Generalized Symmetric Spaces

Oldřich Kowalski

Charles University, Czech Republic

1 Introduction

Let (M, g) be a Riemannian symmetric space. Then for any x ∈ M, there exists an isometry s_{x} : M ⟶ M such that x is an isolated fixed point of s_{x} and s_{x}^{2} = Id. Then we have

(s_{x})_{∗x} = (−Id)_{x}
v_{x} ↦ −v_{x}

Now we consider a generalization of the notion of Riemannian symmetric spaces: Let (M, g) be a Riemannian manifold. We consider only the following condition.

∀x ∈ M, ∃s_{x} : M ⟶ M : isometry s.t. x : isolated fixed point

Such an isometry s_{x} is called a generalized symmetry. And a set of generalized symmetries {s_{x} | x ∈ M} is called a (Riemannian) s-structure.

Lemma 1. If {s_{x} | x ∈ M} on (M, g) is an s-structure, then the closure Cl({s_{x}}) of the group generated by all s_{x}, x ∈ M, in the Lie group I(M, g) (the full isometry group) acts transitively on (M, g).

Theorem 1 (Brickel). Each (M, g) admitting an s-structure is homogeneous and thus real analytic.

Here, “homogeneous” means that ∀x, y ∈ M, ∃φ : M ⟶ M : isometry, such that φ(x) = y.

The proof is elementary and interesting, but not short.

Affine case analogue:

Theorem 2 (Ledger). Let (M, ∇) be an affine manifold (with an affine connection ∇). Let there exists a family {s_{x} | x ∈ M} of generalized affine symmetries. If the map x ↦ s_{x} is smooth, then (M, ∇) is a homogeneous affine manifold.

In general, it is an open problem whether (M, ∇) is still homogeneous if the map x ↦ s_{x} is not smooth. (Elementary but interesting question).

Regularity condition:

s_{x} ∘ s_{y} = s_{z} ∘ s_{x} z = s_{x}(y) ∀x, y ∈ (M, g)

This condition is satisfied for each symmetric space and its standard symmetries s_{x}!

Example. In the case ℝ^{2}, we can illustrate the above condition by a picture. (In general, it is not easy to check).

[figure: x; ū = s_{x}(u); u; z = s_{x}(y); y; u′′ = s_{x}(u′); u′ = s_{y}(u)]

Definition 1. Tangent symmetry field S of {s_{x} | x ∈ M}

S_{x} := (s_{x})_{∗x} ∀x ∈ (M, g)

Lemma 2. The regularity condition for {s_{x} | x ∈ M} is satisfied if and only if S is invariant with respect to each s_{x}:

(s_{x})_{∗y} ∘ S_{y} = S_{s_{x}(y)} ∘ (s_{x})_{∗y} ∀x, y ∈ (M, g)

In the case of symmetric spaces,

S_{y} = (−Id)_{y}, S_{s_{x}(y)} = (−Id)_{s_{x}(y)},

and the regularity follows.

An s-structure {s_{x} | x ∈ M} which satisfies the regularity condition is called a regular s-structure. (It is a very strong condition).

Example of regular s-structures: M = ℝ^{2}, s_{x} = rotation arround x with constant angle α. It is obvious from Lemma 2 that this s-structure is regular because each tangent symmetry S_{x} is also a rotation. (This direct proof of regularity is a good excercise for high school students).

[figure: x; y]

Proposition 1. If {s_{x} | x ∈ M} is a regular s-structure on (M, g), then the tangent field is analytic and the map (x, y) ↦ s_{x}(y) is also analytic.

Proposition 2. If (M, g) admits an s-structure (resp. regular s-structure) {s_{x} | x ∈ M}, then (M, g) admits an s-structure (resp. regular s-structure) {s′_{x} | x ∈ M} of finite order. It means that ∃k ∈ ℤ, k ≥ 2 s.t.

(s′_{x})^{k} = Id for ∀x ∈ M.

Remark: It it not true for the affine case.

If there exists a regular s-structure {s_{x} | x ∈ M} of order k on (M, g), then (M, g) is called a k-symmetric space. (Without regularity, a pointwise k-symmetric space).

