Virtual billiards in pseudo-Euclidean spaces: discrete Hamiltonian and contact integrability

The aim of the paper is to unify the efforts in the study of integrable billiards within quadrics in flat and curved spaces and to explore further the interplay of symplectic and contact integrability. As a starting point in this direction, we consider virtual billiard dynamics within quadrics in pseudo--Euclidean spaces. In contrast to the usual billiards, the incoming velocity and the velocity after the billiard reflection can be at opposite sides of the tangent plane at the reflection point. In the symmetric case we prove noncommutative integrability of the system and give a geometrical interpretation of integrals, an analog of the classical Chasles and Poncelet theorems and we show that the virtual billiard dynamics provides a natural framework in the study of billiards within quadrics in projective spaces, in particular of billiards within ellipsoids on the sphere $\mathbb S^{n-1}$ and the Lobachevsky space $\mathbb H^{n-1}$.


Introduction
It is well known that the billiards within ellipsoids are the only known integrable billiards with smooth boundary in constant curvature spaces [1,7,5,6,11,27,34,35,38]. The elliptical billiards in pseudo-Euclidean spaces are also integrable [25,12]. We will try to present all these integrable models through a unified perspective, within the framework of the virtual billiard dynamic (see [23]).
A pseudo-Euclidean space E k,l of signature (k, l), k, l ∈ N, k + l = n, is the space R n endowed with the scalar product x i y i (x, y ∈ R n ).
A point x ∈ Q n−1 is singular, if a normal EA −1 x at x ∈ Q n−1 is light-like: (EA −2 x, x) = 0, or equivalently, the induced metric is degenerate at x.
In the case that A is positive definite, following Khesin and Tabachnikov [25] and Dragović and Radnović [12], we define a billiard flow inside the ellipsoid (1) in E k,l as follows. Between the impacts, the motion is uniform along the straight lines. If x ∈ Q n−1 is non-singular, then the normal EA −1 x is transverse to T x Q n−1 and the incoming velocity vector w can be decomposed as w = t + n, where t is its tangential and n the normal component in x. The velocity vector after reflection is w 1 = t − n. If x ∈ Q n−1 is singular, the flow stops.
Let φ : (x j , y j ) → (x j+1 , y j+1 ) be the billiard mapping, where x j ∈ Q n−1 is a sequence of non-singular impact points and y j is the corresponding sequence of outgoing velocities (in the notation we follow [38,36,16], which slightly differs from the one given in [30], where y j is the incoming velocity). As in the Euclidean case (see [36,30,16]), the billiard mapping φ is given by: where the multipliers are determined from the conditions (A −1 x j+1 , x j+1 ) = (A −1 x j , x j ) = 1, y j+1 , y j+1 = y j , y j .
From the definition, the Hamiltonian H = 1 2 y j , y j is an invariant of the mapping φ. Therefore, the lines l k = {x k + sy k | s ∈ R} containing segments x k x k+1 of a given billiard trajectory are of the same type: they are all either space-like (H > 0), time-like (H < 0) or light-like (H = 0). Also, the function J j = (A −1 x j , y j ) is an invariant of the billiard mapping (see Lemma 3.1 in [23]).
Note that the billiard mapping (3), (4) is well defined for arbitrary quadric Q n−1 given by (1) and not only for ellipsoids. In that case, the outgoing velocity (directed from x k to x k+1 ) is either y k or −y k , while the segments x k−1 x k and x k x k+1 determined by 3 successive points of the mapping (3), (4) may be: (i) on the same side of the tangent plane T x k Q n−1 ; (ii) on the opposite sides of the tangent plane T x k Q n−1 . Figure 1. A segment of a virtual billiard trajectory within hyperbola (a 1 > 0, a 2 < 0) in the Euclidean space E 2,0 . The caustic is an ellipse.
