 0,X ∗ X = 0} (2.15) taken projectively, that is, each matrix is identiﬁed with the vector subspace consisting of its real non-zero scalar multiples. We shall abuse the notation and also denote by X the 2 class of this point. The lines in OP are the same objects (same notation). A point X lies on a line Y if (X,Y ) = 0.

A matrix X that satisﬁes X2 = X is called an idempotent matrix. The next lemma 2 provides an useful characterization of matrices in OP .

Lemma 2.2 ([13], Lemma 1). An element X ∈ H3(O) satisﬁes X ∗ X = 0 if and only if X = λX0, where X0 is an idempotent matrix with trace 1 and λ ∈ R. 2 Remark 2.2. Considering Lemma 2.2, OP is usually identiﬁed with the set (2.14). Lemma 2.3 ([13], Lemmas 4 and 5).

1. If X,Y ∈ H3(O) and X ∗ X = Y ∗ Y = 0, X and Y are linearly independent over O, then X ∗ Y 6= 0.

2. If X,Y,Z ∈ H3(O), X ∗ X = Y ∗ Y = Z ∗ Z = 0, (X,Y ) = (X,Z) = 0, Y and Z are linearly independent over O, then Y ∗ Z = λX for some non-zero real λ. Theorem 2.5. The structure in Deﬁnition 2.6 satisﬁes axioms 1.1.

Proof. If X and Y are distinct points, by item 1 of Lemma 2.3, X ∗ Y 6= 0. Since

(X ∗ Y,Y ) = (X,Y,Y ) = (X,Y ∗ Y ) = 0 and (Y ∗ X,X) = (Y,X,X) = (Y,X ∗ X) = 0, then X ∗ Y is a line passing through X and Y . If Z is a line through X and Y , then by item 2 of Lemma 2.3, Z = X ∗ Y . Thus Axiom 1 is verified. If X and Y are distinct lines, by the same argument, X ∗ Y is the unique point in both of them. Considering solutions for a system of equations in O, Axiom 3 is readily verified. When defining projective spaces as a subset of hermitian matrices with entries in F, 2 we obtain the general form of such matrices (Section 1.5). The same occurs with OP . 48 2.3. THE OCTONIONIC PROJECTIVE PLANE

2 Theorem 2.6 ([26], Lemma 14.90). Every element X ∈ OP has a representative of the form  2  |z1| z1z2 z1z3 t 2 zz = z2z1 |z2| z2z3  , (2.16) 2 z3z1 z3z2 |z3| 3 where z = (z1, z2, z3) ∈ O lies on an associative subalgebra. Consequently, every element 1 X ∈ OP , has a representative of the form

2 t |z1| z1z2 zz = 2 . (2.17) z2z1 |z2|

Proof. Let   r1 x1 x2 X = x1 r2 x3 ∈ H3(O). x2 x3 r3 By Lemma 4.2, X2 = λX with tr(X) = λ 6= 0. We may suppose λ = 1. From this facts, the following equations are obtained:

(r1 + r2)x1 + x2x3 = x1 ⇒ r3x1 = x2x3

(r1 + r3)x2 + x1x3 = x1 ⇒ r2x2 = x1x3

(r2 + r3)x3 + x1x2 = x1 ⇒ r1x3 = x1x2

One of the ri is nonzero, otherwise the matrix would be null. Thus, x1, x2, and x3 belong to an associative subalgebra A of O. If r1 6= 0, we may suppose r1 > 0 and set

2 r1 = |z1| ; x1 = z1z2; x2 = z1z3, for some z1, z2, z3 in A. Using associativity,

2 2 r2 = |z2| ; r3 = |z3| ; x3 = z2z3.

2 If r1 = 0, then set r2 = |z2| . If r1 = r2 = 0, r3 must be the unique non-null entry. Thus, the representative has the desired form. The consequence is veriﬁed considering the following embedding of H2(O) in H3(O): 0 0 0 x z 7−→ 0 x z . z y   0 z y

2 Theorem 2.6 shows that the points of OP are given by elements in an associative 2 subalgebra of O. Conversely, another useful model to OP is obtained from an associative subalgebra of O3. Deﬁne

U1 = {1} × O × O, U2 = O × {1} × O, U3 = O × O × {1}, U = U1 ∪ U2 ∪ U3. CHAPTER 2. THE OCTONIONIC PROJECTIVE PLANE 49

Two triples z = (z1, z2, z3) and w = (w1, w2, w3) in U are related if and only if there exists λ ∈ O∗ such that z = wλ and we denote z ∼ w. In O3 this relation is not a equivalence relation but in U it is, due to Artin’s theorem. Indeed, only transitivity need to be checked. If (1, z2, z3) ∼ (w1, 1, w3) ∼ (u1, u2, 1), −1 take λ = u1 and we obtain (1, z2, z3) ∼ (u1, u2, 1). The class of (z1, z2, z3) is denoted by [z] = [z1, z2, z3]. The map

2 ψ : U/ ∼ −→ OP (2.18) z 7−→ zzt

2 provides an isomorphism from U/ ∼ to OP . In [10] H. Aslaksen gives a projective structure to the second model and D. Allcock in [9] shows that the map ψ also identiﬁes this structure and that of Deﬁnition 2.6. Consider the set

3 3 O0 = {(z1, z2, z3) ∈ O : z1, z2, z3 lie in some associative subalgebra of O}. (2.19) 3 Two triples z, w ∈ O0 are called equivalent if there exists a nonzero λ ∈ O which belongs in the same associative subalgebra that z satisfying w = zλ. This deﬁnes an equivalence relation, which we will also denote by ∼. Note that

3 2 O0/ ∼ = U/ ∼ = OP . (2.20) Thus, we may recover a projectivization map

3 2 p: O0 −→ OP z 7−→ zzt

2 1 Theorem 2.7. The lines in OP are isomorphic to OP . Proof. Given a point Y on a line X, by the map ψ in (2.18), X = xxt and Y = yyt can be chosen with at least one of the xi equals to 1 and the same for yi. By the Lemma in [9], x1y1 + x2y2 + x3y3 = 0. This equation shows that the two remaining entries are related by a linear equation. Thus, 1 the point Y is determined by an octonion, say y0. Map Y to [1, y0], X ⊂ OP . Conversely, 1 given [1, y1] or [y2, 1] in OP , y3 can be determined to the above equation be satisﬁed. 1 Hence Y = yyt is a point of X and OP ⊂ X. Both the inclusions are injective, then the 1 line determined by X is isomorphic to OP . 2 Theorem 2.8. The octonionic projective plane OP is a 16-dimensional compact simply connected diﬀerentiable manifold. 2 Proof. The following three charts cover OP and are homeomorphisms: 2 16 ϕ1 : U1/ ∼ −→ O ' R [1, z2, z3] 7−→ (z2, z3) 2 16 ϕ2 : U2/ ∼ −→ O ' R [z1, 1, z3] 7−→ (z1, z3) 2 16 ϕ3 : U3/ ∼ −→ O ' R [z1, z2, 1] 7−→ (z1, z2) 50 2.4. RIEMANNIAN METRIC

−1 16 16 The transition functions ϕi ◦ ϕj : R −→ R are:

−1 −1 −1 −1 −1 −1 ϕ1 ◦ ϕ2 (z1, z2) = ϕ1([1, z1 , z2z1 ]) = (z1 , z2z1 ) = ϕ2 ◦ ϕ1 (z1, z2); −1 −1 −1 −1 −1 −1 ϕ1 ◦ ϕ3 (z1, z2) = ϕ1([1, z2z1 , z1 ]) = (z2z1 , z1 ) = ϕ3 ◦ ϕ1 (z1, z2); −1 −1 −1 −1 −1 −1 ϕ2 ◦ ϕ3 (z1, z2) = ϕ2([z1z2 , 1, z2 ]) = (z1z2 , z2 ) = ϕ3 ◦ ϕ2 (z1, z2).

8 2 The sets Ui ' R , for i = 1, 2, 3, are closed, so U is also closed. Then OP ' U/ ∼ 2 is a closed set and Theorem 2.6 shows that it is also limited, hence OP is compact. For simply connectedness, the argument is similar to the projective case. We only note 8 8 that Ui ∩ Uj ' R × (R − {0}) is path-connected. The result follows from van Kampen theorem.

Remark 2.3.

1. Considering the projective strucuture given by Aslaksen in [10] and Alcock’s iden- tiﬁcation via the map (2.18), the projective structure in Deﬁnition 2.15 is the same for the matrix models in Section 1.5.

2 2 2. Since H3(F) is a Jordan subalgebra of H3(O), FP ⊂ OP . On the other hand, 2 the charts of FP are exactly the restrictions to F of the charts {Ui, ϕi} above. 2 2 1 Therefore, FP is a smooth submanifold of OP , as well as FP .

2.4 Riemannian Metric

2 We study a metric for OP following [17] and [56]. We also derive a formula for 2 distance. This metric generalizes the Fubini-Study metric for FP . Recall the form (X,Y ) = 1/2(tr(XY + YX).

2 Deﬁnition 2.7. The metric at a point ζ ∈ OP is deﬁned by 1 ds2 = [tr(Z)tr(dZ) − (Z, dZ)] , (2.21) (tr(Z))2

3 t t where z ∈ O0, with p(z) = ζ, Z = zz , and dZ = dzdz . Proposition 2.1 ([17], Theorem 3.3). The expression (2.21) deﬁnes a Riemannian metric 2 on OP .

3 We shall detail the expression on the coordinate chart U1/ ∼. Let z = (1, z2, z3) ∈ O , then dz = (0, dz2, dz3). Moreover,

2 2 2 2 2 2 2 2 2 2 tr(Z)tr(dZ) = |dz2| + |dz3| + |z2| |dz2| + |z3| |dz3| + |z2| |dz3| + |z3| |dz2| . 2 2 2 2 (Z, dZ) = |z2| |dz2| + |z3| |dz3| + Re[(z3z2)(dz2dz3)] + Re[(z2z3)(dz3dz2)].

Recall the identities in Proposition 0.3. It follows that

Re[(z3z2)(dz2dz3)] = Re[(dz2dz3)(z3z2)]

= Re[(dz2dz3)(z3z2)]

= Re[(z2z3)(dz3dz2)]. CHAPTER 2. THE OCTONIONIC PROJECTIVE PLANE 51

Therefore

2 3 2 2 2 2 2 |dz2| + |dz3| + |z2| |dz3| + |z3| |dz2| − 2Re[(z2z3)(dz3dz2)] ds = 2 2 2 . (2.22) (1 + |z2| + |z3| )

2 2 2 Restricting (z2, z3) ∈ F , by (2.22), the embeddings FP ⊂ OP are Riemannian. The 1 diﬀeomorphism OP ' S8 is, in fact, an isometry, as shows the next proposition.

1 Proposition 2.2. Let F = C, K, or O. Then the projective line FP is isometric to the euclidean sphere Sd with constant curvature 4.

