Fabrice Theoleyre

Fabrice Valois

A Virtual Structure for Hybrid Networks

Hybrid networks are heterogeneous networks merging both wireless and ad hoc nodes and where the interconnection to IP world is an important topic through gateways called AP (access point). Indeed, each node can be contacted and can contact another node in Internet. To reach that, architectures to support mobility management will be studied. The solutions inspired by wired networks are not particularly suited to hybrid networks. We propose to use a virtual dynamic infrastructure including both backbone and clusters. A backbone is suited to spare energy, optimize control traffic diffusion and hierarchize participants. The clusters are intended to create services areas and to handle particularly the mobility management. We present algorithms to both construct and maintain such structure. This dynamic topology is robust according to mobility, and is well suited to implement mobility management and localization procedure. Finally, the number of backbone members and clusters are completely parameterizable according to the environment.

I. INTRODUCTION

Mobile Ad hoc networks (also named MANET) could be defined by spontaneous wireless networks, in which neither wireless nor wired infrastructures exists. All communicating objects are mobile and organize themselves to set up an efficient network. Hybrid networks are ad hoc networks connected to Internet, thanks to APs. The key issue in such network is the routing problem. The routing (proactive [START_REF] Clausen | Optimized link state routing protocol[END_REF] and reactive [START_REF] David | Dynamic source routing in ad hoc wireless networks[END_REF]) is based on high amount of broadcasted control traffic flooding the network. But, in MANET, broadcasts constitute the broadcast storm problem [START_REF] Ni | The broadcast storm problem in a mobile ad hoc network[END_REF]: redundancy of transmissions and reliability problems due to collisions. Hence, we propose to structure hybrid networks using a distributed approach with a virtual structure.

We propose to first create a virtual backbone [START_REF] Liang | Virtual backbone generation and maintenance in ad hoc network mobility management[END_REF]. Some nodes in the network will be elected to serve as masters. This election could take into account batteries longevity, mobility, etc., these parameters being adaptable. The creation of a dynamic backbone have some advantages. First, the backbone can spare control bandwidth for broadcast packets, all packets being sent only to backbone members. Moreover, we can add wireless-routers which are automatically integrated in the hybrid network. Finally, the backbone can help us to manage nodes mobility within wireless hybrid network. Using virtual backbone, we propose further to create services areas thanks to clusters. We can hence create a hierarchy in the hybrid network. This can help clusterhead deliver a service like mobility management and addresses attribution. Moreover, clusters introduce stability in the hybrid network by hiding some properties of the dynamic environment to higher levels. Both backbone and clusters are built and maintained simultaneously to reduce overhead and combine benefits.

Next, we will expose related work of creation of virtual structures in hybrid networks. Section III will expose our solution merging both backbone and clusters, with their construction and maintenance procedures, rarely described due to their complexity. Section IV will present simulations results: the stability of our structure, the persistence of masters, the impact of mobility and degree are discussed. Finally, section V will expose some perspectives.

II. RELATED WORK

A. Backbone

A MANET can be modeled by a graph where the vertices are the communicating objects, and the edges the links between two nodes in communication. Backbone in such model could be represented by a k mcds -Minimal Connected Dominating Set (k mcds -MCDS). In this structure, it exists a maximal distance, k mcds , between any node of the graph and the MCDS. The MCDS members are named dominators and the other nodes dominatees. The dominators give a connected structure. The cardinality of dominators must be minimalized. Many articles deal with the backbone construction using a 1-MCDS approximation. The construction can often be divided in two steps ( [START_REF] Butenko | On the construction of virtual backbone for ad hoc wireless networks[END_REF], [START_REF] Cardei | Connected domination in ad hoc wireless networks[END_REF], [START_REF] Liang | Virtual backbone generation and maintenance in ad hoc network mobility management[END_REF]). If a leader exists, it initiates the construction, otherwise it could be elected. The first step elects some nodes as dominator, their neighbors becoming their dominatees. The election is often based on the lowest-id or on the highest-degree. The second step is the interconnection of these dominators, some dominatees becoming dominators. [START_REF] Cardei | Connected domination in ad hoc wireless networks[END_REF] explores each dominatee and choose firstly the dominatee which has the highest number of dominator neighbors. In [START_REF] Alzoubi | Distributed heuristics for connected dominating set in wireless ad hoc networks[END_REF], each connected dominator sends broadcasted invitations in order to invite other dominators to connect themselves via it, the leader being initially the only connected dominator. However, these algorithms propose a 1-MCDS construction and are not well suited to k mcds -MCDS. The exploration method, for instance, requires a high delay before the end of MCDS construction. Furthermore, we think that a k mcds -MCDS is more suitable in order to have less dominators in the network, and to allow more nodes to spare their energy in reducing their participation in the network control. In addition, the maintenance is a crucial procedure in a dynamic environment. The backbone members must be constantly updated to have a stable, efficient backbone. We think that a backbone should be generally constructed once, and maintained all the rest of the time. Nevertheless, no article proposes a maintenance procedure for MCDS in MANET context. [START_REF] Liang | Virtual backbone generation and maintenance in ad hoc network mobility management[END_REF] introduces a maintenance, but the dominators are not connected to each other, the overhead generated is important (dominators can be (2k+1)-hops far), and virtual connection between dominators could be suboptimal when dominators move.

B. Clusters

Many articles propose to construct clusters in ad hoc network to allow quality of service or to provide hierarchical routing. Clustering is cutting the network in zones, with a master, named clusterhead, for each zone. It exists in a k clustercluster a maximal distance, k cluster , between one node and its clusterhead. Clustering tends generally to minimize the global number of clusters. The construction of clusters is usually based on an election. The node which is the strongest of its k cluster -neighborhood during a round elects itself as clusterhead. This election can use the degree, the address [START_REF] Richard | Adaptive clustering for mobile wireless networks[END_REF], [START_REF] Safwat | Infrastructure-based routing in wireless mobile ad hoc networks[END_REF], a mobility metric [START_REF] Basu | A mobility based metric for clustering in mobile ad hoc networks[END_REF]. . . Different approaches exist for the 1-cluster maintenance. The first is to always maintain as clusterhead the strongest node of a cluster, implicating many changes. The second [START_REF] Richard | Adaptive clustering for mobile wireless networks[END_REF] modifies cluster composition only when clusterhead is too far or down, with the creation of new clusters. There exists other propositions to construct k cluster -clusters. In [START_REF] Amis | Max-min dcluster formation in wireless ad hoc networks[END_REF], the construction is in 2 phases. The first propagates the highest identifiers, and the second informs the clusterheads that they have been chosen. The authors of [START_REF] Fernandess | K-clustering in wireless ad hoc networks[END_REF] propose to construct a tree and to prune the branches when they have k cluster members. Here, k cluster -clusters means that a zone must contain at most k cluster members. But no maintenance, in both methods is explained.

III. PROPOSITION

A. Motivations

We propose to construct a structure which combines the assets of both backbones and clusters (fig. 1). First, we construct a k mcds -MCDS approximation, which represents our backbone. In fact, we don't try to minimize the cardinality of dominants: we try rather to limit this number. By this way, we construct more precisely a CDS. This virtual structure allows to better handle control traffic: nodes not in the backbone can spare their energy, the control information is flooded only to backbone's nodes. Secondly, we will integrate in this backbone a structure of k cluster -clusters. These clusters will serve as services areas like mobility management (cluster can represent a localization) or addresses assignment. Moreover, we propose a maintenance procedure for these two merged structures, limiting the induced overhead.

B. Metric

A key issue for such structure is to be as stable as possible. Hence, we introduced a combined metric that we mean representative of the virtual structure stability. This stability weight depends in order of importance on:

• persistence: to force a master to be master as long as possible, and also support stability;

• relative mobility: to support nodes with a stable neighborhood, which can better and longer fulfill their role; • degree: not too small to restrict collisions and enough important to minimize number of masters; • energy: to penalize nodes with too low energy reserve which can soon die. Metrics for virtual topology will be studied in depth in a future article, in order to simulate the behavior in different environments and to prove such a combined metric can be efficient.

C. Backbone 1) Construction:

We propose to construct a CDS of variable diameter, k cds . The construction of our backbone can be divided in two parts. The first step forces all nodes to choose a dominator. The AP of the hybrid network represents the natural leader and initiates the construction, becoming dominator. There exist 4 states:

• dominator: backbone member;

• dominatee: backbone client;

• active: in election state;

• idle: in initialization. For neighborhood discovering, a node sends initially and periodically an hello with its identifier, state (idle, active, dominatee, dominator) and weight, propagated to k hops. Each node knows also its k-neighborhood. When a node changes its state, it advertises it to its neighborhood with a gratuitous hello. The next rules are applied:

• An idle node which receives an hello from a dominator becomes its dominatee, fixing the dominator as its father; • An idle node which receives an hello from a dominatee becomes active; • An active node which has the highest weight in its k cdsneighborhood of active nodes during τ time becomes dominator. This process is asynchronous, active nodes must also wait for the end of one step before becoming perhaps dominator. All candidates must have the time to declare their potential new state. The state message of an active node can be propagated during k cds -1 hops to prevent others that it participates to the election. Also, τ depends on the propagation delay of a message during 2 • (k cds -1) hops. For the backbone's construction, a node must know its k cds -neighborhood, so k ≥ k cds . The kneighborhood discovering could be in the long term integrated to a routing protocol, cooperating to share the amount of information and reducing the overhead.

The second step is the interconnection of dominators. We connect the backbone thanks to the AP, acting as a gateway to Internet: the AP represents also our leader in the interconnection phase. Initially, only the leader is a connected dominator. A connected dominator sends a join-message to other non connected dominators, with a TTL of 2k cds +1 as in [START_REF] Alzoubi | Distributed heuristics for connected dominating set in wireless ad hoc networks[END_REF]. Indeed, a dominator is maximum 2k cds + 1 hops far from another dominator. A non connected dominator will answer with a join-reply, which follows automatically the inverse route, Each dominator maintains the identity of its father which is the next dominator in the CDS toward the root. The father of a dominator is necessarily one of its 1-neighbor in order to reflect the CDS connection. These new connected dominators will finally send a join message, and the process reiterates until total connection of hybrid network. Maintenance procedure begins as soon as construction is locally finished. Finally, each node owning a father, the k cds -CDS is also a model of tree, with a leader as its root.

A node must know for the construction its k cds -neighborhood before changing its active state into dominator state. But the neighborhood knowledge presents a cost. In consequence, we fixed the neighborhood discovering to k=k cds .

2) Maintenance: The maintenance could be sum up in two elementary rules: each node must all the time own one and only one dominator, and the set of dominators must be connected. We also propose periodically hellos packets from the AP, with an unique incremental ap-hello id. Such ap-hellos packets are relayed by dominators, so a backbone member knows if it is already connected. In parallel, each dominator, like other nodes, sends hellos packets to advertise its presence.

We have separated maintenance procedures for dominatees and for dominators. A dominatee d which lost its dominator searches a new dominator in its k cds -neighborhood-table. If a candidate exists, d chooses it as dominator. If it exists many candidates, it chooses the node with the strongest weight. If no candidate is present in the neighborhood-table, d becomes active. A new election occurs, like in construction mode: it avoids multiple neighbors, which lost their dominator, to become dominator simultaneously.

The maintenance process for dominators is more complex due to the necessity to maintain the connectivity of our CDS approximation:

1) A dominator D not yet connected (x ap-hellos missed) sends a reconnect-request with the id of the last ap-hello seen. If D receives further a reconnect-reply, D will act like with a join-message; 2) Only connected dominators can answer to a reconnect-request with a reconnect-reply.

A connected dominator is a node which has received a newer ap-hello (higher ap-hello id);

3) A dominator D after y successive useless reconnect-request breaks its branch of k CDS -CDS. All its sons and descendants receive its break-message and go to idle state, as in initialization. A reconstruction will follow. After a break-message, an area is formed with only nodes in idle state, waiting for an exterior solicitation for reconstruction, as in construction mode. The next rules are also applied in order to reconstruct this zone:

1) If a connected dominator has an idle neighbor in its k cdsneighborhood, it sends a join-message to initiate the area reconstruction, acting as a new temporary leader. 2) A dominatee neighbor of its dominator which has an exactly k cds hops far idle neighbor advertises its dominator that it must send a join-message for the reconstruction. We must create an antagonist procedure in order to avoid a constant increasing backbone. A dominator which has no dominatee at exactly k cds hops and no son is useless: its father can serve this node and all its dominatees. This dominator sends also a useless-message forcing all its dominatees to choose its father as new dominator, the address of its father being contained in hello packets.

Hence, the maintenance allows all nodes to have a dominator, and a connected dominator set. We have also a virtual dynamic backbone, as stable as possible, using for its construction and maintenance a stability metric. This maintenance requires to know the k cds -neighborhood, as in the construction process. The neighborhood discovering is also set up to k cds hops during both construction and maintenance.

D. Clustering 1) Construction:

The clusters are integrated to the backbone structure. Only backbone members participate to the construction of clusters in order to reduce the overhead during both construction and maintenance. Additionally, clusterheads are necessarily backbone members which is an advantage to further exchange control packets with other clusterheads via the backbone. Dominatees don't participate to the election and take automatically the clusterhead of their dominator. Each backbone member discovers its backbone member neighbors, creating a temporary (k cluster -k cds )-cds-neighbors table, a cds-neighbor being its father or son in the k cds -CDS. Thus, a 1-neighbor is not obligatory a cds-neighbor.