In the case (i) we have a part of the usual pseudo-Euclidean billiard trajectory, while in the case (ii) the billiard reflection corresponds to the points x k−1 x k x ′ k−1 , where x ′ k+1 is the symmetric image of x k+1 with respect to x k . In the threedimensional Euclidean case, Darboux referred to such reflection as the virtual reflection (e.g., see [9] and [11], Ch. 5). In Euclidean spaces of arbitrary dimension, such configurations were introduced by Dragović and Radnović in [9]. It appears that a multidimensional variant of Darboux's 4-periodic virtual trajectory with reflections on two quadrics, refereed as double-reflection configuration [11], is fundamental in the construction of the double reflection nets in Euclidean spaces (see [13]) and in pseudo-Euclidean spaces (see [14]). They also played a role in a construction of the billiard algebra in [10]. The 4-periodic orbits of real and complex planar billiards with virtual reflections are also studied in [18]. Definition 1.1. [23] Let Q n−1 be a quadric in the pseudo-Euclidean space E k,l defined by (1). We refer to (3), (4) as the virtual billiard mapping, and to the sequence of points x k determined by (3), (4) as the virtual billiard trajectory within Q n−1 .
The system is defined outside the singular set and it is invariant under the action of a discrete group Z n 2 generated by the reflections We can interpret (3), (4) in the case of non-light-like billiard trajectories as the equations of a discrete dynamical system (see [36,30,38]) on Q n−1 described by the discrete action functional: where x = (x k ), k ∈ Z is a sequence of points on Q n−1 . Note that the virtual billiard dynamics on Q n−1 can have both virtual and real reflections. Motivated by the Lax reprezentation for elliptical billiards with the Hooke's potential (Fedorov [16], see also [20,32]), we proved in [23] that the trajectories (x j , y j ) of (3), (4) outside the singular set (5) satisfy the matrix equation where q λ is given by For a non-symmetric case (τ i a i = τ j a j ) the matrix representation is equivalent to the system up to the Z n 2 -action (6). Further, from the expression one can derive the integrals f i in the form Outline and results of the paper. In Section 2 we describe discrete symplectic (Theorem 2.1) and contact integrability in the light-like case (Theorem 2.2) of the virtual billiard dynamics directly, by the use of the Dirac-Poisson bracket. This is slightly different from the construction within the framework of the symplectic reduction given by Khesin and Tabachnikov [25,26].
In the symmetric case, when a i τ i = a j τ j for some indexes i, j, we further develop the analysis from [23] of geodesic flows on Q n−1 and elliptical billiards. We prove noncommutative integrability of the system (Theorem 3.2, Section 3) and, by a subtle estimate of the number of real zeros in the spectral parameter λ of the rational function det L x,y (λ), give a geometrical interpretation of integrals -an analog of the classical Chasles and Poncelet theorems for symmetric quadrics (Theorems 4.2 -4.6, Section 4). The Poncelet theorem is based on a noncommutative variant of the description of Liouville integrable symplectic correspondences given by Veselov [38,39] (Theorem 3.1, Section 3).
Further, in Section 5 we show that the virtual billiard dynamics provides a natural framework in the study of billiards within quadrics in projective spaces, in particular the billiards within ellipsoids on the sphere S n−1 and the Lobachevsky space H n−1 . It is well known that the ellipsoidal billiards on S n−1 and H n−1 are completely integrable [7,37,34,8]. The "big" n × n-matrix representation of the ellipsoidal H n−1 -billiard, together with the integration of the flow is obtained in [37]. In this paper we provide a "small" 2 × 2-matrix representation (Theorem 5.2), a modification of (7), as well as the Chasles theorem (Theorem 5.4).
2. Symplectic and contact properties of the virtual billiard dynamics 2.1. Hamiltonian description. In the pseudo-Euclidean case it is convenient to use the following symplectic form on R 2n = T E k,l (x, y) (see [25]): obtained after identification T * E k,l (x, p) ∼ = T E k,l (x, y) using the scalar product ·, · . The corresponding Poisson bracket is where Σ is given by (5) and S n−1 h = {y ∈ R n | y, y = h} is a pseudosphere (h = 0) or a light-like cone (h = 0).
Due to {φ 1 , φ 2 } = 4(A −1 x, y) = 0 on M h , it follows that M h is a symplectic submanifold of (R 2n , ω). Recall, for F 1 , F 2 ∈ C ∞ (M h ), the Hamiltonian vector field X Fi is defined by i XF i ω M h = −dF i , while the Poisson bracket is given by Alternatively, we can define the Poisson bracket in redundant variables by the use of Dirac's construction (e.g., see [29,33]). Let The bracket is characterized by (ii) Assume that the quadric is not symmetric. The integrals (11) commute with respect to the Poisson bracket {·, ·} M h . The virtual billiard map is a completely integrable discrete system on the phase space M h , which is almost everywhere foliated on (n − 1)-dimensional Lagrangian invariant manifolds.