1 Proof. A point Z ∈ OP is of the form 0 0 0  Z = 0 |w|2 wz  , 0 zw |z|2 where z, w ∈ O with z 6= 0 or w 6= 0. We suppose w = 1, because the computations for z = 1 are identical. It follows that dz = diag(0, 0, |dz|2) and the metric becomes

|dz|2 ds2 = , (1 + |z|2)

1 which is the same for FP , see (1.13). Recall the diﬀeomorphism from Theorem 2.1. We shall specify the point correspondent to z on Sd via stereographic projection. If d x = (x1, ..., xd+1) ∈ S is the desired point, the line joining x and the north pole (0, ..., 1) d d intersects R ' F in z. Let z = (z1, ..., zd) ∈ R . Then

xi zi = , for i = 1, ..., d. 1 − xd+1

2 2 Since x1 + ... + xd+1 = 1, it holds that

2z |z|2 − 1 x = i , x = , for i = 1, ..., d. i |z|2 + 1 d+1 |z|2 + 1

Whence 2 P 2dzi(|z| + 1) − 4zi ( i zidzi) dxi = , (|z|2 + 1)2 for i = 1, ..., d, and P 4 i zidzi dxd+1 = . (|z|2 + 1)2 We obtain that d X 2 2 2 dxi + dxd+1 = 4ds . i The result follows considering that a multiplication of the metric by a constant factor k implies a multiplication of the curvature by a factor of 1/k.

Although the next proposition uses results from the next sections, it is convenient to state and prove it here. 52 2.4. RIEMANNIAN METRIC

2 3 Proposition 2.3. Let ρ(X,Y ) denote the distance in OP . If z, w ∈ O0, then |(Z,W )| 1/2 cos (ρ(p(z), p(w))) = , (2.23) (Z,Z)(W, W ) where Z = zzt and W = wwt.

2 Proof. The equation is invariant under the action of the isometry group of OP , which acts transitively, and independs on the liftings. Thus, we may suppose Z = E1 = diag(1, 0, 0) and  cos2 α cos α sin α 0 2 W = cos α sin α sin α 0 , 0 0 0 1 for some α ∈ R, with α > 0. Consider the arcwise of RP connecting these points given by  cos2 t cos t sin t 0 2 Γ(t) = cos t sin t sin t0  , for 0 ≤ t ≤ π/2. 0 0 0 The derivative is  sin2 t − sin t cos t 0 0 Γ (t) = − sin t cos t cos2 t 0 . 0 0 0 1 It will be seen (Theorem 2.11) that RP is, up to isometry, the geodesic connecting these points. Thus, in the metric above, we obtain Z α ρ(p(Z), p(W )) = dt = α. 0 In addition, (Z,W ) cos−1 = α, (Z,Z)(W, W ) and the result follows.

m 2 Corollary 2.1. For F ⊂ O and m ≤ 2 the Riemannian embedding FP ⊂ OP is isometric.

Consider again standard liftings given by z = (1, z2, z3) and w = (1, w2, w3). That is, points in the coordinate chart U1/ ∼. We shall derive a more explicit equation for the distance. A direct calculation shows that

2 (Z,W ) = |1 + z2w2 + z3w3| + 2Re[(z2z3)(w3w2)] − 2Re[(z2w2)(w3z3)]. Set R(z, w) = 2Re [(z2z3)(w3w2) − (z2w2)(w3z3)] . The formula becomes

2 1/2 |1 + z2w2 + z3w3| + 2R(z, w) cos (ρ(Z,W )) = 2 2 2 2 . (1 + |z2| + |z3| )(1 + |w2| + |w3| )

For zi, wi ∈ F, the associativity implies R(z, w) = 0, and the formula coincides with (1.16). CHAPTER 2. THE OCTONIONIC PROJECTIVE PLANE 53

2 2.5 OP as a Homogeneous Space

The space F4 / Spin(9) was predicted in Cartan’s classification of symmetric spaces [1, Theorem 6.6]. In 1950, A. Borel identified F4 with the isometry group of a 16- dimensional projective plane. Independently and starting from Chevalley and Shafer work 2 [25], Freudenthal established the identification of F4 / Spin(9) and OP (see [7, Section 4.2] and [12, Section 4.7]). The proof presented here is that of [26]. Recall that Spin(9) is generated by the 8-sphere r Lu 2 2 : r ∈ R, u ∈ O and r + |u| = 1 . (2.24) Lu −r In particular, if r L g = u Lu −r is a Spin(9) generator, then rx1 + ux2 x1 g(x) = for ∈ O ⊕ O (2.25) ux1 − rx2 x2 determines the spinor representation of Spin(9). Also

−1 ρ Lv r Lu ρ Lv r Lu χg = Lv −ρ Lu −r Lv −ρ Lu −r 2 2 (r − |u| )ρ + 2rhu, vi L2ruρ+uvu−r2v = 2 2 L2ruρ+uvu−r2v −(r − |u| )ρ − 2rhu, vi determines the vector representation of Spin(9) on ρ Lv W = : r ∈ R, v ∈ O . (2.26) Lv −ρ r2 v r2 Lv In what follows, ∈ H3(O) is identiﬁed with ∈ EndR(O ⊕ O). Decom- v r3 Lv r3 pose r v t 0 s v 2 = + , (2.27) v r3 0 t v −s where t = (r2 + r3)/2 and s = (r2 − r3)/2. By deﬁning χg(I2) = I2, r2 v t 0 s v χg = + χg . (2.28) v r3 0 t v −s Moreover,

2 2 2 2 2 2 (r − |u| )s + 2rhu, vi + t = r2/2(r − |u| + 1) − r3/2(r − |u| + 1) + 2rhu, vi 2 2 2 = r + 2rhu, vi + r3|u| .

Then the action of Spin(9) on H2(O) is given by

2 2 2 2 r + 2rhu, vi + r3|u| rr2u + uvu − r v − urr3 χg(a) = 2 2 2 . (2.29) rr2u − r v + uvu − rr3u r2|u| − 2rhu, vi + r r3 2 54 2.5. OP AS A HOMOGENEOUS SPACE

  r1 x1 x2 Finally, if A = x1 r2 v  ∈ H3(O), the action of Spin(9) on H3(O) is given by x2 v r3

t ! r g(x) x1 r2 v g(A) = 1 , where ∈ O ⊕ O, and a = . (2.30) g(x) χg(a) x2 v r3

A direct calculation shows that 1 0 0  g(A) = GAG, where G = 0 r u  . (2.31) 0 u −r

The next lemma is used to show that Spin(9) ⊂ F4.

Lemma 2.4 ([26], Lemma 14.104). Suppose U = (uij) ∈ Mn(O), with span{1, uij} contained in a subalgebra A isomorphic to C for all i, j. Then (UAU)(UBU) + (UBU)(UAU) = U(AU 2B)U + U(BU 2A)U (2.32) for all A, B ∈ Mn(O).

Proof. Let z ∈ O, then span{1, uij, z} ⊂ span{A, z}. Since dim(A) = 2, span{A, z}' C or K. Thus span{A, z} is an associative algebra. In particular, if u1, u2, u3 and u4 are entries of the matrix U, then

(u1zu2)(u3zu4) = u1[z(u2u3)z]u4.

Polarization yields

(u1zu2)(u3wu4) + (u1wu2)(u3zu4) = u1[z(u2u3)w]u4 + u1[w(u2u3)z]u4.

This implies the statement.

Corollary 2.2. The spin group Spin(9) is a subgroup of F4.

Proof. Since span{1, u} is subalgebra isomorphic to C or R, the matrix G in (2.31) satisﬁes the conditions of Lemma 2.4. Making U = G and A = B ∈ H3(O) in (2.32) yields g(A)g(A) = (GAG)(GAG) = GA2G = g(A2),

2 since G = 1. Then g ∈ F4 and this proves that Spin(9) is a subgroup of F4. We are now in conditions to prove the main theorem of this section.

2 Theorem 2.9 ([26],Theorem 14.99). Let E1 ∈ OP be the diagonal matrix diag(1, 0, 0). 2 If A ∈ OP there exists h ∈ F4 such that h(A) = E1. In particular, the action of F4 2 on OP is transitive.

The isotropy group at E1 is equal to Spin(9). Thus 2 OP ' F4 / Spin(9) . CHAPTER 2. THE OCTONIONIC PROJECTIVE PLANE 55

Proof. Let

t r1 x 2 x1 r2 x3 A = ∈ OP , where x = and a = . x a x2 r3 x3

Suppose g is a generator of Spin(9). Since tr(χg(a)) = tr(a), χg leaves the subspace

s v V = : s ∈ , v ∈ v −s R O

9 of H2(O) invariant. Identifying V with O ⊕ R ' R and regarding the dot product as 2ha, ai = tr(a ◦ a), 2 2 8 χg(a) ∈ {b ∈ V : hb, bi = s + |v| }' S . Since χ : Spin(9) → SO(9) acts transitively on S8, choose g ∈ Spin(9) such that s v s2 + |v|2 0 χ = , g v −s 0 −(s2 + |v|2) then χg(a) is of the form 0 0 r2 0 a = 0 . (2.33) 0 r3 Using the decomposition in (2.27), the isotropy group at a0 is the same as the isotropy at 1 0 . Recall the generator set of Spin(8) in (0.16) and note that for h ∈ Spin(8), 0 −1

1 0 1 0 χ = , h 0 −1 0 −1 by (2.29). By [26, Theorem 14.69], Spin(8) has orbits S7 × S7, {0} × S7 or S7 × {0} and the action is transitive on each one of these sets. In all cases there exists h ∈ Spin(8) and 2 y = (y1, y2) ∈ R so that h(x) = y and this action preserves the block (2.33). Thus, the orbit of Spin(9) contains a matrix B ∈ H3(R). Hence, there exists g0 ∈ SO(3) so that 2 g0(A) is diagonal. The conditions B = B and tr(B) = 1 imply that B has eigenvalue 1 with multiplicity 1 and eigenvalue 0 with multiplicity 2. A permutation π ∈ SO(3) when aplied to g0(A) provides E1 as desired. Denote the isotropy at E1 by Isot(E1). By (2.30), g(E1) = E1 for all g ∈ Spin(9). Therefore Spin(9) ⊂ Isot(E1). Consider now g ∈ Isot(E1) and   r1 x1 x2 A = x1 r2 v  ∈ H3(O). x2 v r3 Then     0 x1 x1 0 0 0 g(A) = r1E1 + g x1 0 0  + g 0 r2 v  . (2.34) x2 0 0 0 v r3 We shall to verify that the action of g is by blocks.     0 x1 x1 0 0 0 A1 = x1 0 0  and a1 = 0 r2 v  , x2 0 0 0 v r3 2 56 2.5. OP AS A HOMOGENEOUS SPACE

A1 ◦ E1 = A1 and a1 ◦ E1 = a1. Then g(A1) ◦ E1 = g(A1) and g(a1) ◦ E1 = E1. If α β γ  g(a1) = κ θ1 θ2 , then θi = 0 ∀ i. θ3 θ4

α0 β0 γ0 0 If g(A1) = κ θ δ , then only δ and θ can be non-null by the above conditions. 0 δ θ Rewrite t! g(A) = r f(x) , f(x) h(a) where f ∈ EndR(O ⊕ O), r ∈ R and h is linear preserving the trace, by (2.34). If E2 = diag(0, 1, 0) and E3 = diag(0, 0, 1), then I3 = E1 + E2 + E3. Thus

g(E2 + E3) = h(I2) = I2.