A dominatee is, by definition, k cds far from its dominator. The radius of our clusters is also k cluster . The cluster-hellos packets have the format of normal hellos packets, but can't be integrated in other hellomessages because of backbone utilization. This (k clusterk cds )-neighborhood discovering is also independent from the normal neighborhood-discovering. However, this type of packet is only used in the cluster construction, the overhead is also not too increased.

Each node maintains, during the cluster construction phase, the table of its (k cluster -k cds )-cds-neighbors, and the list of these neighbors which have no clusterhead. For each round, if a node N has, during t time (message propagation time), the highest weight in this list, N is elected as clusterhead. All its k cluster -neighbors having no clusterhead choose N as their new clusterhead. The process reiterates until each node in the hybrid network has a clusterhead. Maintenance process begins as soon as construction is locally finished.

2) Maintenance: The maintenance is a key point for both backbone and clusters. The maintenance must try to minimize the changes of cluster composition. We have introduced a clusterhead-hello packet, with the same role as for ap-hellos packet. The dominatees don't participate to the maintenance.

A backbone member which is always connected to its clusterhead (at least one of the last x clusterhead-hellos received) does nothing. A backbone member B which is not yet connected (none of the last x clusterhead-hellos received) broadcasts a reconnect-request on the virtual backbone with the id of the last clusterhead-hello seen. If B receives further a reconnect-reply, it chooses the source as new clusterhead. We force cluster connectivity with two rules: firstly, a node of the same cluster of the source of reconnect-request which has more recent information (higher clusterhead-hello id) sends a reconnect-reply else it relays the reconnect-request to other members of its own cluster; secondly, a node of a different cluster as source, but a neighbor of the source can send a reconnect-reply if it received one of the last x clusterhead-hellos.

We must propose a mechanism avoiding an increasing number of clusterheads, as in backbone maintenance for dominators. C useless is an useless clusterhead if none of its 1neighbors declares C useless as their clusterhead. Because of cluster connectivity, no other nodes in the network can have chosen C useless as clusterhead. An useless clusterhead which receives a clusterhead-hello chooses the source as new clusterhead, losing its role of master.

IV. PERFORMANCE EVALUATION

A. Simulation

We used Opnet Modeler 8.1 to simulate the behavior of our solution. The radio level is represented by the standard IEEE 802.11b of Opnet Modeler. The random waypoint model was used to simulate mobility of each node [START_REF] Camp | A survey of mobility models for ad hoc network research[END_REF]. A node moves on an area of 1900m × 1900m with a 300m radio range. We estimate Fig. 2. Performances according to duration a node disconnected if it missed the 3 last ap-hellos or clusterhead-hellos. Thus, we tend to over-estimate the disconnection time. We assume there is 30 nodes, a speed of 5m.s -1 , a degree of 10, k cds =2 and k cluster =4, only one parameter being changed by simulation. The ap-hellos and cluster-hellos packets are sent every 2 seconds, and the hellos packet every 5 seconds.

B. Results

We simulated the influence of degree and number of participants, but these results are not presented because of the lack of place.

1) Duration: First, we observe the behavior of our structure during 1 hour. A dominator keeps its role during about 2.4 minutes, and a node changes of dominator every 2.8 minutes. A clusterhead remains clusterhead during about 1.4 minutes, and a node changes its clusterhead after 2 minutes. This duration is small to spare the clusterheads energy, but seems sufficient to allow clusterheads to serve as masters. There exists on average 8.6 dominators and 3.3 clusterheads. The nodes with higher weight tend to be chosen as dominators: with an average weight of 57, a dominator has a weight of 71 and a dominator with Fig. 3. Performances according to k cds and k cluster dominatees a weight of 115. The CDS is relatively stable, the breaks corresponding to CDS reconstruction. The connection time are high (92.7% to CDS, 93.9% to clusterhead). In the future, there will exist data traffic, and the dominators will use these packets to maximize the connection time, having more information about their neighbors and ancestors.

2) CDS and Clusters diameters: We simulated the behavior of our solutions when backbone and cluster diameters vary. We observe that the backbone cardinality decreases when k cds increases (fig. 3). Many nodes can spare their energy, but it exists sufficient backbone members to distribute the load. We observed during the simulations that the connection percentage falls when k cds is too high. Indeed, many collisions occurred and the reconnection is more difficult, due to the average distance between two dominators. We observe that there exists less clusterheads when k cluster increases (fig. 3). We can notice the stability of clusterheads, which is an important property.

3) Mobility: We simulated the influence of mobility on the backbone and clusters. The speed varied between 0 and 35m.s -1 . The cardinality of backbone set is relatively stable , the variations being acceptable. The clusterhead set cardinality is almost constant. For a high mobility of 15m.s -1 , there Fig. 4. Performances according to mobility exist a maximum of 3.5 clusterheads (1/8 of nodes) and 11 dominators (1/3 of nodes). The connectivity to the clusterhead, over 92%, remains almost constant, even with high mobilities. The connectivity to backbone decreases, but remains over 90% for usual speeds of less than 10m.s -1 . Finally, a clusterhead keeps its role of clusterhead during about 1 to 2 minutes and a node keeps its clusterhead during 1 to 50 minutes.

V. CONCLUSION AND FUTURE WORK

We propose an integrated virtual and dynamic infrastructure to organize an hybrid network. This structure combines both backbone and clusters, with the introduction of a distributed algorithm for both construction and maintenance. We show the structure is stable and clusterheads keep their role during a long time. Moreover, our algorithms are robust since they present a very good behavior according to high mobilities. Finally, the cardinality of our structure is completely parameterizable according to the environment, the backbone and cluster diameters being parameters. A key contribution of this article is to present a solution organizing an hybrid network. It constitutes a weelsuited framework to implement network functionalities such as the mobility management, disregarding the physical topology. Such a structure will integrate multiple functionalities to mutualize the cost of construction and maintenance. Next step of this study will consist in improving mobility management with a fine or coarse localization solution. In parallel, the clustering potentially allows to implement a solution of sleeping mode so that normal nodes sleep and spare their energy. Besides, we propose a solution with a unique AP which serves as gateway. It represents a single point of failure, multiple AP are also required to have a certain redundancy. Finally it is important to further theoretically investigate our structures to find for example an upper bound of backbone cardinality and analytically prove the efficiency of our proposition.

Fig. 1 .

 1 Fig. 1. Backbone and clusters combination


Marilena Moruz
email: marilena.moruz@gmail.com

Luc Vrancken
email: luc.vrancken@univ-valenciennes.fr

: Lamav

A CLASSIFICATION OF ISOTROPIC AFFINE HYPERSPHERES

Keywords: 2010 Mathematics Subject Classification. 53A15. Key words and phrases. Affine differential geometry, Blaschke hypersurface, affine homogeneous, isotropic difference tensor

 and [?]. We first show that the dimension of an isotropic affine hypersurface is either 5, 8, 14 or 26. Next we assume that M is an affine hypersphere and we obtain in each of the possible dimensions a complete classification.

Introduction

The notion of a submanifold with isotropic second fundamental form was first introduced in [START_REF] Neill | Isotropic and Kaehler immersions[END_REF] by O'Neill for immersions if Riemannian manifolds and recently extended by Cabrerizo et al. in [START_REF] Cabrerizo | Rigidity of pseudo-iotropic immersions[END_REF] for pseudo-Riemannian manifolds. We say that M has isotropic second fundamental form h if and only if for any tangent vector X at a point p we have that < h(X(p), X(p)), h(X(p), X(p)) >= λ(p) < X(p), X(p) > 2 .

If λ is independent of the point p, the submanifold is called constant isotropic. Given the similarities between the basic equations that charactherise the manifolds and the important role played by the difference tensor it is natural to introduce the equivalent notion of isotropy in affine geometry. That is, a hypersphere M has isotropic difference tensor K if and only if for any tangent vector X at a point p we have that h(K(X(p), X(p)), K(X(p), X(p))) = λ(p)h(X(p), X(p)) 2 , where h is the affine metric on the hypersurface. Note that a 2-dimensional affine surface is always isotropic. In case that the affine metric is positive definite such submanifolds have been previously studied in [START_REF] Birembaux | Isotropic Affine Spheres[END_REF] and [?]. In [START_REF] Birembaux | Isotropic Affine Spheres[END_REF], beside a restriction on the dimension, a complete classification was obtained in case that the affine hypersurface is an affine sphere. In [?] a complete classification was given of 5 dimensional positive definite affine hypersurfaces.

In this paper we deal with the case that the induced affine metric has arbitrary signature. We will first show that the restriction of the dimension remains valid in the indefinite case. Even though the proof remains based on the Hurwitz theorem it is essentially different from the proof in the definite case. This is because unlike in the definite case, the unit tangent bundle at a point p is no longer a compact manifold. Instead of this null vectors will play an important role in the proof of the restriction of the dimension.

In the second part of the paper we will then restrict ourselves to the case that M is an affine hypersphere and we will deduce that in that case the immersion also has parallel difference tensor (and is a pseudo-Riemannian symmetric space). We then look at each of the possible dimensions and determine in each case explicitly by elementary means the form of the difference tensor and the possible examples. Note that for this second part also a more involved Lie group approach would be possible. We show the following theorems.

Theorem 1. Let M 5 ne a 5-dimensional affine hypersphere of R 6 . Assume that M is λ-isotropic with λ = 0. Then if we identify R 6 Note that in the positive definite case we had the connected component of the identity.

Theorem 2. Let M 8 ne a 8-dimensional affine hypersphere of R 9 . Assume that M is λ-isotropic with λ = 0. Then if we identify Theorem 3. Let M 8 ne a 14-dimensional affine hypersphere of R 9 . Assume that M is λ-isotropic with λ = 0. Then if we identify Theorem 4. Let M 8 ne a 26-dimensional affine hypersphere of R 9 . Assume that M is λ-isotropic with λ = 0. Then if we identify

Preliminaries

Let f : M -→ R n+1 be a nondegenerate affine hypersurface immersion. Let D be the covariant derivative on R n+1 and Ω the volume form given by Ω(u 1 , . . . , u n+1 ) = det(u 1 , . . . , u n+1 ), such that R n+1 is endowed with its standard equiaffine structure (D, Ω). In a general setting, an affine manifold (M n , ∇) is said to be equiaffine if there exists a volume form ω, i.e. a non-vanishing n-form, on M which is parallel with respect to ∇:

(1) (∇ X ω)(X 1 . . . , X n ) = X(ω(X 1 , . . . , X n )) -ω(∇ X X 1 , . . . , X n ) -. . .

-ω(X 1 , . . . , ∇ X X n ).

In this case we may also say that (∇, ω) is an equiaffine structure on M n . In what follows, we briefly recall the construction of an equiaffine structure on an affine hypersurface M n in R n+1 . For more details we refer to [START_REF] Nomizu | Affine differential geometry[END_REF]. First, let p ∈ M and X, Y ∈ T p M . If we choose an arbitrary transversal vector field η we can decompose

D X Y = ∇ η X Y + h η (X, Y )η.
It is easy to see that ∇ η is a connection on M and h η is a symmetric bilinear form. Note that the fact whether this bilinear form is degenerate or not is independent of the choice of transversal vector field η. As such M is called nondegenerate if and only if this bilinear form is nondegenerate. Hence, locally there exists a volume form on M associated to h η , given by

ω h η (X 1 , . . . , X n ) = | det h η (X i , X j ) |.
Next, we want to introduce a canonical transversal vector field ξ. In order to make a good choice, we define ω η (X 1 , . . . , X n ) := Ω(X 1 , . . . , X n , η), for X 1 , . . . , X n vector fields on M n and we ask that the volume forms ω ξ and ω h ξ coincide and that (∇ ξ , ω ξ ) is an equiaffine structure on M n . Notice that these conditions guarantee the existence of a unique (up to sign) transversal vector field ξ, see [START_REF] Nomizu | Affine differential geometry[END_REF]. It is called the affine normal vector field, or the Blaschke normal vector field. For convenience, we will denote from now on ∇ := ∇ ξ . Finally, in terms of this transversal vector field we get for M the formulas of Gauss and Weingarten, respectively, as follows:

D X Y = ∇ X Y + h(X, Y )ξ, D X ξ = -SX,
where we call ∇ the induced affine connection, h the affine metric, ξ the affine normal field or Blaschke normal field and S the affine shape operator. An affine hypersurface is called a (proper) affine sphere if S is a (non zero) multiple of the identity. Moreover, let R denote the curvature tensor of M n . Then, the following fundamental equations hold with respect to the induced affine connection:

Gauss equation: R(X, Y )Z = h(Y, Z)SX -h(X, Z)SY ; Codazzi equation for h : (∇h)(X, Y, Z) = (∇h)(Y, X, Z); Codazzi equation for S : (∇ X S)Y = (∇ Y S)X; Ricci equation: h(SX, Y ) = h(X, SY );
The Codazzi equation implies that for a proper affine sphere, the multiple of the identity is constant, in which case by applying a homothety of the ambient space, we may assume that S = εI, where ε = ±1. Moreover we have that ξ + εf , where f denotes the position vector, is a constant vector which is called the center of the proper affine hypersphere. By applying a translation in the ambient space we may of course always assume that the center is the origin. As ∇ is not necessarily compatible with the affine metric h, we can consider the difference tensor K, a (1, 2)-type vector field defined as:

K(X, Y ) = ∇ X Y -∇X Y,
where ∇ is the Levi-Civita connection on M . By convention, one may also write K X Y instead of K(X, Y ). The classical Berwald theorem states that K vanishes identically if and only if M is congruent to a nondegenerate quadric.