Proof. (i) Although it is straightforward, we feel that it would be interesting to present a direct proof of the statement. For our convenience we denote x k , y k , µ k ,ν k , x k+1 , y k+1 by x, y, µ, ν,x,ỹ, respectively. As earlier mentioned, (15) (A −1x ,ỹ) = (A −1 x, y).
Notice also that Indeed, due toỹ + y ∈ TxQ n−1 , we have According to (14) it suffices to prove that The proofs of the first and the third relation in (17) are tedious and we will omit them here. Assuming that {x i ,x j } M h = 0, we will prove only the second relation. At the beginning let's show that (18) {x First, owing to {y i , y j } M h = 0 it is Consequently, from (14), (15), (16), we have Now, using (18) and (16) we obtain Therefore, (ii) Note that the only relation between the integrals on M h is Similarly as in the Euclidean space, we have {f i , f j } = 0 (see [25,26]). Further (18), and {y i , y j } M h = 0 imply that the mapping (x, y) → (x, y) is also symplectic on M h .
Remark 2. Note that in the virtual billiard mapping (3), (4) we allow the trajectories both with J > 0 and J < 0 (J = (A −1 x, y) = 0 defines the tangent space T x Q n−1 ). For example, in the ellipsoidal case when A is positive definite, J > 0 means that y is directed outward Q n−1 . It is also natural to consider the dynamics of lines described by Khesin and Tabachnikov within the framework of the symplectic reduction for A being positive definite [25]. In our notation, in the space-like and time-like cases, the dynamics of lines corresponds to the virtual billiard dynamics on M h / ± 1 with identified y and −y, while in the light-like case it corresponds to the induced dynamics onM = M 0 /R * , where we take the projectivization of the light-like cone S n−1 0 . The latter case will be studied in details below.

Contact description.
In the light-like case h = 0 we show the existence of a contact structure associated to M 0 . Let us introduce an action of The action is evidently free and proper, from which we conclude that the orbit spacē ) is a surjective submersion. With the notation above, (M 0 , ω M0 ) is a symplectic Liouville manifold: The associated Liouville vector field and the Liouville 1-form are given by respectively. Then dβ = ω M0 and g * λβ = λβ (e.g, see [24]). It is well known that the orbit spaceM carries the natural contact structure induced byβ (Proposition 10.3, Ch. V, [24]). We describe this contact structure below. Let Theorem 2.2. (i) There exists a unique 1-formβ onM , such that β = π * β . Furthermore, the formβ is contact andR := π * X J is the Reeb vector field on (M ,β), where X J is the Hamiltonian vector field of the function (iii) Assume that the quadric is not symmetric. The functions f i /J 2 descend to the commutative integralsf i , of the contact mappingφ, where [·, ·] is the Jacobi bracket on (M ,β). Further,f i are preserved by the Reeb vector fieldR of (M ,β) and the contact mappingφ is contact completely integrable: the manifoldM is almost everywhere foliated on (n − 1)-dimensional pre-Legendrian invariant manifolds.
whereR is the Reeb vector field on (M ,β),β(R) = 1, iRdβ = 0, and is the contact Hamiltonian vector field off i . Here,H i are the horizontal vector fields,β(H i ) = 0, satisfying for all tangent vectorsX onM .
In addition, having in mind that each tangent vectorX onM has the form X = π * X for some vector field X on M 0 , we have which together with (27) yields (26). In the end, thanks to (24), (26) we have which together with (19) imply that among the integralsf i we have two relations, f 1 + · · · +f n = 0, τ 1 a −1 1f 1 + · · · + τ n a −1 nf n = 1, and that the number of the independent ones is n − 2. According to the theorem on contact integrability, their invariant level-sets almost everywhere define (n − 1)dimensional pre-Legendrian manifolds, which have an additional (n−2)-dimensional Legendrian foliation (see [26,19]).

Noncommutative integrability and symmetric quadrics
3.1. Discrete noncommutative integrability. Recall that a Hamiltonian flow on a 2n-dimensional symplectic manifold (M 2n , ω) (respectively, a contact flow on a 2n + 1-dimensional contact manifold (M 2n+1 , β)) is noncommutatively integrable, if it has a complete set of integrals F . The set F closed under the Poisson bracket (respectively, the Jacobi bracket) is complete, if one can find 2n − r almost everywhere independent integrals F 1 , F 2 , . . . , F 2n−r ∈ F , such that F 1 , . . . , F r Poisson commute with all integrals [31,28] (respectively, F 1 , . . . , F r commute with respect to the Jacobi bracket with all integrals, and the functions in F are integrals of the Reeb flow, as well [19]).