Decompose a ∈ H2(O) as in (2.27). Identifying s v ' (v, s) ∈ 9 v −s R and recovering the dot product by the relation tr(X ◦ Y ) = 2hX,Y i, we have that h ∈ O(9). Suppose that h∈ / SO(9), then det(h) = −1, which implies −h−1 ∈ SO(9). Choose −1 g˜ ∈ Spin(9) so that χg˜ = −h . It follows thatg ˜ ◦ g ∈ IsotE1 and then ﬁxes E2 + E3. On the other hand, −1 g˜ ◦ g(E2 + E3) = −h ◦ h(I2) = −I2, which is a contradiction, then h lies on SO(9). Utilizing an element of Spin(9) we may assume h = 1. To conclude, we shall prove that f = ±Id ∈ Spin(9), in both cases χf = h = 1. Since h = 1, g ﬁxes E2 and E3. If A is of the special form   0 x1 x2 A = x1 0 0  , x2 0 0 we have     0 x1 0 0 0 x2 2A ◦ E2 = x1 0 0 and 2A ◦ E3 =  0 0 0  . 0 0 0 x2 0 0   0 y1 y2 If g(A) = y1 0 0 , the above conditions imply y1 = f1(x1) and y2 = f2(x2). On y2 0 0 the other hand,  2 2  |x1| + |x2| 0 0 2 2 A =  0 |x1| x1x2 , 2 0 x2x1 |x2| which implies g(A2) = A2. The condition g(A)g(A) = g(A2) implies

x1x2 = f1(x1)f2(x2) ∀ x1, x2 ∈ O. (2.35) CHAPTER 2. THE OCTONIONIC PROJECTIVE PLANE 57

−1 2 If f1(1) = u, f2(x2) = x2u . Setting x2 = 1, f1(x1) = x1u. Since f preserves | . | , 2 |u| = 1. We obtain that f(x) = (x1u, x2u). By (2.35),

x1x2 = (x1u)(ux2).

Since x1, x2 ∈ O are arbitrary, u must be a real number. Then u = ±1 and the result follows.

2 The compactness of OP can be recovered from the previous theorem. The conditions 2 over A ∈ OP ⊂ H3(O) in the proof of item 1 are essentials only to obtain the diagonal matrix E1, then a more general result follows:

Theorem 2.10. If A ∈ H3(O), there exists g ∈ F4 such that g(A) is a diagonal matrix.

For a constructive proof of this fact, see [28]. An involutive automorphism σ on F4 can be deﬁned in such way that (F4, Spin(9), σ) is a symmetric pair with orthogonal symmetric 2 Lie algebra (f4, so(9), dσ)[27, Section 2.9]. Therefore, OP is a symmetric space. Since F4 preserves the metric (2.21) and is a simple Lie group which acts efectivelly on H3(O), by Theorem 0.17, 2 Iso0(OP ) = F4 .

2.6 Totally Geodesic Submanifolds

2 One of the most important aspects of OP is that it contains all the geometry of projective planes over F. This is particularly exempliﬁed by Theorem 2.11 presented here. As a sequel of Section 1.6, this section is dedicated to the proof of this result and 2 2 to show the stabilizers groups of FP in OP . First, the same proof of Lemma 1.1 shows 2 Lemma 2.5. A connected submanifold M of OP is a totally geodesic submanifold which 1 is isometric to a sphere if and only if M is totally geodesic in a projective line OP . 2 The classiﬁcation result of totally geodesic submanifolds in OP is the following. 2 Theorem 2.11. Let M be a totally geodesic submanifold of OP . Then M is isometric to

1. Sr (1 ≤ r ≤ 8), or 2 2. FP , for F = R, C, K, or O. Moreover, all of these inclusions exist. Proof. The statement follows by duality and the one-to-one correspondence between Lie triple systems and totally geodesic submanifolds (Theorem 0.19, and Remark 0.2) from Theorem 4.6. The dual of 1 ' 8 in 2 is 8 ' 1 or 8. On the other hand, the HO HR OP S OP RP dual of 1 ⊂ 1 is totally geodesic in the dual of 1 . Since totally geodesic subspaces HF HO HO of 8 are also real projective spaces, we conclude that 1 must be dual to 8 ' 1. RP HO S OP Therefore, the hypebolic spaces r ⊂ 1 are in duality with spheres. The correspon- HR HO dences are described below. 2 so so 2 HR ←→ (1, 2) ←→ (3) ←→ RP ; 2 su su 2 HC ←→ (1, 2) ←→ (3) ←→ CP ; 2 sp sp 2 HK ←→ (1, 2) ←→ (3) ←→ KP , 58 2.6. TOTALLY GEODESIC SUBMANIFOLDS where the space 2 6⊂ 1 . Recall the isometries d ' 1 , where d = dim( ). Thus, for HR HO HR HF F 1 ≤ r ≤ 8, the next diagram covers the remainder cases:

r 1 so so r 1 HR ⊂ HO ←→ (1, r) ←→ (r + 1) ←→ S ⊂ OP .

2 2 Remark 2.4. To obtain the planes FP ⊂ OP as totally geodesic submanifolds via Lie triple systems is necessary an expression for the bracket in f4. In this case, Mostow- Karpelevitch Theorem countours the problem. See also the proof of [18, Lemma 4].

Theorem 2.12 ([46], Theorem 9). The subgroup Z(F) of F4 leaving the subalgebra H3(F) of H3(O) pointwise ﬁxed is:

1. G2, if F = R.

2. SU(3), if F = C.

3. Sp(1), if F = K.

2 2 Let Stab(F) ⊂ F4 be the subgroup which stabilizes the projective plane FP ⊂ OP . It follows that Z(F) / Stab(F). Let SU(3; F) be a copy of SO(3), SU(3), or Sp(3) in Stab(F). Thus one can form the subgroup Z(F) · SU(3; F).

Theorem 2.13 ([48], Table 2.57). The Lie groups below are maximal in F4.

1. Spin(9).

2. Sp(1) · Sp(3).

3. SU(3) · SU(3).

4. G2 · SO(3).

This theorem follows from the classiﬁcation of the maximal subalgebras of f4, for details see [20, Teorema 1.4.6].

Theorem 2.14 ([49], Table 4). The connected subgroup of F4 leaving the projective plane 2 FP invariant modulo Z(F) is

1. SO(3), if F = R.

2. SU(3), if F = C.

3. Sp(3), if F = K.

Proof. By Theorem 2.13, it remains to show the real case. Recall that SO(3) ' SU(2)/Z2, then

G2 × SO(3) ⊂ G2 × (SU(2) ∩ Stab(R)) ⊂ G2 × SU(2).

Therefore, G2 × SU(2) and G2 × SO(3) are the only possibilities to Stab(R). Since the 2 group SU(2) ⊂ SU(3) does not stabilize RP , the result follows. CHAPTER 2. THE OCTONIONIC PROJECTIVE PLANE 59

2.7 Non-existence of Higher-Dimensional Spaces

The first attempt to construct a higher-dimensional octonionic projective space would be to generalize the matrix model. However, the set Hn(O) is not a Jordan Algebra for n ≥ 4 [35, Paragraph 18]. On the other hand, Artin’s Theorem can not be generalized. In this section, the non-existence of such spaces is justified based on Desargues’s axiom, which will connect the geometric and algebraic structures of a projective plane where it is satisfied. In addition, there is a topological reason of homotopical nature (see [37, section 1.3] and [38, section 4.2]).

Deﬁnition 2.8. A projective plane P2 is said to be a Desarguesian plane if for any pair ABC and DEF of two triangles with diﬀerent vertices and sides such that AD, BE and CF intersect at a point O, the points X = AC ∩ DF , Y = BC ∩ EF and Z = AB ∩ DE are collinear.

B A O E

C D

Figure 2.1: Two triangles satisfying the Desargues condition

Given an abstract projective plane P2 and a line L ∈ P2 choose three points and call them 0, 1 and ∞, and set A = L − {∞}. The plane P2 can be coordinatized by A and an algebraic structure can be constructed on this set (for details see [41, Chapter 5]). However, in general, this construction depends on the choices made. The next result is due to D. Hilbert, who started the above construction with a Desarguesian plane [36, Sections 25 and 26]. For a modern treatment, we refer the reader to [39, Sections 6.4 and 6.5].

Theorem 2.15. A projective plane is Desarguesian if and only if can be coordinatized by a division ring D. In other words, Desarguesian planes are completely classiﬁed like those obtained from a right or left vector space D3 for some division ring D. Since R, C, and K are division rings, the three classical projective planes are Desarguesian. We also note that the euclidean plane is not Desarguesian.

2 Corollary 2.3. The octonionic projective plane OP is not a Desarguesian projective plane.

Proof. For suppose it is. Then, by Theorem 2.15, a division ring structure is induced 1 on the non-associative algebra O ' OP − {∞}. This is a contradiction with Theorem 0.8. 60 2.7. NON-EXISTENCE OF HIGHER-DIMENSIONAL SPACES

Hilbert’s work was followed by Ruth Moufang, who started a construction with a condition equivalent to a weaker form of Desargues’s axiom:

Deﬁnition 2.9. A projective plane P2 is said to be a Little Desarguesian plane if for any pair ABC and DEF of two triangles with diﬀerent vertices and sides such that AD, BE and CF intersect at a point O and two of the points X = AC ∩ DF , Y = BC ∩ EF , Z = AB ∩ DE lie on a line l passing through O, then the third also lies on l.

Some of her results culminate in an analogous to Theorem 2.15 relating Little De- sargues’s condition and alternative division rings [40, Corollary 14.25]. Since O is an 2 alternative division ring, then OP satisﬁes Little Desargues’s theorem. The most general algebraic structure related to a projective plane is a ternary ring [42, Theorem 20.5.3].