Proposition 5. We have the following properties for K:

(1) K(X, Y ) = K(Y, X);

(2) for any X we have Y → K X Y is a symmetric linear map and traceK X = 0 (the apolarity condition);

(3) h(K(X, Y ), Z) = h(K(X, Z), Y ).
It is easy to prove that ∇h is related to K by:

∇h(X, Y, Z) = -2h(Z, K(X, Y )).
Moreover, the equations of Gauss, Ricci and Codazzi, respectively, may also be written out with respect to the Levi-Civita connection as follows:

(

2) R(X, Y )Z = 1 2 {h(Y, Z)SX -h(X, Z)SY + h(SY, Z)X -h(SX, Z)Y } -[K X , K Y ]Z, ∇K(X, Y, Z) -∇K(Y, X, Z) = 1 2 {h(Y, Z)SX -h(X, Z)SY -h(SY, Z)X + h(SX, Z)Y, } ( ∇X S)Y -( ∇Y S)X = K(Y, SX) -K(X, SY ), (∇h)(X, Y, Z) = (∇h)(Y, X, Z), h(X, SY ) = h(SX, Y ), where [K X , K Y ]Z = K X K Y Z -K Y K X Z and ∇K(X, Y, Z) = ∇X K(Y, Z) -K( ∇X Y, Z) -K(Y, ∇X Z).
We have the following Ricci identity:

(3) ∇2 K(X, Y, Z, W ) -∇2 K(Y, X, Z, W ) = R(X, Y )K(Z, W ) -K( R(X, Y )Z, W ) -h(Z, R(X, Y )W ).
A nondegenerate hypersurface M of the equiaffine space R n+1 is called locally homogeneous if for all points p and q of M , there exists a neighborhood U p of p in M , and an equiaffine transformation A of R n+1 , i.e. A ∈ SL(n + 1, R) R n+1 , such that A(p) = q and A(U p ) ⊂ M . If U p = M for all p, then M is called homogeneous. Let G be the pseudogroup defined by

G = {A ∈ SL(n + 1, R) R n+1 | ∃U , open in M : A(U ) ⊂ M },
then M is locally homogeneous if and only if G "acts" transitively on M . If M is homogeneous, then G is a group and every element of G maps the whole of M into M . The following proposition is probably well known, however, as we did not find an explicit reference, we include a small proof. Proposition 6. Let M n be a nondegenerate homogeneous affine hypersurface. Assume that G ⊂ SL(n + 1, R). Then M is an affine sphere centered at the origin.

Proof. We denote the immersion by f . Let p and q be in M and let g be the affine transformation which maps p to q. We have that

ξ(g(p)) = dg(ξ(p)), and dg(f (p)) = g(f (p)) = f (q)
Moreover as M is homogeneous we know that the position vector can not be a tangent vector at one point (and therefore at every point). Indeed if that were the case, we would habe a tangent vector field X such that X(p) = f (p). This would imply that D Y X = Y , and therefore h(X, Y ) = 0 for any vector field Y . This implies that the immersion f would be degenerate. Therefore we may write ξ = ρf + Z, where Z is a tangent vector field and ρ a function. As M is locally homogeneous and g belongs to SL(n + 1, R) it follows that ρ is constant. The construction of the affine normal of [START_REF] Nomizu | Affine differential geometry[END_REF] then implies that M is an affine sphere centered at the origin.

The equivalent notion in affine geometry for isotropic submanifolds from Riemannian geometry, which was introduced by O'Neill in [START_REF] Neill | Isotropic and Kaehler immersions[END_REF], corresponds to submanifolds for which the difference tensor K is isotropic, that is, it satisfies:

h(K(X, X), K(X, X)) = λ(p)h(X, X) 2 ,
for some tangent vector field X. Here we will always deal with the case that λ = 0. Therefore, if necessary, by replacing ξ with -ξ, we may assume that λ is positive and therefore there exists a positive function µ such that λ = µ 2 . We will also use the following lemma from [START_REF] Nomizu | Affine differential geometry[END_REF]: Lemma 7. Let F : M → R n+1 be an equiaffine immersion. If the metric on R n+1 is indefinite, then the immersion is isotropic if and only if for any tangent vectors X 1 , X 2 , X 3 , X 4 ∈ T p M, we have that

(4) h(K(X 1 , X 2 ), K(X 3 , X 4 ))+h(K(X 1 , X 3 ), K(X 2 , X 4 ))+h(K(X 1 , X 4 ), K(X 2 , X 3 )) = λ(p){h(X 1 , X 2 )h(X 3 , X 4 ) + h(X 1 , X 3 )h(X 2 , X 4 ) + h(X 1 , X 4 )h(X 2 , X 3 )}.
By using lemma [START_REF] Carmo | Riemannian geometry[END_REF] and property (3) in Proposition 5 we get that an affine submanifold M n in R n+1 is isotropic if and only if for any tangent vectors X 1 , X 2 , X 3 , X 4 ∈ T p M we have that

(5) K X1 K X2 X 3 + K X2 K X1 X 3 + K X3 K X1 X 2 = λ(p) (h(X 2 , X 3 )X 1 + h(X 1 , X 3 )X 2 + h(X 1 , X 2 )X 3 ) .
Theorem 8. ( [START_REF] Chen | Complex extensors and Lagrangian submanifolds in indinite complex Euclidean spaces[END_REF]) Let (M n k , h) be an n-dimensional simply connected pseudo-Riemannian manifold with index k. Let ∇ denote the Levi Civita connection, R its curvature tensor and let T M denote the tangent bundle of

M n k . If K is a T M - valued symmetric bilinear form on M n k satisfying that i) h(K(X, Y ), Z) is totally symmetric ii) ( ∇K)(X, Y, Z) = ∇X K(Y, Z) -K( ∇X Y, Z) -K(Y, ∇X Z) is totally symmetric, iii) R(X, Y )Z = c(h(Y, Z)X -h(X, Z)Y ) + K(K(Y, Z), X) -K(K(X, Z), Y ),
then there exists an affine immersion φ : M n k → R n+1 as an affine sphere with induced difference tensor K and induced affine metric h.

Theorem 9. ([5]) Let φ 1 , φ 2 : M n k -→ R n+1 be two affine immersions of an pseudo-Riemannian n-manifold (M n k , h) with difference tensors K 1 , K 2 , respec- tively. If h(K 1 (X, Y ), φ 1 * Z) = h(K 2 (X, Y ), φ 2 * Z) for all tangent vectors fields X, Y, Z ∈ T p M n k , then there exists an isometry φ of R n+1 such that φ 1 = φ • φ 2 .

Possible dimensions and choice of frame

From now on we will always assume that M n k is an affine isotropic hypersurface in R n+1 . Here n denotes the dimension and k the index of the affine metric. In case that the metric is definite a classification was obtained already in [START_REF] Birembaux | Isotropic Affine Spheres[END_REF], In view of this we will also assume thay M is neither positive nor negative definite, i.e. 1 ≤ k < n. Also recall that because of the properties of K any surface is isotropic. Therefore we will also assume that n > 2. First, we have the following lemma: Lemma 10. Let M n k be an n-dimensional isotropic affine hypersurface and let p ∈ M n k . If for any null vector v ∈ T p M we have that K(v, v) is a null vector such that h(K(v, v), v) = 0, then the difference tensor K vanishes.

As its proof is very similar to the proof of Lemma 3.1 in [START_REF] Li | Minimal Lagrangian isotropic immersions in indefinite complex space forms[END_REF], we omit it here. From now on , we will assume that λ = 0. By Lemma 10, there exists a null vector v 0 such that v 0 and K(v 0 , v 0 ) are linearly independent and h(v 0 , K(v 0 , v 0 )) = 0. Using Lemma 7, we have that for any null vector u

(6) h(K(v 0 , v 0 ), K(v 0 , u)) = λh(v 0 , v 0 )h(v 0 , u) = 0.
As K v0 is a symmetric operator with respect to the metric h, we get that K v0 K v0 v 0 = 0. Moreover, taking in particular u = v 0 in (6), we get that K(v 0 , v 0 ) is a null vector.

We can now take a null frame such that

e 1 = v 0 , e 2 = K v0 v 0 .
By rescaling v 0 if necessary, we may assume that h(K(v 0 , v 0 ), v 0 ) = -4λ 2 . Then we get h(e 1 , e 1 ) = h(e 2 , e 2 ) = 0 h(e 1 , e 2 ) = -4λ 2 , K(e 1 , e 1 ) = e 2 , K(e 1 , e 2 ) = K v0 K v0 v 0 = 0. (7)

Using the isotropy condition in (5) for X 1 = X 2 = e 1 , X 3 = e 3 we get that

K e2 e 2 = -8λ 3 e 1 .
From relation [START_REF] Carmo | Riemannian geometry[END_REF] we can see that the space span{e 1 , e 2 } is invariant under the operator K e1 . As the operator K e1 is symmetric with respect to the metric, it follows that also the space span{e 1 , e 2 } ⊥ is invariant under K e1 . Now we follow precisely the computations of [START_REF] Li | Minimal Lagrangian isotropic immersions in indefinite complex space forms[END_REF]. We get a basis {e 1 , e 2 , u 1 , . . . , u r , ω 1 1 , ω 1 2 , . . . , ω r 1 , ω r 2 }, which satisfies that {u 1 , . . . , u r , ω 1 1 , ω 1 2 , . . . , ω r 1 , ω r 2 } is an orthogonal basis of {e 1 , e 2 } ⊥ and

h(e 1 , e 1 ) = h(e 2 , e 2 ) = 0, h(e 1 , e 2 ) = -4λ 2 , h(u i , u j ) = ε i δ ij , ε i = ±1, h(ω α 1 , ω α 1 ) = 1, h(ω α 2 , ω α 2 ) = -1, (8)
                     K e1 e 1 = e 2 , K e1 e 2 = 0, K e1 u i = λu i , K e1 ω α 1 = -1 2 λω α 1 - √ 3 2 λω α 2 , K e1 ω α 2 = -1 2 λω α 2 + √ 3 2 λω α 1 , K e2 e 2 = -8λ 3 e 1 , K e2 u i = -2λ 2 u i , K e2 ω α 1 = λ 2 ω α 1 - √ 3λ 2 ω α 2 , K e2 ω α 2 = √ 3λ 2 ω α 1 + λ 2 ω α 2 , K e1 K ui u j = δij εi 4λ (2λe 1 -e 2 ), K ω α 1 ω α 1 = L(ω α 1 , ω α 1 ) -1 8λ (2λe 1 -e 2 ), K ω α 2 ω α 2 = L(ω α 1 , ω α 1 ) + 1 8λ (2λe 1 -e 2 ), K ω α 1 ω α 2 = √ 3 8λ (2λe 1 + e 2 ), K ω α k ω β l = L(ω α k , ω β l ), k, l ∈ 1, 2, 1 ≤ α = β ≤ r. (9)
In the above formulas, U and W correspond to the invariant subspaces of K e1 and the operator L is an operator on W × W , defined by ( 10)

L(ω, ω) = K ω ω + 1 4λ 2 h(K ω ω, e 2 )e 1 + 1 4λ 2 h(K ω ω, e 1 )e 2 , ω, ω ∈ W. which is a symmetric operator, satisfies ImL ⊂ U = span{u 1 , . . . , u k } and L(ω α 1 , ω α 1 ) = L(ω α 2 , ω α 2 ), L(ω α 1 , ω α 2 ) = 0, K ω α 1 ω α 2 = √ 3 4 e 1 + 1 2λ e 2 , L(ω α 1 , ω β 1 ) = L(ω α 2 , ω β 2 ), L(ω α 1 , ω β 2 ) = -L(ω α 2 , ω β 1 ). ( 11
)
As in [START_REF] Li | Minimal Lagrangian isotropic immersions in indefinite complex space forms[END_REF] changing the frame by taking ( 12)

f 1 = (2λe 1 -e 2 )/(4µ 3 ), f 2 = (2λe 1 + e 2 )/(4µ 3 ), we get that (13) h(f 1 , f 1 ) = -h(f 2 , f 2 ) = ε 0 , h(f 1 , f 2 ) = 0 and          K f1 f 1 = -µf 1 , K f1 f 2 = µf 2 , K f2 f 2 = -µf 1 , K f1 u i = µu i , K f2 u i = 0, K f1 ω α 1 = -µ 2 ω α 1 , K f1 ω α 2 = -µ 2 ω α 2 , K f2 ω α 1 = √ 3µ 2 ω α 2 , K f2 ω α 1 = - √ 3µ 2 ω α 1 , K ui u j = µε 0 ε i δ ij f 1 , K ω α 1 ω α 2 = √ 3 2 µε 0 f 2 , K ω α 1 ω α 1 = L(ω α 1 , ω α 1 ) -µ 2 ε 0 f 1 , K ω α 2 ω α 2 = L(ω α 1 , ω α 1 ) + µ 2 ε 0 f 1 , K ω α k ω β l = L(ω α k , ω β l ), k, l ∈ 1, 2, 1 ≤ α = β ≤ r. (14)
Therefore, in order to determine the difference tensor explicitly, we only need to determine all the terms

L(ω α k , ω β l ), k, l ∈ {1, 2}, 1 ≤ α, β ≤ r.
In order to do so we will summarize the above properties in a more invariant way.