Regular compact connected invariant manifolds of the system are r-dimensional isotropic tori generated by the Hamiltonian flows of F 1 , . . . , F r , i.e., r+1-dimensional pre-isotropic tori generated by the Reeb vector field and the contact Hamiltonian flows of F 1 , . . . , F r . Here, a submanifold N ⊂ M 2n+1 is pre-isotropic, if it transversal to the contact distribution H = ker β and if G x = T x N ∩ H x is an isotropic subspace of the symplectic linear space (H x , dβ), for all x ∈ N . The last condition is equivalent to the condition that distribution G = x G x defines a foliation [19].
In a neighborhood of a regular torus there exist canonical generalized actionangle coordinates [31] (generalized contact action-angle coordinates [19]), such that integrals F i , i = 1, . . . , r depend only on the actions and the flow is a translation in the angle coordinates. If r = n we have the usual Liouville integrability described in the Arnold-Liouville theorem [2], i.e., contact integrability described in [4,26].
If instead of the continuous flow we consider the symplectic mapping Φ : M 2n → M 2n , Φ * ω = ω (the contact mapping Φ : M 2n+1 → M 2n+1 , Φ * β = β) having the complete set of integrals F , as above, compact connected components of an invariant regular level set are r-dimensional isotropic tori (r + 1-dimensional pre-isotropic tori) and in their neighborhoods there exist canonical generalized (contact) action-angle coordinates. By the same argumentation as given by Veselov [38,39] for the Liouville integrable symplectic correspondences, we have the following description of the dynamics. (29).
Then we have the following commutative diagrams for some vectors a i k ∈ R r(+1) . In particular, if a point [x] ∈ T i k is periodic with a period mq, then all points of T i1 ∪ T i2 ∪ · · · ∪ T iq are periodic with the same period.

Symmetric quadrics.
We turn back to the virtual billiard dynamics and consider the case when the quadric Q n−1 is symmetric. Define the sets of indices I s ⊂ {1, . . . , n} (s = 1, . . . r) by the conditions (30) 1 • τ i a i = τ j a j = α s for i, j ∈ I s and for all s ∈ {1, . . . , r}, Let E k,l = E k1,l1 ⊕ · · · ⊕ E kr ,lr be the associated decomposition of E k,l , where E ks,ls are pseudo-Euclidean subspaces of the signature (k s , l s ) with By ·, · s we denote the restriction of the scalar product to the subspace E ks,ls : 1 (31) x, x s = i∈Is τ i x 2 i , x ∈ E k,l . 1 To simplify the notation, we omitted the projection operator πs : E k,l → E ks,ls at the left hand side of (31).
Let SO(k s , l s ) be the special orthogonal group of E ks,ls . The quadric, as well as the virtual billiard flow, is SO(k 1 , l 1 ) × · · · × SO(k r , l r )-invariant. The integrals (32) Φ s,ij := y i x j − x i y j , i, j ∈ I s are proportional to the components of the corresponding momentum mapping On the other hand, the determinant det L x,y (λ) is an invariant of the flow, and by expanding it in terms of 1/(λ − α s ), 1/(λ − α s ) 2 , we get where the integrals F s , P s are given by: The Hamiltonian is equal to the sum H = 1 2 r s=1 F s , that is, among integrals F s we have the relation s F s = 2h on M h .
For h = 0, byF s ,P s ,Φ s,ij we denote the functions onM obtained from R *invariant integrals F s /J 2 , P s /J 2 , Φ s,ij /J.
Among central functionsF s ,P s there are (N −2)-independent ones and their contact Hamiltonian vector fields, together with the Reeb vector fieldR, generate N − 1dimensional pseudo-isotropic manifolds -regular levels sets of the integralsF .
The first statement is an analog of Theorems 5.1, 5.2 for the the Jacobi-Rosochatius problem [20] and Theorem 4.1 for geodesic flows on quadrics in pseudo-Euclidean spaces [23], where the Dirac construction is applied for the constraints The second statement follows from the same considerations as in the proof of Theorem 2.2. For example, similarly as in (24), we havē The last equality follows from the commuting relations {J, among the integralsF s ,P s onM .