Deﬁnition 2.10 ([40],Deﬁnition 9.1.1). A ternary ring T = (T, τ) is a system where T is a set and τ is a ternary operation in T , that is, τ : T × T × T → T , such that:

1. There exist elements 0 and 1 such that 0 6= 1, τ(0, a, b) = τ(a, 0, b) = b and τ(1, a, 0) = τ(a, 1, 0) = a.

2. Given a, b, c, d ∈ T with a 6= c there exists a unique x ∈ T such that τ(x, a, b) = τ(x, c, d).

3. Given a, b, c ∈ T there exists a unique x ∈ T such that τ(a, b, x) = c.

4. Given a, b, c, d ∈ T there exists a unique pais (x, y) ∈ T × T such that τ(a, x, y) = b and τ(c, x, y) = d.

Definition 2.11 ([41], Definition 5.2). A loop (T, ◦) is a set T on which is defined a binary operation ◦ such that:

1. For all x, y ∈ T , x ◦ y ∈ T .

2. Given a, b ∈ T , each of the equation a ◦ x = b and y ◦ a = b has a unique solution in T .

3. There exists an element e ∈ T such that e ◦ x = x ◦ e = x ∀ x ∈ T .

Given a ternary ring, note that a sum and a multiplication can be recovered making: a + b = τ(a, 1, b) and a.b = τ(a, b, 0). In this case (T, +) and (T − {0},.) are loops. Moreover, multiplicative loops does not have zero divisor, by item 2 of Deﬁnition 2.11. The construction above applied to an arbitrary projective plane induces a ternary ring [41, Example 5.6]. Since Cayley-Dickson algebras of dimension greater than 8 have zero divisors, there does not exist projective planes, and hence higher-dimensional projective spaces, over such structures.

Theorem 2.16 ([39], Theorem 10.9). Desargues’s theorem holds in any higher-dimensional projective space.

Corollary 2.4. There does not exist higher-dimensional octonionic projective spaces.

Proof. If such spaces existed their projective planes would be Desarguesian and copies of 2 OP , which is a contradiction. CHAPTER 2. THE OCTONIONIC PROJECTIVE PLANE 61

While Desarguesian planes are, in some sense, classiﬁed, non-Desarguesian planes can be constructed over diﬀerent structures as well as induce them. However, the next result due to O. Veblen and J. Young shows that the higher-dimensional case is simpler.

Theorem 2.17 ([45], Theorem 6.7.1 ). Suppose Pn is an arbitrary projective space with dimension n ≥ 3. Then Pn is the projective space obtained from a left or right vector space Dn+1 for some division ring D. Since projective planes in a higher-dimensional projective space are coordinatized by a division ring, the non-existence of higher-dimensional octonionic projective spaces can be derived from Theorem 2.17 by the same argument showed in Corollary 2.3. However, due to its historical role, we included Desargues’s axiom here. A history of non-Desarguesian geometry can be found in [44]. Chapter 3

F-Hyperbolic Spaces

Hyperbolic spaces are the non-compact dual of projective spaces. Following the standard structure, we justify this duality and derive the rank of these spaces. We note that n the Bergman metric introduced here is analogous to the Fubini-Study metric for FP . The main references are [21], [56, Chapters 19 and 20], and [57].

3.1 Introduction

Let V = V 1,n(F) be the right vector space Fn+1 equipped with the hermitian form Φ of signature (1, n). The following subsets are considered:

V0 = {z ∈ V : Φ(z, z) = 0},

V− = {z ∈ V : Φ(z, z) < 0},

V+ = {z ∈ V : Φ(z, z) > 0}.

n Recall the projectivization map p: V − {0} → FP .

Deﬁnition 3.1. The -hyperbolic space n is the projectivization of V : F HF −

n HF := p(V−).

If z = (z0, . . . , zn) ∈ V−, then

2 X 2 Φ(z, z) = −|zo| + |zi| < 0. i

Thus, z 6= 0 and a lifting of p ∈ n can be choose always with the form (1, z , . . . , z ). 0 HF 1 n This leads us to identify n n HF = {ζ ∈ F : hζ, ζi < 1}. The coordinates ζ are called non-homogeneous. Then n has a structure of real dif- HF ferentiable manifold of dimension d.n, where d = dim( ). The boundary ∂ n is the F HF projectivization of V0: n ∂HF = p(V0). It can be identiﬁed with the sphere Sdn−1 ⊂ Fn.

62 CHAPTER 3. F-HYPERBOLIC SPACES 63 3.2 Riemannian Metric

In order to introduce the Bergman metric for n following [56], we describe its tangent HF space. Although the description in an abstract way, this metric derives from taking the real part of a restriction of the Hermitian form. We also derive the distance formula. First, we shall describe the structure of principal bundle (see Subsection 0.4). Let V = {z ∈ V : Φ(z, z) = −1}, then p(V ) = p(V ) = n, which we have identified −1 − −1 HF n d−1 with the unitary ball in F . The group S acts on V− leaving V−1 invariant. Similarly to the projective case, the restriction of p to V−1, which we will also denote by p, is a local diffeomorphism. It follows that (V , n, d−1, p) is a principal fiber bundle. −1 HF S Recovering z as a vector in V ' Rnd+d, and derivating the equation Φ(z, z) = −1, the tangent bundle of V−1 is identified with

T V−1 = {(z, v): z ∈ V−1, v ∈ V, Re[Φ(z, v)] = 0} . In this case, Re[Φ(z, v)] = 0 turns out to be the usual real Hermitian form of signature (1, n). The tangent space to the ﬁber at z ∈ V−1 is equal to (1.7). Consider also the F-subspace HzV−1 = {(z, v): v ∈ V, Φ(z, v) = 0} . (3.1) We obtain an orthogonal decomposition

TzV−1 = VzV−1 ⊕ HzV−1.

The action of Sd−1 on the bundle is similar to (1.9). Since Φ(zλ, vλ) = λΦ(z, v)λ, this defines a connection, where (3.1) is the horizontal space at z. By (0.18), it follows that T n is isomorphic to H V . Now, we move on to the Riemannian metric. p(z)HF z −1 Definition 3.2. The metric at a point ζ ∈ n is defined by HF −1 Φ(z, z) Φ(dz, z) ds2 = det , (3.2) (Φ(z, z))2 Φ(z, dz) Φ(dz, dz) where z ∈ V− is a lifting of ζ, that is, p(z) = ζ. Proposition 3.1. The expression (3.2) defines a Riemannian metric for n. HF Proof. First, note that Φ(vλ, vλ) = |λ|2Φ(v, v) for all v ∈ Fn and λ ∈ F, therefore the equation independs on liftings. Let ζ = (ζ1, ..., ζn) a non-homogeneous coordinate function on n and z ∈ V a lifting. Then z = ζ z and dz = dζ z + ζ dz . We obtain that HF − i i 0 i i 0 i 0 2 Φ(z, z) = |z0| Φ(ζ, ζ) 2 2 Φ(dz, z) = |z0| Φ(dζ, ζ) + |z| dz0z0 2 2 Φ(z, dz) = |z0| Φ(z, dz) + |z| z0dz0 2 2 2 2 Φ(dz, dz) = |z0| |dζ| + 2Re[Φ(dζz0, ζdz0)] + |z| |dz0| . Substituting these expressions in (3.2) yields 1 1 ds2 = |dζ|2 + |hdζ, ζi|2 . (3.3) 1 − hζ, ζi 1 − hζ, ζi This implies that (3.2) depends only on the non-homogeneous coordinate ζ in a smooth way. Since hζ, ζi < 1, this expression is also positive definite. It follows that the metric is well-defined on the unitary ball in Fn. 64 3.2. RIEMANNIAN METRIC

Let F0 ⊂ F and m ≤ n, then restricting scalars and set zero to n−m remaining entries, 1,m 0 1,n m n V ( ) ⊂ V ( ). Therefore, 0 is a submanifold of . By (3.3), this embedding is F F HF HF Riemannian. This metric is called the Bergman metric. The space n endowed with this HR Riemannian structure is called the Beltrami-Klein model for the real hyperbolic space. For n = 1, take a standard lifting z = (1, w) ∈ F2. Note that |hdw, wi| = |dw|2|w|2. 1 − hw, wi = 1 − |w|2.

Thus, in this case, the metric is |dw|2 ds2 = . (3.4) (1 − |w|2)2

Recovering w ∈ F ' Rd, it turns out to be the Poincar´eModel for the real hyperbolic space d . It will be seen that these spaces are isometric (see Proposition 4.4). Therefore, the HR hyperbolic lines 1 are isometric to the real hyperbolic space d with constant curvature HF HR −4. Let g denote the associated inner product on the tangent space T n ' H V . Then ζ ζ HF z −1 2 dn+d−1 gζ (dζ, dζ) = ds . Let p(z) = ζ and z ∈ S . We obtain that Φ(z, dz) = 0 and 1 g (v , v ) = [g (v + v , v + v ) − g (v , v ) − g (v , v )], ζ 1 2 2 ζ 1 2 1 2 ζ 1 1 ζ 2 2 for v1, v2 ∈ HzV−1. Therefore

gζ (v1, v2) = Re[Φ(v1, v2)]. (3.5) In [21, Section 2.3], the authors start with this restriction to derive the metric (3.3). Although the next proposition uses results from the next sections, it is convenient to state and prove it here. Proposition 3.2 ([21], Proposition 2.4.4). Let ρ(p, q) denote the distance in n. If HF z, w ∈ V−, then |Φ(z, w)| cosh (ρ(p(z), p(w))) = . (3.6) (Φ(z, z)Φ(w, w))1/2 Proof. The equation is invariant under the transitive action of the isometry group of n, HF and does not depend on representatives of p and q. Therefore, we may assume

z = (1, 0,..., 0), p(z) = 0 w = (1, α, . . . , 0), p(w) = (α, . . . , 0), where α ∈ and α > 0. Consider the arcwise of 1 connecting these points given by R HR γ(t) = (1, t, . . . , 0), for 0 ≤ t ≤ α.

With respect to this metric, 1 t2 |γ0(t)|2 = + 1 − t2 (1 − t2)2 1 = . (1 − t2)2 CHAPTER 3. F-HYPERBOLIC SPACES 65

It will be seen (Theorem 3.12) that γ(t) is, up to isometry, the geodesic connecting p to q. In other words, the distance between any two points can be calculated by translating them to a hyperbolic line. Thus

Z α dt −1 ρ(p, q) = 2 = tanh (x). 0 1 − t

Recovering the identities 1 1 + a tanh−1(a) = ln , 2 1 − a √ cosh−1(a) = ln(a + a2 − 1), and noting that

|Φ(z, w)| 1 cosh−1 = cosh−1 √ . (Φ(z, z)Φ(w, w))1/2 1 − x2 the formula is veriﬁed.

0 m n Corollary 3.1. For ⊂ and m ≤ n, the Riemannian embedding 0 ⊂ is isometric. F F HF HF

In particular, for standard liftings z, w ∈ V−, the formula becomes

1/2 | − 1 + z1w1 + z2w2| cosh (ρ(p(z), p(w))) = 2 2 2 2 . (3.7) (1 − |z1| − |z2| )(1 − |w1| − |w2| )

3.3 The Hyperbolic Spaces as Symmetric Spaces

This section follows the same scheme of Section 1.4. Further, we prove dualities and derive the rank using Lemma 0.2 (see also Remark 0.2). The next theorem implies the transitivity of U(1, n; )-action on n. F HF

Theorem 3.1 ([21], Propositions 2.1.2 and 2.1.3 ). Let V be a right F-vector space.

1. Let W be a subspace of V . Each linear isometry of W into V can be extended to an element of U(1, n; F).

∗ 2. Let z, w ∈ V−. Then there exists g ∈ U(1, n; F) and λ ∈ F such that g(z) = wλ.