Let I be the identity map and define for any w ∈ W

(15) T w = 2 √ 3λ K e1 + 1 2 λI w.
We can easily check that T satisfies

T ω α 1 = ω α 2 , T ω α 2 = -ω α 1 , T 2 w = -w, h(T v, w) = h(v, T w), T ω α 1 = ω α 2 , T ω α 2 = -ω α 1 , h(T w, T v) = -h(w, v), h(T v, w) = h(v, T w),
for w, v ∈ W . In addition, from [START_REF] Montiel | Isotropic totally real submanifolds[END_REF] it follows that L(w, T v) = -L(v, T w) and L(T w, T v) = L(v, w). We also have that L satisfies an isotropy condition. Indeed,

let w = r α=1 a α ω β 2 + r β=1 b β ω β 2 .
By using [START_REF] Nomizu | Affine differential geometry[END_REF] in lemma 7 we have

h(K w w, e 1 ) = n α,β=1 a α a β h(K e1 ω α 1 , ω β 1 ) + n α,β=1 b α b β h(K e1 ω α 2 , ω β 2 ) + n α,β=1 a α b β h(K e1 ω α 1 , ω β 2 ) + n α,β=1 b α a β h(K e1 ω α 2 , ω β 1 ) = - λ 2 n α,β=1 (a α a β -b α b β )δ αβ - √ 3λ 2 n α,β=1 (a α b β + b α a β )δ αβ = - λ 2 h(w, w) + √ 3λ 2 h(w, T w). (16)
Similarly, we obtain

(17) h(K w w, e 2 ) = λ 2 h(w, w) + √ 3λ 2 h(w, T w).
By combining ( 10), ( 16) and ( 17) we get

h(L(w, w), L(w, w)) = h(K w w, K w w) + 1 2λ 2 h(K w w, e 1 )h(K w w, e 2 ) = λh(w, w) 2 + 1 2λ 2 - 1 2 λ 3 h(w, w) 2 + 3 2 λ 3 h(w, T w) 2 = 3 4 λ(h(w, w) 2 + h(w, T w) 2 ). (18)
Liniarizing the previous expression for arbitrary vectors W 1 , W 2 , W 3 , W 4 ∈ W , we obtain:

(19) h(L(W 1 , W 2 ), L(W 2 , W 3 ))+h(L(W 1 , W 3 ), L(W 2 , W 4 ))+h(L(W 1 , W 4 ), L(W 2 , W 3 )) = 3 4 (h(W 1 , W 2 )h(W 3 , W 4 ) + h(W 1 , W 3 )h(W 2 , W 4 ) + h(W 1 , W 4 )h(W 2 , W 3 ) +h(W 1 , T W 2 )h(W 3 , T W 4 )+h(W 1 , T W 3 )h(W 2 , T W 4 )+h(W 1 , T W 4 )h(W 2 , T W 3 )).
Note that given a metric of neutral signature on {f 1 , f 2 } ⊥ and operators T and L satisfying the previous conditions, we can define a frame such that ( 14) holds. We start with a vector u ∈ {f 1 , f 2 } ⊥ with length 1. Then T u has length -1. We now write w = au + bT u. The fact that w has length 1 and is orthogonal to T w. Then implies that

(a 2 -b 2 ) + 2ab < u, T u >= 1 (a 2 -b 2 ) < u, T u > -2ab = 0,
which determines a and b uniquely upto sign. It is then sufficient to take w 1 1 = w and w 1 2 = T w and to complete the construction is an inductive way. In what follows we are going to determine the possible dimensions of the studied submanifold M n . In order to do this, we will use a well known result from the theory of composition of quadratic forms, namely the '1,2,4,8 Theorem' proved by Hurwitz in 1898. One can find it for example in [START_REF] Shapiro | Compositions of Quadratic Forms[END_REF]. It states that there exists an n-square identity over the complex numbers of the form

(20) (x 2 1 + . . . + x 2 n )(y 2 1 + . . . + y 2 n ) = z 2 1 + . . . + z 2 n ,
where X = (x 1 , . . . , x n ) and Y = (y 1 , . . . , y n ) are systems of indeterminates and each z k = z k (X, Y ) is a bilinear form in X and Y , if and only if n = 1, 2, 4 or 8.

We are going to see how this result applies in our case and then determine the values of L on the components of the basis in order to determine the difference tensor of our immersion.

In order to apply the 1,2,4,8 Theorem, we are going to find conveniently defined complex vector spaces and an operator which preserves lengths. First, we denote by U C the complex linear extension of V and by W C the complex linear extension of W . We now take

W 1 = {v + iT v|v ∈ W }, W 2 = {w -iT w|w ∈ W }.
Note that these are indeed complex linear vector spaces as i(v±iT v) = ∓(T v∓iv) = (∓T v ±iT (∓T v)) and we complexify the metric and the previously defined operator L. Note that L is symmetric and that from the properties of L and T it follows that the restriction of L to W 1 × W 1 and W 2 × W 2 vanishes identically. Therefore in order to determine L it is sufficient to study L on

L : W 1 × W 2 → U C L(ω, ω) = K ω ω + 1 4λ 2 h(K ω ω, e 2 )e 1 + 1 4λ 2 h(K ω ω, e 1 )e 2 , (21)
where

U C := span{u 1 , . . . , u r } over C.
Proposition 11. The operator L defined in (21) satisfies:

(1) For any vectors x ∈ W 1 and y ∈ W 2 we have

(22) h(L(x, y), L(x, y)) = 3µ 2 4 h(x, x)h(y, y);
(2) Given x 0 in W 1 such that h(x 0 , x 0 ) = 1, we have that L(x 0 , -) preserves norms in the sense that h(L(x 0 , y), L(x 0 , y)) = 3 2 µ 2 h(y, y), ∀y ∈ W 2 ;

(3) Given x 0 a non-null vector, we have that L(x 0 , -) : W 2 → U C is a bijective operator; (4) For any x, x ∈ W 1 , y, z ∈ W 2 we have that

(23) h(L(x, y), L(x , z)) + h(L(x , y), L(x, z)) = 3 2 µ 2 h(x, x )h(y, z).
Proof.

(1) Take

W 1 = W 3 = ω 1 and W 2 = W 4 = ω 2 in relation (19),
where

ω 1 := v + iT v ∈ W 1 and ω 2 := w -iT w ∈ W 2 .
Using the properties of T in [START_REF] Shapiro | Compositions of Quadratic Forms[END_REF] and the fact that ω 1 and ω 2 are orthogonal, we obtain

h(L(ω 1 , ω 2 ), L(ω 1 , ω 2 )) = 3µ 2 4 h(ω 1 , ω 1 )h(ω 2 , ω 2 ).
(2) This property follows directly from the previously proved one.

(3) We linearize in the second argument in property (22), that is y y + z, for y, z ∈ W 2 and we get for arbitrary x ∈ W 1

(24) h(L(x, y), L(x, z)) = 3 4 µ 2 h(x, x)h(y, z).
Fix x = x 0 , for x 0 arbitrarily chosen in W 1 , and write equation (24) once for y = y 1 and once for y = y 2 . Assuming L(x 0 , y 1 ) = L(x 0 , y 2 ), as h is nondegenerate and x 0 is a non-null vector, we get that L(x 0 , -) is injective. This gives dim Im(L(x 0 , -)) = dim W 2 = r, but, as dim U C = r, we obtain that L is also surjective.

(4) The property in (23) follows immediately by liniarizing in (24) for x x + x , ∀x, x ∈ W 1 .

Theorem 12. Let M n k be a λ-isotropic affine hypersurface. Assume that λ = 0. Then either n = 2, 5, 8, 14 or 26.

Proof. We assume that n > 2. We can write out equation ( 22) for the elements of the bases. For more convenience, choose {e i } i={1,...,r} , {f j } j={1,...,r} , {g k } k={1,...,r} bases for U, W 1 , W 2 , respectively, and let u =

r i=1 u i e i , v = r j=1 v j f j . With this choice, relation (22) becomes (25) (u 2 1 + . . . + u 2 r )(v 2 1 + . . . + v r ) = z 2 1 + . . . + z 2 r ,
where 25) yields an r-square quadratic equation. Thus, we may apply now the theorem of Hurwitz and obtain r = 1, 2, 4, 8, which implies that n = 5, 8, 14, 26.

L(e i , f j ) = l k ij g k and z k = n i,j=1 u i v j l k ij . Equation (

Isotropic affine hyperspheres

From now on, we will assume that M is a λ-isotropic affine hypersphere with λ = 0. Proposition 13. Let n ≥ 3 and M n be an n-dimensional affine λ-isotropic hypersphere in R n+1 . Then M n is constant isotropic.

Proof. Let e 1 := f 1 , e 2 := f 2 , e 3 := u 1 , . . . , e r+2 := u r , e r+3 := ω 1 1 , . . . , e 2r+2 := ω r

1 , e 2r+3 := ω r 2 . Then {e 1 , . . . , e n } is an orthogonal basis with h(e i , e j ) = ε i δ ij , ε i = ±1. We denote by Ric the Ricci tensor of M n with respect to the affine metric h. As M n is an affine sphere, we have that the shape operator is a multiple of the identity, say S = εI. Using as well the Gauss equation in (2), the apolarity condition in proposition 5 and the isotropy condition (4) we have

Ric(e j , e k ) = n i=1 ε i h( R(e i , e j )e k , e i ) = h(ε(ε j δ jk e i -ε i δ ik e j ) -[K e i , K e j ]e k , e i ) = nεε j δ jk -εε j δ jk - i ε i h([K ei , K ej ]e k , e i ). (26)
For k = j we obtain

Ric(e j , e k ) = - n i=1 ε i h([K e i , K e j ]e k , e i ) = n i=1 ε i h(K(e i , e j ), K(e i , e k )) = - 1 2 n i=1 ε i h(K(e i , e i ), K(e j , e k )) = 0 (27)
and for k = j

Ric(e j , e j ) = n i=1 εh(K(e i , e j ), K(e i , e j ))

= n i=1 1 2 [-h(K(e j , e j ), K(e i , e i )) + 2λ(p)δ ij + λ(p)ε i ε j ] = n 2 + 1 ε j λ(p). (28)
Since n ≥ 3, by using the fact that the Levi-Civita connection on M n is torsion free and using the second Bianchi identity, we get that λ is constant.

Similarly to [START_REF] Li | Minimal Lagrangian isotropic immersions in indefinite complex space forms[END_REF], Proposition 3.6, we can prove the following:

Proposition 14. Let M n be an n-dimensional affine submanifold in R n+1 . If M n is constant isotropic with λ = 0, then M n has parallel difference tensor. Proof Since M n is constant isotropic, we have λ = h(K(v, v), K(v, v))
and by taking the derivative, we obtain h( ∇χ

K(v, v), K(v, v)) = 0, ∀p ∈ M n , ∀v, χ ∈ T p M n , h(v, v) = 1.
In the isotropy relation (4) we take

X 1 = ∇χ v, X 2 = X 3 = X 4 = v and obtain h(K( ∇χ v, v), K(v, v)) = λh( ∇χ v, v)h(v, v) = 0, for h(v, v) = 1. This implies (29) h(( ∇K)(χ, v, v), h(v, v)) = 0.
for any v, χ ∈ T p M n such that h(v, v) = 1 and in particular, we have

(30) h(( ∇K)(v, v, v), h(v, v)) = 0,
Further on, we take the derivative with respect to some vector U ∈ T p M n in equation ( 4) for X 1 = X 2 = X 3 = v, X 4 = w and for h(v, w) = 0 and obtain

h(( ∇K)(v, v, v), K(v, w)) -h(( ∇K)(v, v, w), K(v, v)) = 0.
As ∇K is totally symmetric, using also (29) we have

(31) h(( ∇K)(v, v, v, ), K(v, w)) = 0, for any v, χ ∈ T p M n such that h(v, v) = 1. We can write K(v, K(v, v)) = av + bw, for v ∈ T p M n , w an (n -1)-dimensional tangent vector , h(v, w) = 0. Since h(K(v, K(v, v)), v) = h(K(v, v), K(v, v)) = λ, h(K(v, K(v, v)), w) = bh(w, w) = 0, we get a = λ, b = 0 so that K(v, K(v, v)) = λv. If we take w = K(v, v) in equation (31) we get (32) λh(( ∇K)(v, v, v), v) = 0.
As λ = 0, using (32) and the symmetry of ∇K, we also have ∇K = 0.

Proposition 15. Let n ≥ 3 and M n be an n-dimensional λ-isotropic affine hypersphere in R n+1 , such that S = εI, with ε constant. Assume that λ = 0. If R n+1 is endowed with an indefinite metric and M n is not totally geodesic, then M n is a locally symmetric space and λ = -1 2 ε.

Proof. From the previous propositions we conclude that ∇K = 0. Hence, by the Gauss equation we have ∇R = 0, which means that M n is a locally symmetric space. Using the Ricci identity, from ∇K = 0 we also have R.h = 0, that is

(33) R(X, Y )K(Z, W ) -K( R(X, Y )Z, W ) -K(Z, R(X, Y )W ) = 0,
for X, Y, Z, W tangent vector fields. If we take

X = Z = W = f 1 , Y = f 2 , it implies (34) R(f 1 , f 2 )K(f 1 , f 1 ) = 2K( R(f 1 , f 2 )f 1 , f 1 )
and then, from Gauss equation we have

R(f 1 , f 2 )f 1 = -(ε + 2λ)f 2 ,
which together with (34) implies ε + 2λ = 0.

Proposition 16. Let n ≥ 3, f 1 : M n 1 → R n and f 2 : M n 1 → R n be n-dimensional λ-isotropic affine hypersphere in R n+1 , such that S 1 = S 2 = εI, with ε = ±1 constant. Let p 1 ∈ M 1
and p 2 ∈ M 2 and assume that there exists an isometry

A : T p1 M 1 → T p2 M 2 such that AK 1 (v, w) = K 2 (Av, Aw),
i.e. A preserves the difference tensor. Then there exists a local isometry F :

(M 1 , h 1 ) → (M 2 , h 2 ) such that dF (K 1 (X, Y )) = K 2 (dF (X), dF (Y )),
for any vector fields X, Y on M 1 . Moreover the immersions f 1 and f 2 •F are locally congruent.