Remark
3. An example of noncommutatively integrable multi-valued symplectic correspondence is a recently constructed discrete Neumann system on a Stiefel variety [17]. Another example of a discrete integrable contact system is the Heisenberg model in pseudo-Euclidean spaces [21]. We shall discus relationship between the Heisenberg model and virtual billiard dynamics in a forthcoming paper.

The Chasles and Poncelet theorems for symmetric quadrics
4.1. Pseudo-confocal quadrics. There is a nice geometric manifestation of integrability of elliptical billiards in pseudo-Euclidean spaces given by Khesin and Tabachnikov [25]. Consider the following "pseudo-confocal" family of quadrics in E k,l For a nonsymmetric ellipsoid, the lines l k , k ∈ Z determined by a generic spacelike or time-like (respectively light-like) billiard trajectory are tangent to n − 1 (respectively n − 2) fixed quadrics from the pseudo-confocal family (34) (pseudo-Euclidean version of the Chasles theorem, see Theorem 4.9 in [25] and Theorem 5.1 in [12]). A related geometric structure of the set of singular points for the pencil (34) is described in [12,14].
Here we consider the case of symmetric quadrics and further develop the analysis given in [23], where A had been positive definite.
Without loss of generality we assume in the section that The equation (34) has r solutions in the complex plane for a generic x. The following lemma estimates the number of real solutions in certain cases.
(ii) The quadrics passing through arbitrary point x are mutually orthogonal at x.
Proof. (i) We slightly modify the proof of the corresponding Khesin and Tabachnikov statement given for non-symmetric ellipsoids (Theorem 4.5 [25]). Consider the function x, x s α s − λ .
(ii) The second statement has the same proof as in the case when A is positive definite (Theorem 4.5 [25]). Example 1. From Lemma 4.1 it follows that in the Euclidean space E n,0 through a generic point pass r quadrics, while through a generic point in the Lorentz-Poincaré-Minkowski space E n−1,1 pass r or r−2 quadrics from the pseudoconfocal family (34) for arbitrary symmetric quadric Q n−1 (Figures 2 and 3).

Geometrical interpretation of integrals. The condition
(42) det L x,y (λ) = q λ (y, y)(1 + q λ (x, x)) − q λ (x, y) 2 = 0 is equivalent to the geometrical property that the line l x,y = {x + sy | s ∈ R} is tangent to the quadric Q λ (see [29,12]). Therefore, if the line l k determined by the segment x k x k+1 of the virtual billiard trajectory within Q n−1 is tangent to a quadric Q λ * , then det L x k ,y k (λ * ) = 0, implying det L x k ,y k (λ * ) = 0 for all k. Also note that det L x,y (λ) is SO(k 1 , l 1 ) × · · · × SO(k r , l r )-invariant function.
As a result we have: If a line l k determined by the segment x k x k+1 of the virtual billiard trajectory within Q n−1 is tangent to a quadric Q λ * from the pseudo-confocal family (34), then it is tangent to Q λ * for all k ∈ Z. In addition, R(x k ) is a virtual billiard trajectory tangent to the same quadric Q λ * for all R ∈ SO(k 1 , l 1 ) × · · · × SO(k r , l r ).
From (33) follows that for a symmetric quadric (30) we have In particular, K N −1 = 2H = y, y . Thus, the degree of P (λ) is N − 1 for a space-like or time-like vector y, or N − 2 for a light-like y, and for a general point (x, y) ∈ M h , the equation det L x,y (λ) = 0 has either N − 1 (h = 0) or N − 2 (h = 0) complex solutions. As in the lemma above, the number of real solutions can be estimated in certain cases. In [23] we proved: We proceed with the cases mentioned in the Example 3. Proof. The proof is a modification of the idea used in [3,12] and [23] for an analogous assertion in the case of nonsymmetric ellipsoids and symmetric ellipsoids, respectively. We have From the definition of R(λ) we obtain Assume the relation α r > 0. The proof for the case 0 > α 1 is the same.
As a result, we get that in (0, ζ 1 ) there are In the space-like case h > 0, due to (46), we have a root ζ 0 ∈ (α 1 , ∞) of R(λ) and so there are additional δ 1 roots of P (λ) in (ζ 1 , ζ 0 ). Also, according to we have a zero of det L x,y (λ) in (ζ 0 , ∞) as well. Therefore, the number of real roots of P (λ) is N − 1.