Proof. Item 1 is a special case of Witt’s theorem, which asserts that any linear isometry of a subspace of a vector space over a ﬁeld of characterist not 2 is extendable to a isometry of the space. For item 2, deﬁne the map

T : Lz −→ Lw zα 7−→ w(λα),

q Φ(z,z) where λ = Φ(w,w) and Lz and Lw are the spaces generated by z and w, respectively. The map T is an isometry and the result follows from item 1. 66 3.3. THE HYPERBOLIC SPACES AS SYMMETRIC SPACES

3.3.1 The Real Case Theorem 3.2. The real hyperbolic space n is diﬀeomorphic to the coset space O(1, n)/ O(1)× HR O(n).

Proof. See the proof of Theorem 3.7.

Note that O(1, n) SO (1, n) ' 0 . O(1) × O(n) SO(n) Recall the automorphism

σ : SO0(1, n) −→ SO0(1, n)

A 7−→ I1,nAI1,n.

It holds that σ(A) = A if and only if A(e0) = e0, then (SO0(1, n), SO(n), σ) is a symmetric pair. The group PSO (1, n) acts eﬀectively on n and it is a simple group, then 0 HR

n Iso0(HR) = PSO0(1, n). The relation with the compact case is expressed by the following Theorem.

Theorem 3.3. The orthogonal symmetric Lie algebras (o(1, n), dσ) and (o(n+1), dσ) are duals. In particular, the real hyperbolic space n is dual to the euclidean sphere n. HR S Proof. By [3, p.341], the Lie algebra o(1, n) can be identiﬁed with the direct sum of the subspaces 0 0 T l = : X1 ∈ Mn(R),X1 = −X1 , , 0 X1 0 A p = : AT ∈ n . AT 0 R Since o(1) = 0, l ' o(n), and this is the Cartan decomposition with respect to dσ of o(1, n). When applied to o(1, n)∗ = l ⊕ ip the automorphism dσ works as the conjugation ξ. On the other hand, o(n + 1) = l ⊕ p0, where p0 is the subspace of matrices

0 A . −AT 0

This is the Cartan decomposition for o(n + 1) with respect to dσ. The map

0 −iA 0 A T 7−→ T −iA X1 −A X1 identiﬁes (o(1, n)∗, ξ) and (o(n + 1), dσ).

Theorem 3.4. Let so(n + 1) = l ⊕ p0 as before. A maximal abelian subalgebra of p0 has dimension 1. Thefore, the spaces n , n, and n have rank one. HR RP S Proof. See the proof of Theorem 3.7. CHAPTER 3. F-HYPERBOLIC SPACES 67

3.3.2 The Complex Case Theorem 3.5. The complex hyperbolic space n is diffeomorphic to the coset space HC U(1, n)/ U(1) × U(n). Proof. See the proof of Theorem 3.8. We also have U(1, n) SU(1, n) ' . U(1) × U(n) U(n) Considering the automorphism σ on U(1, n), (U(1, n), U(1) × U(n), σ) is a symmetric pair. The group PU(1, n) acts effectivelly on n and it is a simple group, then HC n Iso0(HC) = PU(1, n). The relation with the compact case is expressed by the following Theorem. Theorem 3.6. The orthogonal symmetric Lie algebras (u(n + 1), dσ) and (u(1, n), dσ) are duals. In particular, the complex hyperbolic space n is dual to the complex projective HC n space CP . Proof. By [3, p.341], the Lie algebra u(1, n) can be identified with the direct sum of the subspaces X1 0 ∗ l = : X1 ∈ C,X1 + X1 = 0,X2 ∈ Mn(C),X2 = −X2 , 0 X2

0 A p = : AT ∈ n . A∗ 0 C Since l = u(1)⊕u(n), this is the Cartan decomposition with respectto dσ of u(1, n). When applied to u(1, n)∗ = l ⊕ ip, this automorphism works as the conjugation ξ. On the other hand, u(n + 1) = l ⊕ p0, where p0 is the subspace with matrices of the form 0 A . −A 0 This is the Cartan decomposition for u(n + 1) with respect to dσ. The map X1 iA X1 A ∗ 7−→ iA X2 −AX2 identiﬁes (u(1, n)∗, ξ) and (u(n + 1), dσ). Theorem 3.7. Let u(n + 1) = l ⊕ p0 as before. A maximal abelian subalgebra of p0 has dimension 1. Thefore, the spaces n and n have rank one. HC CP Proof. Consider the subspace V ⊂ p0 of the real matrices  0 0 . . . λ  . ..   . 0 . 0 , −λ 0 ... 0 where λ ∈ R. It holds that V is abelian. For v ∈ V and a ∈ p0, the Lie bracket [v, a] lies on p0 if and only if a ∈ V . Thus V is a maximal abelian subspace of rank one. The result follows from Lemma 0.2. 68 3.3. THE HYPERBOLIC SPACES AS SYMMETRIC SPACES

3.3.3 The Quaternionic Case Theorem 3.8. The quaternionic hyperbolic space n is diﬀeomorphic to the coset space HK Sp(1, n) / Sp(1) × Sp(n).

Proof. If g ∈ Sp(1, n), then g(V−) = V− and g(vλ) = g(v)λ, for λ ∈ K. Thus Sp(1, n) acts on n leaving n invariant. Moreover, by Theorem 3.1, this action is transitive. KP HK n+1 n Let {e0, . . . , en} be the standard basis in K and W = he1, . . . , eni ' K . Suppose that g ﬁxes the point 0 = p(e0). This implies g(e0) = e0λ, for some λ ∈ K. Note that

Φ(λe0, g(ei)) = λΦ(e0, g(ei)) = Φ(e0, ei) = 0,

λ 0 for i = 1, . . . , n. Then g leaves the subspace W invariant and has the form , for 0 A λ ∈ Sp(1) and A ∈ Sp(n). Therefore, the isotropy group of the point 0 is Sp(1) × Sp(n) and the result follows. We also shall consider a diﬀerent version of the automorphism σ:

σ : Sp(1, n) −→ Sp(1, n) A 7−→ KAK, where −1 0   0 Id  K =   .  −1 0  Id Then (Sp(1, n), Sp(1) × Sp(n), dσ) is a symmetric pair. As in the projective case, the group PSp(1, n) acts eﬀectivelly on n and it is simple. By Theorem 0.17, HK

n Iso0(HK) = PSp(1, n). Theorem 3.9 ([3], p.351). The orthogonal symmetric Lie algebras (sp(n + 1), s) and (sp(1, n), s) are duals. In particular, the quaternionic hyperbolic space n is dual to the HK n quaternionic projective space KP . Proof. By [3, p.341], the Lie algebra sp(1, n) can be identiﬁed with the direct sum of the subspaces    X1 0 X2 0    0 Y1 0 Y2 ∗ T  l =   : Xi ∈ C,X1 + X2 = 0,Yi ∈ Mn(C),Y1 + Y2 = 0,Y2 = Y2 , −X2 0 X1 0     0 −Y2 0 Y1 

 0 A 0 B    T  A 0 BT 0   p =   : AT ,BT ∈ n .  0 B 0 −A C  T T    B 0 −A 0  Since l = sp(1)⊕sp(n), this is the Cartan decomposition with respect to automorphism dσ on sp(1, n). When applied to sp(1, n)∗ = l ⊕ ip the automorphism dσ works as the CHAPTER 3. F-HYPERBOLIC SPACES 69 conjugation ξ. On the other hand, sp(n + 1) = l ⊕ p0, where p0 is the subspace of matrices of the form  0 A 0 B T −A 0 BT 0    .  0 −B 0 A  T  −B 0 −AT 0 This is the Cartan decomposition for sp(n + 1) with respect to dσ. The map     X1 iA X2 iB X1 AX2 B  T T   T T   iA Y1 iB Y2  −A Y1 B Y2   7−→   −X2 iB X1 −iA −X2 −B X1 A T T T T iB −Y2 −iA Y1 −B −Y2 −A Y1 identiﬁes (sp(1, n)∗, ξ) and (sp(n + 1), dσ). Theorem 3.10. Let sp(n + 1) = l ⊕ p0 as before. A maximal abelian subalgebra of p0 has dimension 1. Thefore, the spaces n and n have rank one. HK KP Proof. Consider the subspace V ⊂ p0 of the matrices

 0 A 0 0   T 1 ... 0 −A 0 0 0 . .   , where A = . .. 0 .  0 0 0 A   0 ... 0 0 0 −AT 0

It holds that V is abelian and maximal. The result follows from Lemma 0.2.

3.4 Totally Geodesic Submanifolds

In this section, we show the classiﬁcation of totally geodesic submanifolds of n. The HF remarkable Mostow-Karpelevitch Theorem for non-compact symmetric spaces is central for the proof.

0 m Theorem 3.11. Let ⊂ and m ≤ n. Then the hyperbolic spaces 0 are totally F F HF geodesics submanifolds of n. HF Proof. Since the subgroups U(1, m; F0) ⊂ U(1, n; F), where m ≤ n and F0 ⊂ F, are semisimple, the result follows from Theorem 0.20 applied to the point p(e ) ∈ n, where 0 HF n+1 e0 = (1,..., 0) ∈ F . The next lemma follows by duality from Corollary 1.2.

0 n m Lemma 3.1. Let ⊂ . Then is not a totally geodesic submanifold of 0 for any F F HF HF m, n. Theorem 3.12 ([21], Proposition 2.5.1). Let M be a totally geodesic submanifold of n. HF Then M is isometric to 1. r (1 ≤ r ≤ n), for = ; or HF F R 2. r (1 ≤ r ≤ n), for = , ; or HF F R C 70 3.4. TOTALLY GEODESIC SUBMANIFOLDS

3. r (1 ≤ r ≤ n), for = , , . HF F R C K Proof. Let M be a totally geodesic submanifold of n. By the characterization of totally HF geodesic subspaces via symmetric pairs (Remark 0.1), it follows that M is a symmetric space of non-compact type and rank one. The classification of such symmetric spaces m 0 (Theorem 0.23) shows that M is a hyperbolic space 0 . By Lemma 3.1, ⊂ . Now we HF F F r+1 r shall see the condition over r. Suppose 0 totally geodesic in . In details, HF HF r+1 ⊂ r ; HR HC r+1 ⊂ r ; HC HK r+1 ⊂ r . HR HK Since we may suppose r+1 ⊂ r , the third case is covered by the first. Considering the HR HC isometries 2 ' 1 and 4 ' 1 , we also may assume r > 1. For r = 3, 4 ' 1 would HR HC HR HK HR HK be totally geodesic in 3 , which is a contradiction with Lemma 3.1. This also covers the HC first case with r > 3. It remains to see the situation for r = 2. Suppose 3 is a totally HR geodesic subspace of 2 . By Lemma 1.1, the dual of 3 in 2 must be 3. This HC HR CP RP duality is preserved if we consider the embedding 2 ⊂ 2 . In particular, a isometric HC HK copy of 3 is not contained in any quaternionic hyperbolic line. This is a contradiction HR with the correspondences (1.25) and the result follows. Chapter 4

The Octonionic Hyperbolic Plane

The octonionic hyperbolic plane is the non-compact dual of the octonionic projective 2 plane. This space is also constructed as a subset of OP , but coordinatized by a diﬀerent algebra. Therefore, although the exceptionality, it has similarities with the F-hyperbolic spaces. Mutatis mutandis, the ﬁrst section and the proofs of Theorem 4.3 and Proposition 4.2 follow the same lines as the proofs in Chapter 2. The main references are [56, Chapters 19 and 20], [14], [23], [22], and [57, Chapter 9]. In particular, the metric introduced for 2 OP is analogous to that given by D. Mostow in [56].