Proof. From the previous propositions we know that λ is a constant, and that with respect to the Levi Civita connection M 1 and M 2 are locally symmetric spaces whose difference tensor is parallel with respect to the Levi Civita connection.

We take p 1 ∈ M 1 and we take a basis {e 1 , . . . , e n } of T p1 M 1 . As A is an isometry we take as basis of T p2 M 2 the vectors {Ae 1 , . . . , Ae n }. By the initial conditions we have that h 1 (e i , e j ) = h 2 (Ae i Ae j ) (isometry)

h 1 (K 1 (e i , e j ), e k ) = h 2 (AK 1 (e i , e j ), Ae k ) = h 2 (K 2 (Ae i , Ae j ), Ae k ).
We now extend {e 1 , . . . , e n } to a local differential basis {X 1 , . . . , X n } by parallel translation along geodesics with respect to the Levi Civita connection of the affine metric. In the same way we extend {Ae 1 , . . . , Ae n } to local vector fields {Y 1 , . . . , Y n }. As the difference tensors are parallel, we have that the components of the difference tensor stay constant along geodesics. Therefore by construction, we have that

h 1 (X i , X j ) = h 2 (Y i , Y j ), h 1 (K 1 (X i , X j ), X k ) = h 2 (K 2 (Y i , Y j ), Y k )
Hence by the lemma of Cartan, see [START_REF] Carmo | Riemannian geometry[END_REF] we know that there exists a local isometry F such that dF (X i ) = Y i . In order to complete the proof it is now sufficent to apply Theorem 9.

So in order to complete the classification it is now sufficient to determine, up to isometries, the possible forms of the difference tensor and for each of those forms obtained to determine an explicit example of an affine hypersphere with isotropic difference tensors. This is done explicitly for the 4 remaining dimensions 5, 8, 14 and 26 in the next sections.

Affine hyperspheres of dimension 5

5.1. The form of L, dim U = 1. We start with ω = v + iT v ∈ W 1 , a vector of length 2. As the length of ω is 2 it follows that v has unit length and is orthogonal to T v. So we can take ω 1 1 = v and ω 1 2 = T v. Note that by the properties of L we have that L(v + iT v, v -iT v) is a real vector in U C whose square length is 3µ 2 . Hence we can pick a unit vector u 1 in U such that

L(v + iT v, v -iT v) = √ 3µu 1 .
By the properties of L this implies that

L(ω 1 1 , ω 1 1 ) = √ 3 2 µu 1 .
From the properties of T we see

L(ω 1 1 , ω 1 2 ) = 0 and L(ω 1 1 , ω 1 1 ) = L(ω 1 2 , ω 1
2 ), hence L is completely determined. Therefore L and also K are completely determined and the signature of the metric, if necessary after replacing ξ by -ξ in order to make λ > 0, equals 2.

A canonical example.

We consider R 6 = s(3) as the set of all symmetric 3 × 3 matrices and we take as hypersurface M those symmetric matrices with determinant 1. We define an action σ of SL(3, R) on M as follows

σ : SL(3, R)× → M, such that (g, p) → σ g (p) = gpg T .
Note that M has two connected components and that the action is transitive on each of the connected components. The connected component of I has been studied in [START_REF] Birembaux | Isotropic Affine Spheres[END_REF] where it was shown that it gives a positive definite isotropic affine hypersurface. It also appears in [START_REF] Hu | Locally strongly convex affine hypersurfaces with parallel cubic form[END_REF]. Here we are interested in the component of the matrix

A =   1 0 0 0 -1 0 0 0 -1   , which we denote by M 1 . So M 1 = {gAg T |g ∈ SL(3, R)}.
The isotropy group of A consists of the matrices g of determinant 1 such that gAg T = A. This Lie group is congruent to SO(2, 1) and therefore, by Theorem 9.2 of [START_REF] Boothby | An Introduction to Differentiable Manifolds and Riemannian Geometry[END_REF], we know that M 1 is locally isometric with SL(3,R) SO(2,1) . Note that, of course, every element of SL(3, R) acts at the same time also on s(3) and that this action belongs to SL(6, R), see [START_REF] Nomizu | Affine differential geometry[END_REF]. This implies that M 1 is at the same time an homogeneous affine hypersurface and by Proposition 6 an equiaffine sphere centered at the origin.

In order to determine the tangent space at p = gAg T , we look at the curves in M 1 γ(s) = ge sX Ae sX T g T .

These are indeed curves in M 1 , provided that e sX ∈ SL(3, R) or, equivalently, provided that Tr X = 0. Note that γ (s) = ge sX (XA + AX T )e sX T g T , where v = (XA + AX T ) is a symmetric matrix. So by using a dimension argument we see that the tangent space is given by

{gvg T |v = 2XA, XA = AX T , Tr X = 0, X ∈ R 3×3 } = T p M 1 .
Working now at the point A, taking g = I and X ∈ so(2, 1) = {X ∈ R 3×3 | Tr X = 0, XA = AX T } we see that

∇ γ (s) γ (s) + h(γ , γ )γ = γ (s) = e sX (4X 2 A)e sX T = e sX ((4X 2 -4
3 Tr(X 2 )I)A)e sX T + 4 3 Tr(X 2 )e sX Ae sX T = e sX ((4X 2 -4 3 Tr(X 2 )I)A)e sX T + 4 3 Tr(X 2 )γ(s). As the matrix (4X 2 - 4 3 Tr(X 2 )I) commutes with A, we can decompose the above expression into a tangent part and a part in the direction of the affine normal given by the position vector, and therefore we find that h(γ (s), γ (s)) = 4 3 Tr(X 2 ). So we see that s is a constant length parametrisation of the curve γ and therefore we have that h(γ , ∇γ γ ) = 0 and

h(γ , ∇ γ γ ) = h(γ , K(γ , γ ))
In addition, we have

γ (s) = ∇ γ (s) ∇ γ (s) γ (s) + h(γ , γ )γ + h(γ , K(γ , γ )γ = e sX (8X 3 A)e sX T = e sX ((8X 3 -8
3 Tr(X 3 )I)A)e sX T + 8 3 Tr(X 3 )γ(s). So therefore working at s = 0 and writing v = 2XA as tangent vector, we obtain that h(v, v) = 4 3 Tr(X 2 ), h(v, K(v, v)) = 8 3 Tr X 3 . Linearising the above expressions, i.e. writing v = α

1 v 1 + α 2 v 2 , respectively v = α 1 v 1 + α 2 v 2 + α 3 v 3 , for v = 2X i A, i = 1, 2, 3
, and looking at the coefficient of α 1 α 2 , respectively α 1 α 2 α 3 , we obtain that

h(v 1 , v 2 ) = 4 3 Tr(X 1 X 2 ) = 4 3 Tr(X 2 X 1 ) 6h(K(v 1 , v 2 ), v 3 ) = 8 3 (Tr X 1 X 2 X 3 + Tr X 3 X 1 X 2 + Tr X 2 X 3 X 1 + Tr X 1 X 3 X 2 + Tr X 3 X 2 X 1 + Tr X 2 X 1 X 3 ) = 8(Tr X 1 X 2 X 3 + Tr X 2 X 1 X 3 ).
So we see that

K(v 1 , v 2 ) = 2(X 1 X 2 + X 2 X 1 -2 3 Tr(X 1 X 2 )I)A. Indeed, we have that (X 1 X 2 +X 2 X 1 -2
3 Tr(X 1 X 2 )I) has vanishing trace, commutes with A and therefore K(v 1 , v 2 ) is indeed the unique tangent vector such that

h(K(v 1 , v 2 ), v 3 ) = 4 3 (Tr(X 1 X 2 X 3 ) + Tr(X 2 X 1 X 3 )).
As by Cayley Hamilton, for a matrix X with vanishing trace, we have that X 3 = 1/2 Tr(X 2 )X + det(X)I, we deduce that Tr X 4 = 1 2 (Tr X 2 ) 2 , and therefore we have that

h(K(v, v), K(v, v)) = 4 3 Tr(2X 2 -2 3 Tr X 2 I) 2 = 4 3 (4 Tr X 4 + 4 9 (T rX 2 ) 2 T rI -8 3 (Tr X 2 ) 2 ) = 8 9 (T rX 2 ) 2 = 1 2 (h(v, v)) 2 .
Hence M 1 is isotropic with positive λ. A straightforward computation also shows that the index of the metric is 2. Combining therefore the results in this section with Proposition 16 and the classification result of O. Birembaux and M. Djoric, see [START_REF] Birembaux | Isotropic Affine Spheres[END_REF] in the positive definite case, we get the following theorem:

Theorem 17. Let M be a 5-dimensional λ-isotropic affine hypersphere, with λ = 0. Therefore, if necessary, by changing the sign of the affine normal, we may assume that λ is positive. Then either

A =   1 0 0 0 -1 0 0 0 -1   , of R 6 ≡ s(3) ⊂ R 3×3 .
6. Affine hyperspheres of dimension 8

6.1. The form of L, dim U = 2. Let W = span{ω 1 1 , ω 1 2 , ω 2 1 , ω 2 2 } and W 1 = span{ω 1 1 +iω 1 2 , ω 2 1 +iω 2 2 }, W 2 = span{ω 1 1 - iω 1 2 , ω 2 1 -iω 2 2 }.
Remark that all the bases are orthogonal and in addition h(ω

1 1 + iω 1 2 , ω 1 1 + iω 1 2 ) = h(ω 2 1 + iω 2 2 , ω 2 1 + iω 2 2 ) = 2, h(ω 1 1 -iω 1 2 , ω 1 1 -iω 1 2 ) = h(ω 2 1 -iω 2 2 , ω 2 1 -iω 2 2 ) = -2. Then, straightforward computations lead to        L(ω 1 1 + iω 1 2 , ω 1 1 -iω 1 2 ) = 2L(ω 1 1 , ω 1 1 ), L(ω 1 1 + iω 1 2 , ω 2 1 -iω 2 2 ) = 2L(ω 1 1 , ω 2 1 ) -2iL(ω 1 1 , ω 2 2 ), L(ω 2 1 + iω 2 2 , ω 1 1 -iω 1 2 ) = 2L(ω 1 1 , ω 2 1 ) + 2iL(ω 1 1 , ω 2 2 ), L(ω 2 1 + iω 2 2 , ω 2 1 -iω 2 2 ) = 2L(ω 2 1 , ω 2 1 ).
(35)

Notice that the vector L(ω

1 1 + iω 1 2 , ω 1 1 -iω 1
2 ) is a real vector of length 3µ 2 . So we can pick u 1 ∈ U, h(u 1 , u 1 ) = 1 such that (36)

L(ω 1 1 + iω 1 2 , ω 1 1 -iω 1 2 ) = √ 3µu 1 .
With this choice, from property (24) we obtain that L(ω

1 1 + iω 1 2 , ω 2 1 -iω 2 2
) is orthogonal to u 1 . Moreover as its length is a real number, we must have that Re(L(ω

1 1 + iω 1 2 , ω 2 1 -iω 2 2 ) and Im(L(ω 1 1 + iω 1 2 , ω 2 1 -iω 2
2 ) are orthogonal to each other. As they are also both orthogonal to u 1 , one of them has to vanish. Therefore, we get two cases:

Case II-1. Re(L(ω 1 1 + iω 1 2 , ω 2 1 -iω 2 2 )) = 0 Now we obtain that L(ω 1 1 , ω 2 1 ) = 0 and L(ω 1 1 + iω 1 2 , ω 2 1 -iω 2
2 ) is an imaginary vector of length 3µ 2 , orthogonal to u 1 . Thus, we can pick u 2 ∈ U in the direction of

L(ω 1 1 + iω 1 2 , ω 2 1 -iω 2 2 ) such that h(u 2 , u 2 ) = -1 and such that (37) L(ω 1 1 + iω 1 2 , ω 2 1 -iω 2 2 ) = i √ 3µu 2 . Consider now L(ω 2 1 + iω 2 2 , ω 2 1 -iω 2 2
). It is a real vector orthogonal to u 2 , of length 3µ 2 and thus we can write L(ω

2 1 + iω 2 2 , ω 2 1 -iω 2 2 ) = ± √ 3µu 1 .
Furthermore, from (35), (37) and Proposition 11-(4), we obtain

3µ 2 = -h(L(ω 1 1 + iω 1 2 , ω 2 1 -iω 2 2 ), L(ω 2 1 + iω 2 2 , ω 1 1 -iω 1 2 )) = h(L(ω 1 1 + iω 1 2 , ω 1 1 -iω 1 2 ), L(ω 2 1 + iω 2 2 , ω 2 1 -iω 2 2 )) = √ 3µh(u 1 , L(ω 2 1 + iω 2 2 , ω 2 1 -iω 2 2 )). So we get that L(ω 2 1 + iω 2 2 , ω 2 1 -iω 2 2 ) = √ 3µu 1 .
In this case, the signature of the metric is 4. Case II-2. Im(L(ω

1 1 + iω 1 2 , ω 2 1 -iω 2
2 )) = 0 Reasoning in a similar way, we choose u 2 ∈ U a real vector in the direction of

L(ω 1 1 + iω 1 2 , ω 2 1 -iω 2 2 ), with h(u 2 , u 2 ) = 1 such that L(ω 1 1 + iω 1 2 , ω 2 1 -iω 2 2 ) = √ 3µu 2 . We find L(ω 2 1 + iω 2 2 , ω 2 1 -iω 2 2 ) = - √ 3µu 1
and in this case the signature of the metric is 3.