On the other hand, the analysis above in the space-like case h > 0 implies at least N − 3 real roots of P (λ). The analysis for the light-like case h = 0 is the same as in the proof of (i).
Remark 4. In the previous proof we considered the case when 1 < g < r. The borderline cases g = 1 and g = r have similar analysis. Moreover, we have better estimates of the number of quadrics for the assumptions (40) and δ g = 1: if EA is positive (negative) definite and g = 1 (g = r), then the signature of the space is (1, n−1) (respectively (n−1, 1)) and there are N −1 caustics for billiard trajectories with h = 0 and N − 2 caustics for h = 0. This situation appears in Theorem 4.5.
Example 5. Next, we take E 2,1 and a nonsymmetric quadric defined by A = diag(a 1 , a 2 , a 3 ), α 1 = −a 3 > α 2 = a 2 > α 3 = a 1 > 0. According to Lemma 4.1, through the points x = (x 1 , x 2 , x 3 ) outside the coordinate planes (x 1 · x 2 · x 3 = 0) pass 3 quadrics from the pseudo-confocal family (34). The discriminant of the  polynomial It is obvious that in the time-like case the discriminant is positive and we always have two real roots. From Theorem 4.4 (i) follows that D > 0 in the space-like case, too. In the light-like case, the real root is Let us consider the signature (n−1, 1) in general situation. Suppose (35) and let g ∈ {1, . . . , r} be the index, such that n ∈ I g . In order to simplify the formulation of the theorem we additionally assume that δ g = 1, i.e., I g = {n}. for h > 0, h < 0 and h = 0, respectively. If we suppose 0 ∈ (α r , α r−1 ), then there are N − 1 quadrics for h > 0, as well. If α r > 0 (α 1 < 0) and h > 0, h < 0, h = 0, the number of caustics is at least Proof. Let us prove the item (i). The proof of the other statements is similar.

The Poncelet porism.
Here, we suppose that one of the following conditions holds: (i) The signature is arbitrary, A is positive definite. (ii) The signature is (n, 0), A is arbitrary. (iii) The signature is arbitrary, EA is positive or negative definite and the assumption (38) is satisfied.
Then τ i a i = τ j a j only if a i = a j , τ i = τ j , and the symmetry group is Similarly, Theorem 4.6 applies also in all cases described in Theorem 4.4 (ii) and Theorem 4.5 with maximal number of caustics.

Pseudo-Euclidean billiards in projective spaces
5.1. Billiards on sphere and Lobachevsky space. It is well-known that the billiards within an ellipsoid E n−2 on the sphere S n−1 and the Lobachevsky space H n−1 are completely integrable [7,37,34,8]. The ellipsoid E n−2 can be defined as a intersection of a cone (55) K n−1 : where (56) A = diag(a 1 , . . . , a n ), 0 < a 1 , a 2 , . . . , a n−2 , a n−1 < −a n , respectively. The induced metrics on S n−1 and H n−1 (a model of the Lobachevsky space) are Riemannian with constant curvatures +1 and −1, while geodesic lines are simply intersections of S n−1 and H n−1 with two-dimensional planes through the origin. Together with billiards on S n−1 and H n−1 , let us consider the following virtual billiard dynamic: where the multipliers are now determined from the conditions (A −1 x j+1 , x j+1 ) = (A −1 x j , x j ) = 0, y j+1 , y j+1 = y j , y j , that is, the impact points x j belong to the cone (55). Again, the dynamics is defined outside the singular set As a slight modification of Veselov's description of billiard dynamics within E n−2 [37] we have the following Lemma.
Lemma 5.1. Assume that the signature of the pseudo-Euclidean space E k,l is (n, 0) or (n − 1, 1), respectively. Let (x j , y j ) be a trajectory of the billiard mapping φ given by (59), (60), where A is given by (56). Then the intersections z j of the sequence of the lines span {x j } with the ellipsoid E n−2 determine the billiard trajectory within E n−2 on the sphere S n−1 and the Lobachevsky space H n−1 , respectively.
Proof. Firstly, we prove that the virtual billiard mapping φ defines the dynamics of the lines span {x j }, i.e, the dynamics of the 2-planes π j = span {x j , y j } through the origin.