4.1 The Jordan Algebra H1,2(O)

Consider the diagonal matrix H = diag(−1, 1, 1). The subset H1,2(O) ⊂ M3(O) consists of the H-hermitian matrices, that is,

∗ H1,2(O) = {X ∈ M3(O): XH = HX }. (4.1)

The general form of an element in H1,2(O) is   r1 x1 x2 −x1 r2 x3 , (4.2) −x2 x3 r3 where ri ∈ R and xi ∈ O. For X,Y ∈ H1,2(O), note that 1 (X ◦ Y )H = [(XY ) + (YX)] H 2 1 = (XHY ∗ + Y HX∗) 2 1 = (HX∗Y ∗ + HY ∗X∗) 2 = H(X ◦ Y )∗.

Albert’s theorem [16, Principal Theorem] also implies that H1,2(O) is an exceptional Jordan Algebra of dimension 27. The bilinear forms in Definition2 .4 are analogously defined on H1,2(O). A matrix X ∈ H1,2(O) satisfies the same characteristic equation (2.3), see [23, p.39]. The automorphisms of H1,2(O) are also linear maps in GL(H1,2(O)) that preserve the Jordan product. Besides the compact form of the Lie algebra of type F4 represented by the group F4, we have the following:

71 72 4.2. THE OCTONIONIC HYPERBOLIC PLANE

Theorem 4.1 ([27], 2.14). The automorphism group of the Jordan algebra H1,2(O) is a * non-compact simply connected real Lie group of type F4 and it is denoted by F4.

* The group F4 is usually denoted by F4(−20). The same proof of 2.1 shows that the * group F4 also preserves the trace of a matrix in H1,2(O). Similarly to the compact case, * the groups SO(1, 2), SU(1, 2), and Sp(1, 2) are subgroups of F4. In particular, we decribe the embedding of SO(1, 2).

* Lemma 4.1. The group SO(1, 2) ⊂ F4.

t Proof. If g ∈ SO(1, 2), then g HgH = I3. For A ∈ H1,2(O), deﬁne g(A) = (gH)A(gtH).

Note that

g(A)∗H = (Hg)(A∗H)(gtH) = (Hg)(HA)(gtH) = Hg(A).

Thus g(A) ∈ H1,2(O). Moreover, g(A)g(A) = (gH)A(gtH)(gH)A(gtH) = g(A2).

This implies the result.

4.2 The Octonionic Hyperbolic Plane

2 The octonionic projective plane OP can be also deﬁned as the set

{X ∈ H1,2(O): X 6= 0,X ∗ X = 0}

2 up to real scalar multiplication. In what follows, OP is supposed to be coordinatized by 2 H1,2(O). Recall that an element X ∈ H1,2(O) is idempotent if X = X and nilpotent if X2 = 0.

Lemma 4.2 ([13], Lemma 1). An element X ∈ H1,2(O) satisﬁes X ∗ X = 0 if and only 2 if X = 0 or X = λX0 where X0 is an idempotent matrix with trace 1 and λ ∈ R.

The diﬀerence to H3(O) is the existence of nilpotent elements. For example, the matrix  1 0 1  A =  0 0 0  (4.3) −1 0 −1

2 satisﬁes A2 = 0 and so A ∈ OP . Proposition 4.1 ([14], 12.1, 12.2).

2 1. If X is an idempotent point in OP , then the number of nilpotent lines passing though X is either zero or at least two. CHAPTER 4. THE OCTONIONIC HYPERBOLIC PLANE 73

2. In duality with the previous item, an idempotent line contains either no nilpotent lines or at least two.

3. If X1 and X2 are nilpotents, then X1 ∗ X2 is idempotent.

4. If X is nilpotent, then the line determined by X is the unique nilpotent line passing though X.

We consider three disjoint subsets of {X ∈ H1,2(O): X 6= 0,X ∗ X = 0}:

The nilpotent points, denoted by V0. This set is also called the conic.

The idempotent points such that no line containing them is nilpotent, denoted by V−. This set is also called the inner points.

The idempotent points contained in at least one nilpotent line, denoted by V+. This set is also called the outer points.

2 Since OP is the projectivization of the above set, by Lemma 4.2 and items 1 and 2 of Proposition 4.1, it can be divided in three disjoint sets:

2 OP = p(V0) ∪ p(V−) ∪ p(V+).

2 The conic V0 induces a partition of OP in two connected components. If X is a nilpotent element, item 4 of Proposition 4.1 motivates us to call X a tangent line to the conic. The projectivization map p sends a matrix to the vector subspace consisting of its real non-zero scalar multiples.

Deﬁnition 4.1. The octonionic hyperbolic plane 2 is the projectivization of inner points HO 2 in OP , that is, 2 HO := p(V−). An octonionic hyperbolic line 1 consists of the projectivization of inner points on a line HO that intersects the conic.

There is also a geometric interpretation of hyperbolic points. Consider a point X ∈ 2 OP such that every line containing X intersects the conic. We claim that X is an inner point. Indeed, suppose that X lies on a nilpotent line Y and X0 is a conic element also on Y . Then, the line Y ∗ X0 = Y is idempotent by item 3 of Proposition 4.1, which is a contradiction. The converse is also true [14, 16.2]. Thus, another description is:

The octonionic hyperbolic plane 2 is the set of points in 2 such that every line HO OP containing them intersects the conic.

If X is an outer point, then X lies on tangent lines or lines that intersect the conic or not. A more detailed description of this conﬁguration is given by the next result.

Lemma 4.3 ([14], 18). If X ∈ 2 and Y is a line containing X, then Y is an outer HO point. In other words, the lines cointaining inner points are determined by outer points (see Figure 4.1). 74 4.2. THE OCTONIONIC HYPERBOLIC PLANE

p(V+) Z Y

X p(V−)

p(V0)

2 Figure 4.1: Conﬁguration of OP coordinatized by H1,2(O)

2 Proposition 4.2. Every element X ∈ OP has a representative of the form  2  −|z1| z1z2 z1z3 T 2 zz H = −z2z1 |z2| z2z3 , (4.4) 2 −z3z1 −z3z2 |z3|

3 where (z1, z2, z3) ∈ O lies on an associative subalgebra. 2 Proof. Suppose X ∈ OP given by (4.2). By Lemma 4.2, X2 = λX with tr(X) = λ or X2 = 0. We may suppose λ = 1. From this facts, we derive the following equations:

r3x1 = x2x3, r2x2 = x1x3, −r1x3 = x3x2.

One of the ri is nonzero, otherwise the matrix would be null. Thus, x1, x2, and x3 belong to an associative subalgebra A of O. If r1 6= 0, we may suppose r1 < 0 and set 2 r1 = −|z1| ; x1 = z1z2; x2 = z1z3, for some z1, z2, z3 in A. Using associativity, 2 2 r2 = |z2| ; r3 = |z3| ; x3 = z2z3.

2 If r1 = 0, then set r2 = |z2| . The representative has the desired form and the result follows. If X is a conic element, choose a representative zzT H as above. It holds that (zzT H)2 = 0, which implies tr(zzT H) = 0, and hence tr(X) = 0. Recall the set

3 3 O0 = {(z1, z2, z3) ∈ O : z1, z2, z3 lie in some associative subalgebra of O}, (4.5) with the equivalence relation in (2.20). Moreover, if z ∈ O3 lies on an associative subalgebra, then every nonzero octonionic multiple zλ, where λ belongs in the same associative 2 subalgebra, deﬁnes the same point in OP as z. Finally, considering that T 2 2 2 tr(zz H) = −|z1| + |z2| + |z3| , we associate

3 T V0 = {(z1, z2, z3) ∈ O0 : tr(zz H) = 0}; 3 T V− = {(z1, z2, z3) ∈ O0 : tr(zz H) < 0}; 3 T V+ = {(z1, z2, z3) ∈ O0 : tr(zz H) > 0}, CHAPTER 4. THE OCTONIONIC HYPERBOLIC PLANE 75 and consequently 3 T p(V0) = {[z1, z2, z3] ∈ O0/ ∼: tr(zz H) = 0}; 3 T p(V−) = {[z1, z2, z3] ∈ O0/ ∼: tr(zz H) < 0}; (4.6) 3 T p(V+) = {[z1, z2, z3] ∈ O0/ ∼: tr(zz H) > 0}. The identification above allows us to identify the conic and the octonionic hyperbolic plane with the 15-dimensional sphere and the unitary ball in O2 ' R16, respectively. In addition, restricting the entries to , the hyperbolic planes 2 are obtained as submani- F HF folds of 2 . HO * Furthermore, since the trace is invariant under the action of F4, we conclude that this group preserves the conic, outer, and inner points. The matrix in (4.3) is an example of a conic point. Examples of inner and outer points are given below. 2 Example 4.1. The matrix E1 = diag(1, 0, 0) ∈ H1,2(O) satisfies E1 = E1. Hence E1 ∈ 2 3 OP . If Y is a line passing through E1, then tr(E1 ◦ Y ) = 0. Taking a lifting on O0, Y is of the form 0 0 0  2 0 |z2| z2z3  2 0 −z3z2 |z3| . Note that tr(Y ) 6= 0. Thus, no line containing X is nilpotent and X ∈ 2 . This result HO also agrees with Lemma 4.3. 2 Example 4.2. The matrix E3 = diag(0, 0, 1) ∈ H1,2(O) satisfies E3 = E3. Hence E3 ∈ 2 OP . If Y is a line passing through E3, then tr(E3 ◦ Y ) = 0. Taking a standard lifting on 3 O0, Y is of the form  2  −|z1| z1z2 0 2 −z2z1 |z2| 0 , 0 0 0 in such way that Y can be a nilpotent line or not and E3 is an outer point. Theorem 4.2. An octonionic hyperbolic line 1 is diffeomorphic to the 8-dimensional HO real hyperbolic space 8 . HR Proof. Recall Lemma 4.3 and consider the outer point E3 = diag(0, 0, 1). The hyperbolic line determined by E3 is the subset of points (matrices) of the form  2  −|z1| z1z2 0 2 −z2z1 |z2| 0 , (4.7) 0 0 0 where −|z |2 + |z |2 < 0. We may suppose z ∈ . Thus, E ∩ 2 is the projectivization 1 2 1 R 3 HO of negative vectors in R1,8 ' R ⊕ O. That is, 2 8 E3 ∩ HO ' HR. Since projective lines are diffeomorphic and the group F* preserves 2 , octonionic hyper- 4 HO bolic lines are also diffeomorphic and the result follows. Lemma 4.2 allows us another identification. The hyperbolic plane 2 consists of the HO 2 elements in OP represented by matrices X satisfying X2 = λX with tr(X) = λ < 0. Equivalently, it is identified with the set 2 2 {X ∈ OP : X = −X and tr(X) = −1} (4.8) up to real scalar multiplication. 76 4.3. RIEMANNIAN METRIC

4.3 Riemannian Metric

We study a metric for 2 following [17] and [56]. We also derive a formula for dis- HO tance. This metric generalizes the Bergman metric for 2 . Recall the form (X,Y ) = HF 1/2(tr(XY + YX).