Two canonical examples.

First we consider R 9 as the set of Hermitian symmetric matrics Y ∈ C 3×3 . We take as hypersurface M those Hermitian symmetric matrices with determinant 1. We define an action σ of SL(3, C) on M as follows

σ : SL(3, C) × M → M, such that (g, p) → σ g (p) = gpḡ T .
Note that M has two connected components and that the action is transitive on each of the connected components. The connected component of I has been studied in [START_REF] Birembaux | Isotropic Affine Spheres[END_REF], where it was shown that it gives a positive definite isotropic affine hypersurface. It also appears in [START_REF] Hu | Locally strongly convex affine hypersurfaces with parallel cubic form[END_REF]. Here we are interested in the component of the matrix

A =   1 0 0 0 -1 0 0 0 -1   , which we denote by M 1 . So M 1 = {gAḡ T |g ∈ SL(3, R)}.
The isotropy group consists of the matrices g of determinant 1 such that gAḡ T = A. This Lie group is congruent to SU (2, 1) and therefore, by Theorem 9.2 of [START_REF] Boothby | An Introduction to Differentiable Manifolds and Riemannian Geometry[END_REF], we know that M 1 is locally isometric with SL (3,C) SU [START_REF] Birembaux | Isotropic Affine Spheres[END_REF][START_REF] Baez | The octonions[END_REF] . Note that of course every element of SL(3, C) acts at the same time also on R 9 in a linear way and that, therefore, this action belongs to GL(9, R). A straightforward computation shows that this action actually belongs to SL(9, R). This implies that M 1 is at the same time an homogeneous affine hypersurface and, by Proposition 6, an equiaffine sphere centered at the origin. So, in order to determine the properties of M 1 , it is sufficient to look at a single point.

In order to determine the tangent space at the point p = gAḡ T , we look at the curves in M 1 γ(s) = ge sX Ae s XT ḡT .

These are indeed curves in M 1 provided that e sX ∈ SL(3, C) or equivalently provided that Tr X = 0. Note that γ (s) = ge sX (XA + A XT )e s XT ḡT , where v = (XA + A XT ) is a Hermitian symmetric matrix. So by using a dimension argument we see that the tangent space is given by

{gvḡ T |v = 2XA, XA = A XT , Tr X = 0, X ∈ C 3×3 } = T p M 1 .
Working now at the point A, taking g = I and

X ∈ su(2, 1) = {X ∈ C 3×3 | Tr X = 0, XA = A XT } we see that ∇ γ (s) γ (s) + h(γ , γ )γ = γ (s) = e sX (4X 2 A)e sX T = e sX ((4X 2 -4 3 Tr(X 2 )I)A)e s XT + 4 3 Tr(X 2 )e sX Ae s XT = e sX ((4X 2 -4
3 Tr(X 2 )I)A)e s XT + 4 3 Tr(X 2 )γ(s). As the matrix (4X 2 -4 3 Tr(X 2 )I) commutes with A, we can decompose the above expression into a tangent part and a part in the direction of the affine normal given by the position vector, and therefore we find that h(γ (s), γ (s)) = 4 3 Tr(X 2 ).

So we see that s is a constant length parametrisation of the curve γ and therefore we have that h(γ , ∇γ γ ) = 0 and

h(γ , ∇ γ γ ) = h(γ , K(γ , γ )).
As

γ (s) = ∇ γ (s) ∇ γ (s) γ (s) + h(γ , γ )γ + h(γ , K(γ , γ )γ = e sX (8X 3 A)e s XT
= e sX ((8X 3 -8 3 Tr(X 3 )I)A)e s XT + 8 3 Tr(X 3 )γ(s), working at s = 0 and writing v = 2XA as tangent vector, we have that

h(v, v) = 4 3 Tr(X 2 ), h(v, K(v, v)) = 8
3 Tr X 3 . Linearising the above expressions, i.e.

writing v = α 1 v 1 + α 2 v 2 , respectively v = α 1 v 1 + α 2 v 2 + α 3 v 3 , for v = 2X i A, i = 1, 2, 3
, and looking at the coefficient of α 1 α 2 , respectively α 1 α 2 α 3 , we obtain that

h(v 1 , v 2 ) = 4 3 Tr(X 1 X 2 ) = 4 3 Tr(X 2 X 1 ), 6h(K(v 1 , v 2 ), v 3 ) = 8 3 (Tr X 1 X 2 X 3 + Tr X 3 X 1 X 2 + Tr X 2 X 3 X 1 + Tr X 1 X 3 X 2 + Tr X 3 X 2 X 1 + Tr X 2 X 1 X 3 ) = 8(Tr X 1 X 2 X 3 + Tr X 2 X 1 X 3 ).
So we see that

K(v 1 , v 2 ) = 2(X 1 X 2 + X 2 X 1 -2 3 Tr(X 1 X 2 )I)A. Indeed, we have that (X 1 X 2 + X 2 X 1 -2
3 Tr(XY )I) has vanishing trace, commutes with A and therefore K(v 1 , v 2 ) is indeed the unique tangent vector such that

h(K(v 1 , v 2 ), v 3 ) = 4 3 (Tr(X 1 X 2 X 3 ) + Tr(X 2 X 1 X 3 )).
As by Cayley Hamilton for a matrix X with vanishing trace we have that X 3 = 1/2 Tr(X 2 )X + det(X)I, we deduce that Tr X 4 = 1 2 Tr X 2 , and therefore, we have that

h(K(v, v), K(v, v)) = 4 3 Tr(2X 2 -2 3 Tr X 2 I) 2 = 4 3 (4 Tr X 4 + 4 9 (T rX 2 ) 2 T rI -8 3 (Tr X 2 ) 2 ) = 8 9 (T rX 2 ) 2 = 1 2 (h(v, v)) 2 .
Hence M 1 is isotropic with positive λ. A straightforward computation also shows that the index of the metric is 4.

Next, we consider R 9 = R 3×3 . We take as hypersurface M 2 those matrices with determinant 1. We define an action σ of SL(3, R) on M 2 as follows

σ : SL(3, R) × M 2 → M 2 , such that (g, p) → σ g (p) = gp.
The isotropy group of the identity matrix consists only of the identity matrix. Therefore, by Theorem 9.2 of [START_REF] Boothby | An Introduction to Differentiable Manifolds and Riemannian Geometry[END_REF] we know that M 2 is locally isometric with SL(3, R).

Note that, of course, every element of SL(3, R) acts at the same time also on R 9 in a linear way and that therefore this action belongs to GL(9, R). A straightforward computation shows that this action actually belongs to SL(9, R). This implies that M 2 is at the same time an homogeneous affine hypersurface and, by Proposition 6, an equiaffine sphere centered at the origin. So in order to determine the properties of M 2 it is sufficient to look at a single point.

In order to determine the tangent space at a point p, we look at the curves in M 2 γ(s) = e sX p. These are indeed curves in M 2 , provided that e sX ∈ SL(3, R) or equivalently, provided that Tr X = 0. Note that γ (s) = e sX Xp, so by using a dimension argument we see that the tangent space is given by

{Xp| Tr X = 0, X ∈ R 3×3 } = T p M 2 .
Working now at the point I and

X ∈ sl(3, R) = {X ∈ R 3×3 | Tr X = 0}, we see that ∇ γ (s) γ (s) + h(γ , γ )γ = γ (s) = e sX X 2 = e sX (X 2 -1 3 Tr(X 2 )I) + 1 3 Tr(X 2 )e sX = e sX (X 2 -1 3 Tr(X 2 )I) + 1 3 Tr(X 2 )γ(s). As the matrix (X 2 -1
3 Tr(X 2 )I) commutes with e sX and has vanishing trace, we can interprete e sX (X 2 -1 3 Tr(X 2 )I) as a tangent vector at the point e sX . By decomposing the above expression into a tangent part and a part in the direction of the affine normal given by the position vector, we deduce that h(γ (s), γ (s)) = 1 3 Tr(X 2 ). So we see that s is a constant length parametrisation of the curve γ and therefore we have that h(γ , ∇γ γ ) = 0 and

h(γ , ∇ γ γ ) = h(γ , K(γ , γ )). As γ (s) = ∇ γ (s) ∇ γ (s) γ (s) + h(γ , γ )γ + h(γ , K(γ , γ )γ = e sX X 3
= e sX (X 3 -1 3 Tr(X 3 )I)e sX T + 1 3 Tr(X 3 )γ(s), working at s = 0 and writing v = X as tangent vector, we have that

h(v, v) = 1 3 Tr(X 2 ), h(v, K(v, v)) = 1 3 Tr X 3 . Linearising the above expressions, i.e. writing v = α 1 v 1 + α 2 v 2 , respectively v = α 1 v 1 + α 2 v 2 + α 3 v 3 , for v i = X i , i = 1, 2, 3
, and looking at the coefficient of α 1 α 2 , respectively α 1 α 2 α 3 we obtain that

h(v 1 , v 2 ) = 1 3 Tr(X 1 X 2 ) = 1 3 Tr(X 2 X 1 ), 6h(K(v 1 , v 2 ), v 3 ) = 1 3 (Tr X 1 X 2 X 3 + Tr X 3 X 1 X 2 + Tr X 2 X 3 X 1 + Tr X 1 X 3 X 2 + Tr X 3 X 2 X 1 + Tr X 2 X 1 X 3 ) = (Tr X 1 X 2 X 3 + Tr X 2 X 1 X 3 ).
So we see that

K(v 1 , v 2 ) = 1 2 (X 1 X 2 + X 2 X 1 -2 3 Tr(X 1 X 2 )I). Indeed, we have that (X 1 X 2 + X 2 X 1 -2
3 Tr(X 1 , X 2 )I) has vanishing trace and therefore K(v 1 , v 2 ) is indeed the unique tangent vector such that

h(K(v 1 , v 2 ), v 3 ) = 1 6 (Tr(X 1 X 2 X 3 ) + Tr(X 2 X 1 X 3 )).
As by Cayley Hamilton for a matrix X with vanishing trace we have that X 3 = 1/2 Tr(X 2 )X + det(X)I, we deduce that

Tr X 4 = 1 2 (Tr X 2 ) 2 ,
and therefore we have that

h(K(v, v), K(v, v)) = 1 3 Tr(X 2 -1 3 Tr X 2 I) 2 = 1 3 (Tr X 4 + 1 9 (T rX 2 ) 2 T rI -2 3 (Tr X 2 ) 2 ) = 1 18 (T rX 2 ) 2 = 1 2 (h(v, v)) 2 .
Hence M 1 is isotropic with positive λ. A straightforward computation also shows that the index of the metric is 3.

Combining therefore the results in this section with Proposition 16 and the classification result of O. Birembaux and M. Djoric, see [START_REF] Birembaux | Isotropic Affine Spheres[END_REF] in the positive definite case, we get the following theorem: Theorem 18. Let M be a 8-dimensional λ-isotropic affine hypersphere, with λ = 0. Therefore if necessary by changing the sign of the affine normal we may assume that λ is positive. Then either We start with w 1 ∈ W a vector with length 1. As L(w 1 + iT w 1 , w 1 -iT w 1 ) is a real vector in U with length 3µ 2 there exists a real unit length vector u 1 in U such that L(w 1 + iT w 1 , w 1 -iT w 1 ) = √ 3µu 1 .

We now complete u 1 to a basis of U by choosing orthogonal u 2 , u 3 , u 4 in {u 1 } ⊥ such that h(u k , u k ) = ε k , where ε k = ±1. We also introduce δ k , for k=2,3,4, by

δ k = i, if ε k = -1 and δ k = 1, if ε k = 1.
Now we apply Proposition 11, which tells us that we can find vectors w 2 , w 3 , w 4 such that

L(w 1 + iT w 1 , w 2 -iT w 2 ) = √ 3µδ 2 u 2 , L(w 1 + iT w 1 , w 3 -iT w 3 ) = √ 3µδ 3 u 3 , L(w 1 + iT w 1 , w 4 -iT w 4 ) = √ 3µδ 4 u 4 .
The first two properties of Proposition 11 then tells us that {w 1 , T w 1 , . . . , w 4 , T w 4 } is a basis of W , as in Lemma 7. Of course the previous equations also imply that

L(w k + iT w k , w 1 -iT w 1 ) = √ 3µ δk u k .
We now look at L(w 2 + iT w 2 , w 3 -iT w 3 ). From the last part of Proposition 11 it follows that this vector is orthogonal to L(w 2 + iT w 2 , w 1 -iT w 1 ), L(w 1 + iT w 1 , w 3 -iT w 3 ) and L(w 1 + iT w 1 , w 1 -iT w 1 ). So this implies that there exists a complex number b 4 such that

L(w 2 + iT w 2 , w 3 -iT w 3 ) = b 4 u 4 .
Similarly, we have that

L(w 2 + iT w 2 , w 4 -iT w 4 ) = b 3 u 3 , L(w 3 + iT w 3 , w 4 -iT w 4 ) = b 2 u 2 .
Using again Proposition 11 we see that there exists real numbers c k such that

h(L(w k + iT w k , w k -iT w k ), L(w k + iT w k , w k -iT w k )) = c k u k . From (38) h(L(w k + iT w k , w k -iT w k ), L(w 1 + iT w 1 , w 1 -iT w 1 )) = -h(L(w k + iT w k , w 1 -iT w 1 ), L(w 1 + iT w 1 , w k -iT w k )), it follows that c k = - √ 3µε k .
Next we use the fact that for different indices k and we have that

(39) h(L(w k + iT w k , w k -iT w k ), L(w + iT w , w -iT w )) = -h(L(w k + iT w k , w -iT w ), L(w + iT w , w k -iT w k )).
Expressing this for the different possibilities for k and we find that

3µ 2 ε 2 ε 3 = -|b 4 | 2 ε 4 3µ 2 ε 2 ε 4 = -|b 3 | 2 ε 3 3µ 2 ε 4 ε 3 = -|b 2 | 2 ε 4 .
Hence, up to permuting the vectors, we see that there are two possibilities. Either ε 2 = ε 3 = ε 4 = -1, in which case the index of the metric is 8 or ε 2 = -1 and ε 3 = ε 4 = 1, in which case the index of the metric is 6.