In [8], Cayleys type conditions for periodical trajectories of the ellipsoidal billiard on the Lobachevsky space H n−1 are derived using the "big" n × n-matrix representation obtained by Veselov [37]. Here, as a simple modification of the Lax representation (7), we obtain the following "small" 2 × 2-matrix representation of billiards within E n−2 . Note that the relationship between the projective equivalence of the Euclidean space with the Beltrami-Klein model of the Lobachevsky space and integrability of the corresponding ellipsoidal billiards is obtained independently in [34] and [8].
Theorem 5.2. The trajectories of the mapping (59), (60) satisfy the matrix equation with 2 × 2 matrices depending on the parameter λ, where q λ is given by (8) and J j , I j , ν j by (9).
5.2. Billiards in projective spaces. Next, we consider the mapping (59), (60) in the pseudo-Euclidean spaces E k,l of arbitrary signature and without the assumption (56). We also suppose the symmetries (30). Note that Theorem 5.2 still applies and from the expression we get the integrals: They satisfy the relation (65)F 1 + · · · +F r = 0.
Further, as in the proof of Lemma 5.1, if (x ′ j , y ′ j ) is the image of (x j , y j ) by the transformation (62) and (x j+1 , y j+1 ) = φ(x j , y j ), (x ′ j+1 , y ′ j+1 ) = φ(x ′ j , y ′ j ), then the 2-planes spanned by x j+1 , y j+1 and x ′ j+1 , y ′ j+1 coincides. Also, the part of the singular set {(EA −2 x, x) = 0} ∪ {(A −1 x, y) = 0} in (61) is invariant with respect to the transformation (62). If (A −1 y j , y j ) = 0, then we can apply the transformation (62) to obtain (A −1 y ′ j , y ′ j ) = βγ(A −1 x j , y j ) = 0. Thus, if necessary, we can replace y j by y ′ j in order to determine x j+1 . Therefore, the dynamics (59), (60) induces a well defined dynamics of the lines span {x j }, i.e., the points of the (n − 1)-dimensional projective space P(E k,l ) z j = [x j ] ∈ Q n−2 outside the singular set Ξ = {[x] ∈ P(E k,l ) | (EA −2 x, x) = 0}, where Q n−2 is the projectivisation of the cone (55) within P(E k,l ). Definition 5.3. We refer to a sequence of the points (z j ) as a billiard trajectory within the quadric Q n−2 in the projective space P(E k,l ) with respect to the metric induced from the pseudo-Euclidean space E k,l .
Theorem 5.4. Let (z k ) be a sequence of the points of a billiard trajectory within quadric Q n−2 in the projective space P(E k,l ). If a projective line is tangent to a quadric P λ * then it is tangent to P λ * for all k ∈ Z.
Further, if detL x k ,y k (λ * ) = 0 for a given (x k , y k ), it will be zero for all k ∈ Z under the mapping φ (Theorem 5.2), while from the description of the billiard dynamics, the projectivisation of π k = span {x k , y k } equals l k for all k ∈ Z.
Then, in view of (67), the set of the 2-planes π = span {x, y} that are tangent to K λ * is described by the following quadratic equation in terms of the Plücker coordinates π i,j = x i y j − x j y i , 1 ≤ i < j ≤ n of π In order to determine the number of caustics one should provide an additional analysis. The following situation leads to the statement analogous to Theorems 4.3 and 4.4.
As in the case of the ellipsoidal billiards on a sphere S n−1 and a Lobachevsky space H n−1 , we assume the relation (56). Then τ i a i = τ j a j only if a i = a j , τ i = τ j , i, j < n. As above, let δ s = 2 for |I s | ≥ 2, δ s = 1 for |I s | = 1, and N = δ 1 + · · · + δ r .
Theorem 5.5. The lines l k = z k z k+1 determined by a generic billiard trajectory within Q n−2 are tangent to N − 2 fixed quadrics from the projectivisation of the confocal family (66). In particular, the trajectories of billiards within ellipsoid E n−2 , with the above symmetry, on the sphere (57) and the Lobachevsky space (58) are tangent to N − 2 fixed cones from the confocal family (66).
From (55) we have detL x,y (0) < 0 and following the lines of the proof of Theorem 4.4, it can be proved that the equationP (λ) = 0 has N − 2 real solutions, for a generic (x, y).