Deﬁnition 4.2. The metric at a point ζ ∈ 2 is deﬁned by HO −1 ds2 = [tr(Z)tr(dZ) − (Z, dZ)] , (4.9) (tr(Z))2

t t 3 where Z = zz H and dZ = dzdz H for a lifting z ∈ O0 of ζ. Proposition 4.3 ([17], Theorem 3.3). The expression (4.9) deﬁnes a Riemannian metric on 2 . HO 3 For a standard lifting (1, z2, z3) ∈ O0, the expression is |dz |2 + |dz |2 − |z |2|dz |2 − |z |2|dz |2 + 2Re[(z z )(dz dz )] ds2 = 2 3 2 3 3 2 2 3 3 2 . 2 2 2 (−1 + |z2| + |z3| )

2 Restricting (z2, z3) ∈ F , the metric above is exactly the same as in (3.3), and the embeddings 2 ⊂ 2 are Riemannian. The diﬀeomorphism 1 ' 8 is, in fact, an isometry, HF HO HO HR as shows the next proposition.

Proposition 4.4. Let = , , or . Then the hyperbolic line 1 is isometric to the F C K O HF real hyperbolic space d with constant curvature −4. HR Proof. Let 1 the hyperbolic line determined by E = diag(0, 0, 1). A point Z ∈ 1 has HO 3 HO the form (4.7). Moreover, we may suppose w = 1. Then dZ = diag(0, |dz|2, 0) and

tr(Z)tr(dZ) = (−1 + |z|2)|dz|2 (Z, dZ) = |z|2|dz|2.

The metric becomes |dz|2 ds2 = (4.10) (1 − |z|2)2 Restricting z to , the expression above coincides with the reduced metric for 1 , see F HF (3.4). Recall that 1 and d are balls of dimension d. We shall construct an isometry. HF HR Consider the map

d 1 f : HR −→ HF ! ζ1 ζd ζ = (ζ1, ..., ζd) 7−→ , ..., . 1 + p1 − |ζ|2 1 + p1 − |ζ|2

The inverse is

−1 1 d f : HF −→ HR 2ω 2ω ω = (ω , ..., ω ) 7−→ 1 , ..., d 1 d 1 + |ω|2 1 + |ω|2 CHAPTER 4. THE OCTONIONIC HYPERBOLIC PLANE 77

Both f and f −1 are smooth functions, then f is a diﬀeomorphism. Let f(ζ) = ω. We have that dζ (1 − |ζ|2 + p1 − |ζ|2) + ζ hζ, dζi dω = i i . i 2 p1 − |ζ|2 1 + p1 − |ζ|2 Therefore |dζ|2(1 − |ζ|2 + p1 − |ζ|2) + 2hζ, dζi2 + |ζ|2hζ, dζi2 |dω|2 = . 4 (1 − |ζ|2) 1 + p1 − |ζ|2 Also, 2 p1 − |ζ|2 2 1 − |ω| = . 1 + p1 − |ζ|2 Substituting these expressions in (4.10) yields (3.3). The result follows considering that a multiplication of the metric by a constant factor k implies a multiplication of the curvature by a factor of 1/k. Actually, the proof above exhibit an isometry between Poincar´eand Beltrami-Klein models of real hyperbolic space. Although the next proposition uses results from the next sections, it is convenient to state and prove it here. Proposition 4.5. Let ρ(X,Y ) denote the distance in 2 . If z, w ∈ V , then HO − (Z,W ) 1/2 cosh (ρ(p(z), p(w))) = , (4.11) (Z,Z)(W, W ) where Z = zztH and W = wwtH. Proof. The equation is invariant under the action of the isometry group of 2 , which HO acts transitively, and independs on the liftings. Thus, we may suppose z = (1, 0, 0), w = (1, α, 0) ∈ V−, for some α ∈ R and α > 0. Then p(z) = Z = diag(−1, 0, 0) and −1 α 0 p(w) = W = −α α2 0 . 0 0 0

Consider the arcwise of 1 connecting these points given by HR −1 t 0 Γ(t) = −t t2 0 , for 0 ≤ t ≤ α. 0 0 0 Thus Γ0(t) = diag(0, 1, 0). It will be seen that Γ(t) (Theorem 4.6) is, up to isometry, the geodesic between Z and W . In order to ﬁnd the norm of this tangent vector, we have: tr(Γ(t))tr(Γ0(t)) = −1 (Γ(t), Γ0(t)) = t2. Therefore, 1 |Γ0(t)|2 = , (1 − t2)2 and the result follows as in the proof of Proposition 3.2. 78 4.4. 2 AS A HOMOGENEOUS SPACE HO

Consider again standard liftings given by z = (1, z2, z3) and w = (1, w2, w3). We shall derive a more explicit equation for the distance. A direct calculation shows that

2 (Z,W ) = | − 1 + z1w1 + z2w2| + 2Re[(z2z3)(w3w2) − (z2w2)(w3z3)].

Set R(z, w) = Re[(z2z3)(w3w2) − (z2w2)(w3z3)]. Thus 2 1/2 | − 1 + z1w1 + z2w2| + 2R(z, w) cosh (ρ(Z,W )) = 2 2 2 2 . (4.12) (1 − |z1| − |z2| )(1 − |w1| − |w2| )

For zi, wi ∈ F, the associativity implies R(z, w) = 0, and the formula coincides with (3.7).

4.4 2 as a Homogeneous Space HO In this section, the proof of Theorem 2.9 is adapted to obtain 2 as a homogeneous HO space. The duality between 2 and 2 is brieﬂy discussed. OP HO Theorem 4.3. Let E ∈ 2 be the diagonal matrix diag(1, 0, 0). 1 HO The action of F* on 2 is transitive. 4 HO

The isotropy group at E1 is equal to Spin(9). Thus 2 * HO ' F4 / Spin(9) . Proof. The proof is very similar to that of Theorem 2.9. First, recall that F* preserves 2 4 HO by the identiﬁcation (4.8). Moreover, the action of Spin(9) on H1,2(O) is the following. If g ∈ Spin(9) is a generator of the form

r L u , Lu −r and   r1 −x1 −x2 X = x1 r2 x3  . x2 x3 r3 then the action is given by

t ! r g(x) x1 r2 v g(X) = 1 , where x = ∈ O ⊕ O, and a = . −g(x) χg(a) x2 v r3

−1 0 0  If G =  0 r u  the following identity holds 0 u −r

g(A) = GAG.

2 * Since G = 1, Lemma 2.4 shows that Spin(9) ⊂ F4. The computation of Isot(E1) is equal 2 to the projective case. For transitivity we proceed as follows. Let A ∈ OP be an inner CHAPTER 4. THE OCTONIONIC HYPERBOLIC PLANE 79 point represented by an idempotent matrix A. By an action of Spin(8) ⊂ Spin(9), we may suppose A ∈ H1,2(R). Consider the following embedding in SO(1, 2): det(h) 0 O(2) ' : h ∈ O(2) . 0 h

Let g ∈ O(2) as above. Then

t λ λg(x) x1 2 r2 x3 h(A) = t , x = ∈ R , a = . −λg(x) gag x2 x3 r3 The matrix a is diagonalizable by a matrix in O(2). So we may suppose   r1 −x1 −x2 A = x1 r2 0  . x2 0 r3

The condition of A to be idempotent implies that x1 = 0 and r2 = 0 or x2 = 0 and r3 = 0. By an action of −1 0 0   0 0 −1 ∈ SO(1, 2), 0 −1 0 if necessary, we may assume −1 −z 0 A =  z z2 0 , (4.13) 0 0 0 where (1, z) ∈ 2 is a negative vector That is, A ∈ 1 . For given any other inner point R HR −1 −w 0 B, we assume B =  w w2 0. Consider now the following embedding in SO(1, 2): 0 0 0

h 0 SO(1, 1) ' : h ∈ SO(1, 1) . 0 1

a a We know from Chapter 3 (Theorem 3.1) that there exists h = 1 2 ∈ SO(1, 1) such a2 a1 that a1 + a2z λ ∗ h(1, z) = = , λ ∈ R . a2 + a1z −wλ The action of SO(1, 1) on (4.13) will recover this transitivity:       −a1 a2 0 −1 −z 0 −a1 a2 0 2 g(A) = −a2 a1 0 .  z z 0 . −a2 a1 0 . 0 0 0 0 0 0 0 0 0

It follows that g(A) = |λ|2B, therefore the action of F* on 2 is transitive. 4 HO As in the projective case, the proof can be simpliﬁed considering that Spin(9) is a * maximal compact subgroup of F4, because the complexiﬁcations of the Lie algebras are the same as F4. Moreover, the choice of E1 is essential, because the connected isotropy group of an outer point is SO0(1, 8) [22, Proposition 4.1]. 80 4.5. TOTALLY GEODESIC SUBMANIFOLDS

Note that if g fixes a point Y , then g preserves the line determined by Y . Let E3 = diag(0, 0, 1). Thus, the group SO(1, 8) acts on projective negative vectors of R ⊕ O ' R9 as usual, and has orbit SO (1, 8)/ SO(8) ' 1 . 0 HO The duality between 2 and 2 follows from Cartan’s classification of rank one OP HO symmetric spaces (Theorem 0.23). However, we give a more detailed discussion. N. C Jacobson showed that there are three real forms of the complex Lie algebra f4 [47, Section 12]. The correspondent Lie groups are the automorphisms of Albert algebras. We have seen two of them. The third is associated with the algebra of split octonions. A suitable change in the definition of the Cayley-Dickson construction (Definition 0.3) yields an 8-dimensional algebra O˜ which is non-associative, non-commutative, alternative, ˜ and have zero divisors when applied to O. The set H3(O) of hermitian matrices with ˜ C entries in O is a Jordan algebra and the third real form of F4 is its automorphism group, denoted by F4(4). Theorem 4.4. The octonionic hyperbolic plane 2 is the non-compact dual of the octo- HO 2 nionic projective plane OP . 2 Proof. The associated orthogonal symmetric Lie algebra of OP is (f4, so(9), σ), where C σ denotes an involution of F4. The complexification of dual Lie algebra is f4 . By the considerations above, there are two possibilities. Since the maximal compact subgroup of F4(4) has dimension 24 [52, Section 94.33] and dim(Spin(9)) = 36, the associated dual * space must be F4 / Spin(9), as stated.