Computing the length of L(w 2 + iT w 2 , w 3 -iT w 3 ) we have in both cases that b 2 4 ε 4 = 3µ 2 . So if necessary by changing the sign of u 4 and w 4 , we may assume that b 4 = √ 3µ. We now complete the argument by looking at h(L(w 2 +iT w 2 , w 3 -iT w 3 ), L(w 1 +iT w 1 , w 4 -iT w 4 ) = -h(L(w 1 +iT w 1 , w 3 -iT w 3 ), L(w 2 +iT w 2 , w 4 -iT w 4 ) This yields that b 3 = -√ 3µ. Interchanging the indices 2 and 3 in the formula above finally gives that b 2 = -√ 3µ in the first case, and -√ 3µi in the second case.

Two canonical examples.

First we look at the following example. We identify R 15 with the set of all skew symmetric matrices in R 6×6 . So an element p ∈ R 15 is of the form

p =         0 a 1 a 2 a 3 a 4 a 5 -a 1 0 a 6 a 7 a 8
a 9 -a 2 -a 6 0 a 10 a 11 a 12 -a 3 -a 7 -a 10 0 a 13 a 14 -a 4 -a 8 -a 11 -a 13 0 a 15 -a 5 -a 9 -a 12 -a 14 -a 15 0

        .
We take as hypersurface M in R 15 the skew symmetric matrices with determinant 1. Let G = SL(6, R). Then, we have an action ρ of G on M by ρ(g)(p) = gpg T Here we are interested in the connected component of the matrix

I 0 =         0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0        
.

If necessary, we restrict now M to the orbit of I 0 . Its isotropy group consists of the matrices g of determinant 1 such that gI 0 g T = I 0 . This Lie group is congruent to Sp(6) and therefore by Theorem 9.2 of [START_REF] Boothby | An Introduction to Differentiable Manifolds and Riemannian Geometry[END_REF] we know that M 1 is locally isometric with SL(6,R) Sp [START_REF] Dillen | Hypersurfaces with parallel difference tensor[END_REF] . Note that of course every element of SL(6, R) acts at the same time also on R 15 in a linear way and that therefore this action belongs to GL(15, R). A straightforward computation shows that this action actually belongs to SL(15, R). This implies that M is at the same time an homogeneous affine hypersurface and by Proposition 6 an equiaffine sphere centered at the origin. So in order to determine the properties of M it is sufficient to look at a single point.

In order to determine the tangent space at a point p = gI 0 g T , we look at the curves in M γ(s) = ge sX I 0 e sX T g T .

These are indeed curves in M 1 , provided that e sX ∈ SL(6, R) or equivalently, provided that Tr X = 0. Note that γ (s) = ge sX (XI 0 + I 0 X T )e sX T g T , where v = (XI 0 + I 0 X T ) is a symmetric matrix. So by using a dimension argument we see that the tangent space is given by

{gvg T |v = 2XI 0 , XI 0 = I 0 X T , Tr X = 0, X ∈ R 6×6 } = T p M.
In fact, such a matrix X is of the form

X =         a 1 a 2 a 3 a 4 a 5 a 6 b 1 b 2 b 3 b 4 0 -a 5 c 1 c 2 -a 1 -b 2 0 -b 4 -a 4 d 1 d 2 0 -a 1 -b 2 b 3 a 3 e 1 0 -d 2 c 2 b 2 a 2 0 -e 1 -d 1 c 1 b 1 a 1        
.

Working now at the point I 0 , taking g = I and X ∈ {X ∈ R 6×6 | Tr X = 0, XI 0 = I 0 X T } we see that

∇ γ (s) γ (s) + h(γ , γ )γ = γ (s) = e sX (4X 2 I 0 )e sX T = e sX ((4X 2 -4 6 Tr(X 2 )I)I 0 )e sX T + 4
6 Tr(X 2 )e sX I 0 e sX T = e sX ((4X 2 -4 6 Tr(X 2 )I)I 0 )e sX T + 4 6 Tr(X 2 )γ(s). As the matrix (4X 2 -4 6 Tr(X 2 )I) commutes with I 0 , we can decompose the above expression into a tangent and a part in the direction of the affine normal given by the position vector, and therefore we find that h(γ (s), γ (s)) = 4 6 Tr(X 2 ). So we see that s is a constant length parametrisation of the curve γ and therefore we have that h(γ , ∇γ γ ) = 0 and

h(γ , ∇ γ γ ) = h(γ , K(γ , γ )) As γ (s) = ∇ γ (s) ∇ γ (s) γ (s) + h(γ , γ )γ + h(γ , K(γ , γ )γ = e sX (8X 3 I 0 )e sX T = e sX ((8X 3 -8 6
Tr(X 3 )I)I 0 )e sX T + 8 6 Tr(X 3 )γ(s), working at s = 0 and writing v = 2XI 0 as tangent vector, we have that

h(v, v) = 4 6 Tr(X 2 ), h(v, K(v, v)) = 8
6 Tr X 3 . Linearising the above expressions, i.e. writing

v = α 1 v 1 + α 2 v 2 , respectively v = α 1 v 1 + α 2 v 2 + α 3 v 3 , for v i = 2XI 0 , i = 1, 2, 3
, and looking at the coefficient of α 1 α 2 , respectively α 1 α 2 α 3 we obtain that

h(v 1 , v 2 ) = 4 6 Tr(X 1 X 2 ) = 4 6 Tr(X 2 X 1 ), 6h(K(v 1 , v 2 ), v 3 ) = 8 3 (Tr X 1 X 2 X 3 + Tr X 3 X 1 X 2 + Tr X 2 X 3 X 1 + Tr X 1 X 3 X 2 + Tr X 3 X 2 X 1 + Tr X 2 X 1 X 3 ) = 4(Tr X 1 X 2 X 3 + Tr X 2 X 1 X 3 ). So we see that K(v 1 , v 2 ) = 2(X 1 X 2 + X 2 X 1 -2 6 Tr(X 1 X 2 )I)I 0 . Indeed we have that (X 1 X 2 + X 2 X 1 -2 6
Tr(X 1 X 2 )I) has vanishing trace, commutes with I 0 and therefore K(v 1 , v 2 ) is indeed the unique tangent vector such that

h(K(v 1 , v 2 ), v 3 ) = 2 3 (Tr(X 1 X 2 X 3 ) + Tr(X 2 X 1 X 3 ))
. By straightforward computations we deduce that Tr X 4 = 1 4 (Tr X 2 ) 2 , and therefore we have that

h(K(v, v), K(v, v)) = 4 6 Tr(2X 2 -2 3 Tr X 2 I) 2 = 4 6 (4 Tr X 4 + 4 9 (T rX 2 ) 2 T rI -8 3 (Tr X 2 ) 2 ) = 2 9 (T rX 2 ) 2 = 1 2 (h(v, v)) 2 .
Hence M 1 is isotropic with positive λ. A straightforward computation also shows that the index of the metric is 4. Next, the following example ilustrates the case when the signature of the indefinite metric on M 14 is 8. First we identify R 15 with the set of matrices a We take as hypersurface M in R 15 all such matrices with determinant 1. Let G = SU * (6). Then, we have an action ρ of G on M by ρ(g)(p) = gpḡ T . Note that M has two connected components and that the action is transitive on each of the connected components. The connected component of I has been studied in [START_REF] Birembaux | Isotropic Affine Spheres[END_REF],

= { E F -F Ē , E = ĒT , F = -F T } ⊂ C 6×6 . An element in a is of the form p :=         a 1 a 2 + ia 3 a 4 +
where it was shown that it gives a positive definite isotropic affine hypersurface.

Here we are interested in the connected component M 1 containing the matrix

I 0 =        
1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1

        .
Its isotropy group consists of the matrices g of determinant 1 such that gI 0 ḡT = I 0 . This Lie group is congruent to Sp(1, 2) and therefore, by Theorem 9.2 of [START_REF] Boothby | An Introduction to Differentiable Manifolds and Riemannian Geometry[END_REF], we know that M is locally isometric with SU * (6)

Sp [START_REF] Baez | The octonions[END_REF][START_REF] Birembaux | Isotropic Affine Spheres[END_REF] . Note that of course every element of SU * (6) acts at the same time also on R 15 in a linear way and that therefore this action belongs to GL(15, R). A straightforward computation shows that this action actually belongs to SL(15, R). This implies that M is at the same time an homogeneous affine hypersurface and, by Proposition 6, an equiaffine sphere centered at the origin. So in order to determine the properties of M 1 , it is sufficient to look at a single point.

In order to determine the tangent space at a point p = gI 0 ḡT , we look at the curves in M γ(s) = ge sX I 0 e s XT ḡT .

These are indeed curves in M 1 , provided that Tr X = 0 and XJ = J X, for J = 0 I n -I n 0 . Note that γ (s) = ge sX (XI 0 +I 0 XT )e s XT ḡT . So by using a dimension argument, we see that the tangent space is given by

{gvḡ T |v = 2XI 0 , XI 0 = I 0 XT , Tr X = 0, XJ = J X, X ∈ C 6×6 } = T p M 1 .
In fact, such an X if of the form

        -x -x 0 x 1 + iy 1 x 2 + iy 2 0 x 3 -iy 3 x 4 -iy 4 x 1 -iy 1 x x 5 + iy 5 -x 3 + iy 3 0 -x 6 + iy 6 -x 2 + iy 2 -x 5 + iy 5 x 0 x 4 -iy 4 x 6 -iy 6 0 0 -x 3 -iy 3 -x 4 -iy 4 -x -x 0 x 1 -iy 1 -x 2 -iy 2 x 3 + iy 3 0 -x 6 -iy 6 x 1 + iy 1 x -x 5 + iy 5 x 4 + iy 4 x 6 + iy 6 0 x 2 + iy 2 x 5 + iy 5 x 0        
.

Working now at the point I 0 , taking g = I and X ∈ C 6×6 satisfying XI 0 = I 0 XT , Tr X = 0, XJ = J X, we see that ∇ γ (s) γ (s) + h(γ , γ )γ = γ (s) = e sX (4X 2 I 0 )e s XT = e sX ((4X 2 -4 6 Tr(X 2 )I)I 0 )e s XT + 4 6 Tr(X 2 )e sX I 0 e s XT = e sX ((4X 2 -4 6 Tr(X 2 )I)I 0 )e s XT + 4 6 Tr(X 2 )γ(s). As the matrix (4X 2 -4 6 Tr(X 2 )I) has the same properties as X, we can decompose the above expression into a tangent part and a part in the direction of the affine normal given by the position vector, and therefore we find that h(γ (s), γ (s)) = 4 6 Tr(X 2 ). So we see that s is a constant length parametrisation of the curve γ and therefore we have that h(γ , ∇γ γ ) = 0 and h(γ , ∇ γ γ ) = h(γ , K(γ , γ )). As γ (s) = ∇ γ (s) ∇ γ (s) γ (s) + h(γ , γ )γ + h(γ , K(γ , γ )γ = e sX (8X 3 I 0 )e s XT = e sX ((8X 3 -8 6 Tr(X 3 )I)I 0 )e s XT + 8 6 Tr(X 3 )γ(s), working at s = 0 and writing v = 2XI 0 as tangent vector, we have that h(v, v) = 4 6 Tr(X 2 ), h(v, K(v, v)) = 8 6 Tr X 3 . Linearising the above expressions, i.e. writing v = α 1 v 1 + α 2 v 2 , respectively v = α 1 v 1 + α 2 v 2 + α 3 v 3 , for v i = 2XI 0 , i = 1, 2, 3, and looking at the coefficient of α 1 α 2 , respectively α 1 α 2 α 3 we obtain that

h(v 1 , v 2 ) = 4 6 Tr(X 1 X 2 ) = 4 6 Tr(X 2 X 1 ), 6h(K(v 1 , v 2 ), v 3 ) = 8 3 (Tr X 1 X 2 X 3 + Tr X 3 X 1 X 2 + Tr X 2 X 3 X 1 + Tr X 1 X 3 X 2 + Tr X 3 X 2 X 1 + Tr X 2 X 1 X 3 ) = 4(Tr X 1 X 2 X 3 + Tr X 2 X 1 X 3 ).
So we see that

K(v 1 , v 2 ) = 2(X 1 X 2 + X 2 X 1 -2 6
Tr(X 1 X 2 )I)I 0 . Indeed we have that (X 1 X 2 + X 2 X 1 -2 6 Tr(X 1 X 2 )I) has vanishing trace, commutes with I 0 and therefore K(v 1 , v 2 ) is indeed the unique tangent vector such that h(K(v 1 , v 2 ), v 3 ) = 2 3 (Tr(X 1 X 2 X 3 ) + Tr(X 2 X 1 X 3 )). By straightforward computations we deduce that Tr X 4 = 1 4 (Tr X 2 ) 2 , and therefore we have that h(K(v, v), K(v, v)) = 4 6 Tr(2X 2 -2 3 Tr X 2 I) 2 = 4 6 (4 Tr X 4 + 4 9 (T rX 2 ) 2 T rI -8 3 (Tr X 2 ) 2 ) = 2 9 (T rX 2 ) 2 = 1 2 (h(v, v)) 2 . Hence M 1 is isotropic with positive λ. A straightforward computation also shows that the index of the metric is 8.