4.5 Totally Geodesic Submanifolds

As in the projective case, one of the important properties of 2 is to contain all the HO geometry of F-hyperbolic planes. In addition to some remarks, the main argument in the proof of classiﬁcation is the same of Section 3.4. This is also true for stabilizers of 2 , HF 2 2 which are in duality with stabilizers of FP ⊂ OP . Theorem 4.5. Let = , , , or . The hyperbolic line 1 and the hyperbolic planes F R C K O HO 2 are totally geodesic submanifolds of 2 . HF HO

Proof. The subgroups SO0(1, 2), SU(1, 2), Sp(1, 2), and SO0(1, 8) are simple and have orbits 2 , 2 , 2 , and 1 ' 8 , respectively. The statement follows from Theorem HR HC HK HO HR 0.20. Theorem 4.6 ([22], Proposition 3.1). If M is a totally geodesic submanifold of 2 , then HO M is isometric to 1. r (1 ≤ r ≤ 8); or HR 2. 2 , for = , , , or . HF F R C K O Proof. The proof follows the same lines of Theorem 3.12. It remains to show the suitable dimension of inclusions. It is enough to see that 3 and 9 are not totally geodesic HC HR subspaces of 2 . For the ﬁrst case, it would imply 2 totally geodesic in 1 ' 8 , HO HC HO HR which is a contradiction with Lemma 3.1. For the second case, as an irreducible symmetric space of non-compact type and rank one, the associated symmetric pair must be ∗ (SO0(1, 9), SO(9), σ). But the Lie algebra so(1, 9) can not be a subalgebra of f4, because contains so(9), which is maximal (see Theorem 4.8). This concludes the proof. CHAPTER 4. THE OCTONIONIC HYPERBOLIC PLANE 81

The next three theorems are analogous to the compact case. We ﬁrst remark that the maximilaties in Theorem 4.8 also follows from complexiﬁcation of non-compact algebras. Moreover, Theorem 4.7 is valid for all three exceptional Albert algebras.

* Theorem 4.7 ([46], Theorem 9). The subgroup Z(F) of F4 leaving the subalgebra H1,2(F) of H1,2(O) pointwise ﬁxed is

1. G2, if F = R.

2. SU(3), if F = C.

3. Sp(1), if F = K.

* Theorem 4.8 ([48], Table 2.57). The Lie groups below are maximal in F4. 1. Spin(9).

2. Sp(1) · Sp(1, 2).

3. SU(3) · SU(1, 2).

4. G2 · SO(1, 2).

* Theorem 4.9 ([22], Proposition 4.3). The connected subgroup of F4 leaving the hyperbolic plane 2 invariant modulo Z( ) is HF F

1. SO0(1, 2), if F = R.

2. SU(1, 2), if F = C.

3. Sp(1, 2), if F = K. Appendix A

Automorphisms of H3(F) and H1,2(F)

Recall that the sets

∗ H3(F) = {A ∈ M3(F): A = A} ; ∗ H1,2(F) = {A ∈ M3(F): A H = HA} , are Jordan algebras with Jordan product given by X ◦Y = 1/2(XY +YX). Motivated by the proofs of Theorems 2.9 and 4.3 and using the matrix models of 2 and 2 , we shall FP HF derive their connected automorphism groups Aut0(H3(F)) and Aut0(H1,2(F)). First, we need a lemma.

Lemma A.1. Let M be a connected manifold and N a closed embedded submanifold whithout boundary. If dim(M) = dim(N), then N = M.

Proof. The submanifold N being embedded means that the inclusion N,→ M is a smooth embedding. Then the diﬀerential is injective at each point. The condition over the dimensions shows that it is bijective. Since N has no boundary, it follows from the inverse function theorem that N is also open in M. Therefore, N is open and closed in the connected manifold M, this implies M = N.

A.1 Compact Case

A matrix in H3(F) ⊂ H3(O) also satisﬁes (2.9). Thus, a proof similar to that of Lemma 2.1 shows that Aut(H3(F)) preserves trace. Recall the identiﬁcation between the dot product and the trace in (2.8). Hence, we have the following inclusion as closed subgroup:

Aut(H3(F)) ⊂ O(3d + 3).

By Theorem 0.10, this implies that Aut(H3(F)) is a Lie group. For A ∈ H3(F) the action of g ∈ U(3; F) is given by g(A) = gAg∗.

∗ 2 Since gg = I3, then g(A ) = g(A)g(A). However, the group PU(3; F) that acts eﬀectivelly, thus

PU(3; F) ⊂ Aut0(H3(F)).

82 APPENDIX A. AUTOMORPHISMS OF H3(F) AND H1,2(F) 83

We shall prove that these groups are equal. Also, consider the following embeddings of three important subgroups of PU(3; F):

det(h) 0 O(2) ' : h ∈ O(2) (A.1) 0 h det(h) 0 U(2) ' : h ∈ U(2) (A.2) 0 h λ 0 Sp(1) × Sp(2) ' , λ ∈ Sp(1) : h ∈ Sp(2) . (A.3) 0 h

  r1 x1 x2 Their action on A = x1 r2 x3 ∈ H3(F) is of the form x2 x3 r3

t! r1 λg(x) x1 2 r2 x3 g(A) = ∗ , x = ∈ F , a = . λg(x) gag x2 x3 r3

Theorem A.1. The connected automorphism group Aut0(H3(F)) is:

1. SO(3), if F = R;

2. PSU(3), if F = C;

3. PSp(3), if F = K.

2 Proof. Recall the matrix model for FP (Section 1.5). Then the connected automorphism 2 group acts leaving the projective plane FP invariant. A matrix in H3(F) is always 2 diagonalizable. The conditions A = A and tr(A) = 1 imply that for all A ∈ H3(F) there exists g ∈ U(3; F) such that g(A) = E1. Hence the action is transitive. Note that the 2 subgroups explicited above ﬁx the point E1 ∈ FP . Let g ∈ Isot(E1). The same argument in the proof of Theorem 2.9 shows that g is of the form

t! g(A) = r1 f(x) . f(x) h(a)

Since h preserves trace, h ∈ O(d + 1). For F = R, by an action of O(2) ⊂ SO(3), we may assume h = 1. For F = C, K, d + 1 is odd and we deduce that h ∈ SO(d + 1). Recall that

Spin(3) ' SU(2); Spin(5) ' Sp(2).

By an action of these groups, we also may assume h = 1. Again, f(x) = (f1(x1), f2(x2)) satisﬁes

x1x2 = f1(x1)f2(x2) for all x1, x2 ∈ F. We also obtain that f(x) = (x1u, x2u) for some u ∈ U(1; F). In this case, since F is associative, u is not necessarily ±1. For F = R, C, a diagonal action of U(1; F) ⊂ U(2; F) allows us to conclude f ∈ U(2; F). For F = K, we use an element of 84 A.2. NON-COMPACT CASE

Sp(1). In this case, note that the action of Sp(1) is not recovered by a diagonal action of Sp(1) ⊂ Sp(2). Therefore

Isot(E1) = U(2; F), for F = R, C Isot(E1) = Sp(1) × Sp(2), for F = K.

On the other hand, 2 FP = Aut0 (H3(F)) /Isot(E1).

Since dim(PU(3; F)) = dim(Aut0(H3(F))), the result follows from Lemma A.1. For alternative proofs, see [27, Propositions 2.11.1 and 2.12.1]. Another reference is [12, p.165].

A.2 Non-compact Case

A matrix in H1,2(F) ⊂ H1,2(O) also satisﬁes (2.9). Thus, a proof similar to 2.1 shows that Aut(H1,2(F)) preserves trace. A similar identiﬁcation between the Hermitian form of signature (1, n) and the trace also holds as in (2.1). Hence, we have the following inclusion as closed subgroup:

Aut(H1,2(F)) ⊂ O(1, 3d + 2).

This implies that Aut(H1,2(F)) is a Lie group (Theorem 0.10). For A ∈ H1,2(F) the action of g ∈ U(1, 2; F) is given by

g(A) = (gH)A(g∗H).

∗ 2 Since g HgH = I3, then g(A ) = g(A)g(A). However, the group PU(1, 2; F) that acts eﬀectivelly, thus

PU(1, 2; F) ⊂ Aut0(H1,2(F)).   r1 x1 x2 The groups (A.1) are subgroups of PU(1, 2; F) and acts on A = −x1 r2 x3 ∈ H3(F) −x2 x3 r3 as t r1 λg(x) x1 2 r2 x3 g(A) = ∗ , x = ∈ F , a = . −λg(x) gag x2 x3 r3 In addiction, we have the subgroup

h 0 U(1, 1; ) ' : h ∈ U(1, 1; ) . F 0 1 F

Theorem A.2. The connected automorphism group Aut0(H1,2(F)) is:

1. SO0(1, 2), if F = R;

2. PSU(1, 2), if F = C;

3. PSp(1, 2), if F = K. APPENDIX A. AUTOMORPHISMS OF H3(F) AND H1,2(F) 85

Proof. The matrix model for 2 yields also a matrix model for 2 by restricting the HO HF algebra to F. Let   r1 x1 x2 A = −x1 r2 x3 ∈ H3(F) −x2 x3 r3 be an inner point represented by an idempotent matrix A. An action of U(2; F) diagonal- r x izes the block a = 2 3 . As in Theorem 4.3, we may suppose x3 r3

−1 z 0 A = −z |z|2 0 , 0 0 0 where (1, z) ∈ 2 is a negative vector. That is, A ∈ 1 . For given any other inner point F HF B, we may suppose B in the form

−1 w 0 −w |w|2 0 . 0 0 0 a1 a2 From Chapter 3 (Theorem 3.1), there exists g = ∈ U(1, 1; F) such that a2 a1 a1 a2 1 λ ∗ g(1, z) = . = , λ ∈ F . a2 a1 z wλ

This transitivity will be recovered considering the action of U(1, 1; F) ⊂ PU(1, 2; F) on (A.2):       −a1 a2 0 −1 z 0 −a1 a2 0 2 g(A) = −a2 a1 0 −z |z| 0 −a2 a1 0 , 0 0 1 0 0 0 0 0 1

2 which yields g(A) = |λ| B. The computation of Isot(E1) is the same as before and also gives Isot(E1) = U(2; F), for F = R, C, or Sp(1) × Sp(2). On the other hand,

2 HF = Aut0(H1,2(F))/Isot(E1).

Since dim(PU0(1, 2; F)) = dim(Aut0(H1,2(F))), the result follows from Lemma A.1. (...) Que sei eu do que serei, eu que nãosei o que sou? Ser o que penso? Mas penso ser tanta coisa! E hátantos que pensam ser a mesma coisa que nãopode haver tantos! (...)

Fernando Pessoa

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