Affine hyperspheres of dimension 26

8.1. The form of L, dim U = 8. Before treating each case of the signature for the metric, we first will give some lemmas which will be very useful in order to simplify the proof significantly. We start with an arbitrary vector w + iT w ∈ W 1 with length 2 and define a real vector √ 3µu = L(w + iT w, w -iT w). We call w 1 1 = w and w 1 2 = T w. Next, we choose arbitrary orthogonal vectors u 2 , . . . , u 8 such that u 1 , u 2 , . . . , u 8 forms an orthonormal (real) basis in U, that is h(u j , u k ) = ε j δ jk , where ε j = ±1 indicate the length of u j . As the operator L(ω 1 1 + iω 1 2 , -) is bijective, for every u j we find ω 1 j , ω 2 j , such that (40) L(ω 1 1 + iω 1 2 , ω j 1 -iω j 2 ) = √ 3µδ j u j , where δ j = 1, if ε = 1 i, if ε = -1.

Lemma 19. For the previously defined vectors, L satisfies 2 . Thus, the signature of the metric on M is 16. When the indefinite signature on U is 3, we have the following example. Consider the set of Hermitian matrices with entries in the split-octonions space endowed with the Jordan multiplication •, as previously defined. For {1, i, j, k, li, lj, lk} an orthogonal basis of the split-octonion space, the length of a vector x = x 0 + x 1 i + x 2 j + x 3 k + x 4 l + x 5 li + x 6 lj + x 7 lk is given by h(x, x) = xx = (x 2 0 + x 2 1 + x 2 2 + x 2 3 ) -(x 2 4 + x 2 5 + x 2 6 + x 2 7 ). We define the manifold in a similar way as in the previous example and, by similar arguments, we get that M is an isotropic affine hypersphere of dimension 26 for which the signature of the metric is 12.

L(ω k 1 + iω k 2 , ω k 1 -iω k 2 ) = - √ 3µε k u 1 .

  with symmetric 3 × 3 -matrices, then M is congruent with the connected component of A =

( 1 )

 1 the metric is positive definite, M is isometric with SL(3,R) SO(3) and M is affine congruent to an open part of the hypersurface {gg T |g ∈ SL(3, R)} of R 6 ≡ s(3) ⊂ R 3×3 , or (2) the metric has signature 2, M is isometric with SL(3,R) SO(2,1) and M is affine congruent to an open part of the hypersurface {gAg T |g ∈ SL(3, R)}, where

( 1 ) 1 . 7 . 14 7. 1 .

 117141 the metric is positive definite, M is isometric with SL(3,C) SU (3) and M is affine congruent to an open part of the hypersurface {gḡ T |g ∈ SL(3, C)} of R 9 , identified with the space of Hermitian symmetric matrices, or (2) the metric has signature 4, M is isometric with SL(3,C) SU (2,1) and M is affine congruent to an open part of the hypersurface {gA|g T |g ∈ SL(3, C)} of R 9 , , identified with the space of Hermitian symmetric matrices, or (3) the metric has signature 3, M is isometric with SL(3, R) and M is affine congruent with SL(3, R) considered as a hypersurface in R 9 identified with R 3×3 Affine hyperspheres of dimension The form of L, dim U C = 4.

G 7 i=0y 7 i=0y 7 i=0y 1 

 7771 = {N ∈ h(O) 3 |det(N ) = 1}. We define an action of G on M 1 = { N AN T |N ∈ G} by ρ : G × M 1 -→ M 1 ρ(N )X = N XN T , By construction, this action is transitive and therefore M 1 is congruent with G/H, where H = {N ∈ G|A N AN T = I}. Note that ρ(N ) can be seen as a linear transformation acting on R 27 and a straightforward computation shows that ρ(N ) ∈ SL(27, R). Therefore, M 1 is an homogeneous affine hypersphere in R 27 . It is now sufficient to work around a point. We introduce local coordinates around a point p ∈ M 1 by taking y 1 , • • • , y 26 such that ξ1 = 1,ξ 2 = y 1 , ξ 3 = y 2 , x 1 = 3+i e i , x 2 = 11+i e i , x 3 = 19+i e i ,for {e 0 , • • • , e 7 } a basis of O. Therefore, the parametrization for our hypersurface is given byF : R 26 → R 27 p -→ g(p) -13 (1, p), where p = (y 1 , • • • , y 26 ) and g(p) := det N. By using the multiplication table for octonions, we can determine g(p) and then, straightforward computations around the point N = allow us to find that the isotropy condition holds for λ = 1

  ia 5 0 a 6 + ia 7 a 8 + ia 9 a 2 -ia 3 a 10 a 11 + ia 12 -a 6 -ia 7 0 a 13 + ia 14 a 4 -ia 5 a 11 + ia 12

						
	0 a 6 -ia 7	a 15 -a 6 + ia 7 -a 8 + ia 9 0 -a 13 + ia 14 a 2 + ia 3 -a 8 -ia 9 -a 13 -ia 14 a 1 a 2 -ia 3 a 10	0 a 4 -ia 5 a 11 -ia 12	      	.
	a 8 -ia 9 a 13 -ia 14	0	a 4 + ia 5	a 11 + ia 12	a 15

Proof. The result is straightforward, by properties (23) and (24):

(41)

Lemma 20. Let u j and u k determine ε j and ε k such that ε j = ε k , for k, j > 1.

Then L(ω k 1 + iω k 2 , ω j 1 -iω j 2 ) is an imaginary vector. Proof. Let us define the orthonormal basis of U given by    u * k = cos(t)u k + sin(t)u j , u * j = -sin(t)u k + cos(t)u k , u * l = u l , l = k, j. By relation (40), we compute

and therefore we find ω * k 1 +iω * k 2 = cos(t)(ω k 1 +iω k 2 )+sin(t)(ω j 1 +iω j 2 ) and ω * j 1 +iω * j 2 = -sin(t)(ω k 1 + iω k 2 ) + cos(t)(ω j 1 + iω j 2 ) such that L(ω 1 1

Next, by lemma (19) we may write

and using the bilinearity of L, we get the conclusion.

Lemma 21. Let u j and u k determine ε j and ε k such that ε j = -1 and

2 ) is a real vector. Proof. First, define an orthonormal basis of U given by

) to be the unique vectors satisfying

Finally, using the bilinearity of L, the conclusion follows easily from

In what follows, we will study the remaining cases for the metric on U. First we deal with the case that the signature of the metric is 4, 5 or 6. Let u 1 be defined as in the beginning of this section. Next, choose u 2 ⊥ u 1 such that h(u 2 , u 2 ) = -1 and w 2 1 and w 2 2 such that L(ω

Then, we look at the vector L(ω 2 1 + iω 2 2 , ω 3 1 -iω 3 2 ) and see, by property (23), that it is orthogonal to u 1 , u 2 and u 3 and has length 3µ 2 and by lemma (20), that it is an imaginary vector. Therefore, we define u 4 of length -1 such that

Next, by surjectivity of L(ω

2 ) are real, of positive length, mutually orthogonal and orthogonal to u 1 , u 5 . Therefore, the choice of u 1 , . . . , u 5 implies that the metric on {u 2 , u 3 , u 4 } ⊥ is positive definite. Therefore, the cases when the metric has signature 4, 5 or 6 cannot happen.

In case that the index is 0, we proceed as follows. Let u 1 be defined as before, choose u 2 ⊥ u 1 of length 1 and obtain the existence of ω 2 1 , ω 2 2 such that L(ω

2 ) is an imaginary vector, orthogonal on u 1 , u 2 , u 3 (by relation ( 23)) and therefore, we get the existence of a unit vector of negative length, u 4 , such that

This contradicts the fact that the index equals 0. Next, we start anew, with different choices of vectors in order to eliminate the case when the signature of the metric is 1. Let u 1 be defined as before, choose u 2 ⊥ u 1 of length -1 and obtain the existence of

2 ) is a real vector, orthogonal on u 1 , u 2 , u 3 (by relation ( 23)) and therefore, we get the existence of a unit vector of positive length, u 4 , such that L(ω

Finally, by lemma (20) and the property in (23), we see that the vectors

2 ) are orthogonal imaginary vectors. This implies that the index of the metric is at least 2. Now, we will prove that the metric on U cannot have signature 2. Let u 1 be defined as in (??), choose

2 ) is an imaginary vector, orthogonal on u 1 , u 2 , u 3 (by relation ( 23)). So we have that

, where u 4 has negative length and belongs to {u 1 , u 2 , u 3 } ⊥ , where the metric is positive definite, which is a contradiction.

Next we deal with the case that the index of the metric equals 7. So on {u 1 } ⊥ the metric is negative definite. We may take u 2 ∈ U such that h(u 2 , u 2 ) = -1 and h(u 1 , u 2 ) = 0.

As L(ω 1 1 + iω 1 2 , -) is a surjective operator, we can pick w 1 1 and w 1 2 = T w 1 1 such that

and by the lemma we have

Next, we take u 3 ∈ U such that h(u 3 , u 3 ) = -1. In a similar way as before, we define ω 3 1 and ω 3 2 and obtain

By the lemma this implies that L(ω 3 1

2 ) is an imaginary vector which is orthogonal to u 1 , u 2 and u 3 such that we may write L(ω

2 ). We immediately obtain that it is an imaginary vector of length 3µ 2 which is orthogonal to u 1 , u 2 and u 4 . As

Remember that so far we have defined u 1 , u 2 , u 3 and u 4 ∈ U and ω

We take now some arbitrary u 5 ∈ {u 1 , u 2 , u 3 , u 4 } ⊥ such that h(u 5 , u 5 ) = -1 and use again the surjectivity of L(ω 1 1 + iω 1 2 , -) to define w 5 1 and

(45)

Next, we proceed with the computations as we did, for instance, for L(ω

2 ) and define u 6 , u 7 , u 8 ∈ U such that

Given u 6 , u 7 , u 8 , we use the surjectivity of L(ω 1 1 + iω 1 2 , -) and just like previously done, we define ω k 1 , ω k 2 ∈ U, for k = 6, 7, 8 and determine

Next, we find

. Then, we compute similarly as for L(ω

2 ) in order to determine ( 46)

As for the vectors

2 ), by using property (24) and the determined vectors so far, we see they are in the directions of u 8 , u 7 and u 8 respectively. We can easily determine their components by following the same procedure as for L(ω

). Thus, we may write

where ε, ε 1 , ε 2 = ± √ 3µi. Further on, in order to determine L(ω 3 1 + iω 3 2 , ω 7 1 -iω 7 2 ), we first see by property (24) that it is orthogonal to {u 1 , u 2 , u 3 , u 4 , u 7 , u 8 }. Next, as

2 )) = 0 we find

2 ) is coliniar with u 6 . From (23) we obtain

Using similar methods we consecutively obtain that

Note that by applying (??) on

2 )), we see that = -2 . Using similar arguments, we proceed to find that

Moreover it now immediately follows that 2 = 1.

At last, we will study the solution given by the case when the metric on U has signature 3. Start with u 1 defined as in (40), choose u 2 ⊥ u 1 of length -1 and by surjectivity of L(ω

Then, by lemma (20) we can see that the vector L(ω

Moreover, we find the unique vectors ω 4 1 and ω 4 2 such that L(ω

)) and, as the metric on {u 2 , u 3 , u 4 } ⊥ is positive definite, we find

Next, we choose u 5 ⊥ u 1 , u 2 , u 3 , u 4 of length 1 and find ω 5 1 , ω 5 2 such that L(ω

Then, we notice by property (23) that L(ω 2 1 + iω 2 2 , ω 5 1iω 5 2 ), L(ω 3 1 + iω 3 2 , ω 5 1 -iω 5 2 ) and L(ω 4 1 + iω 4 2 , ω 5 1 -iω 5 2 ) are real vectors and satisfy the orthogonality conditions which allow us to pick u 6 , u 7 , u 8 of length 1, in their directions respectively, and complete {u 1 , u 2 , u 3 , u 4 } to an orthonormal basis, that is

Notice that, by lemmas (20) and property (40) we obtain

In the following, we determine

, as it is a real vector of length 3µ 2 , orthogonal on u 1 , u 2 , u 3 , u 4 , u 6 , and given that its component in the direction of u 5 is -√ 3µ ( by property ( 23)). Furthermore, we find

2 ), for k = 2, . . . , 6 and ε 1 = ±1. Similarly, we determine for ε j = ±1, j = 2, . . . , 8 the following vectors Then, we can easily find the relations between the coefficients ε j using property (23): ε 2 = -ε 1 , ε 4 = -ε 3 , ε 6 = -ε 5 and ε 7 = -1, ε 8 = -i. Moreover, we can find ε 1 = -1, ε 3 = 1 and ε 5 = -1 by applying property (23) successively to L(ω When the indefinite signature on U is 7, we have the following example. Let h 3 (O) denote the set of Hermitian matrices with entries in O, the space of octonions endowed with the Jordan multiplication •:

By definition, we have that the determinant of N ∈ h

For more details for the space of octonions see [START_REF] Baez | The octonions[END_REF]. Next, we define