Thus if there exists a regular s-structure on (M, g), then for some ∃k, (M, g) is a ksymmetric space.

Regularity condition is really an additional condition but in a nontrivial sense:

1) (R.A. Marinosci [4]) If dim(M, g) ≤ 5, then every (M, g) admitting an s-structure also admit a regular s-structure.

2) S^{9} = SU(5)/SU(4) S^{9} is a geodesic sphere in (P^{5}, Fubini-Study metric) with the induced metric, which is not the standard sphere. This S^{9} admits an s-structure of order 4 but no regular s-structure at all.

There are finitely many other examples S^{13}, S^{17}, · · ·.

In the following, we always assume regularity!

(M, g) is called a generalized symmetric space if it admits a regular s-structure. (M, g) is called of order k (k ≥ 2) if it admits a regular s-structure of order k but not of

any lower order l < k.

2 Examples and Classifications

Dimension n = 3

G = ∥ \begin{matrix}e^{−z} & 0 & x \\ 0 & e^{z} & y \\ 0 & 0 & 1 \end{matrix} ∥

(solvable)

G ≅ ℝ^{3}(x, y, z)

Special invariant metrics on G:

g = e^{2z}dx^{2} + e^{−2z}dy^{2} + λ^{2}dz^{2}

where λ > 0 is an arbitrary parameter. Symmetry of order 4 at the origin:

x′ = −y, y′ = x, z′ = −z

(See B-Spaces by Takahashi).

Thses are all generalized symmetric spaces of dimension n = 3 which are not locally symmetric!

One kind of symmetric spaces:

G: compact connected Lie group Consider the coset space (G × G)/Δ(G × G), where Δ(G × G) is the diagonal of G × G, i.e., Δ(G × G) = {(g, g) | g ∈ G}. G × G/Δ(G × G) is diffeomorphic to G via the map

(g_{1}, g_{2}) ↦ g_{1}g_{2}^{−1}.

G × G acts on G by

(g_{1}, g_{2})(y) = g_{1}yg_{2}^{−1}.

The isotropy group at the origin e ∈ G is Δ(G × G).

Now define σ : G × G ⟶ G × G by

σ(g_{1}, g_{2}) = (g_{2}, g_{2}),

which is an involutive automorphism. The fixed point set is (G×G)^{σ} = Δ(G×G). σ induces a map s : G ⟶ G defined by

s(g) = g^{−1} for ∀g ∈ G.

Take a bi-invariant Riemannian metric Φ on G. Then Φ is invariant with respect to the action of G × G on G. It is also invariant with respect to s : G ⟶ G.

Then (G, Φ) is a Riemannian symmetric space in which the symmetry with respect to e is just the map

s : g ↦ g^{−1}.

The symmetry s_{x} at general x ∈ G is given by

g ↦ xg^{−1}x.

Generalization(Ledger + Obata)

G^{k+1}/ΔG^{k+1} ≅ G^{k}

via π : G^{k+1} ⟶ G^{k}, where

π(g_{1}, ⋯, g_{k+1}) = (g_{1}g_{k+1}^{−1}, ⋯, g_{k}g_{k+1}^{−1}).

Further define σ : G^{k+1} ⟶ G^{k+1} by

σ(g_{1}, ⋯, g_{k+1}) = (g_{k+1}, g_{1}, ⋯, g_{k}).

Then σ is an automorphism of order k + 1. It induces a map s : G^{k} ⟶ G^{k} defined by

s(g_{1}, ⋯, g_{k}) = (g_{k}^{−1}, g_{1}g_{k}^{−1}, ⋯, g_{k−1}g_{k}^{−1}).

Let Φ be a bi-invariant metric on G. Then Φ generates a bi-invariant metric Φ^{k+1} on G^{k+1} such that

(G^{k+1}, Φ^{k+1}) ≅ \underbrace{(G, Φ) × ⋯ × (G, Φ)}_{k+1} .

Then Φ^{k+1} induces a G^{k+1}-invariant metric Φ^{[k]} on G^{k}. The Riemannian manifold (G^{k}, Φ^{[k]}) is a (k + 1)-symmetric space. (Here Φ^{[k]} ≠ Φ^{k}). (G^{k}, Φ^{[k]}) is not a symmetric space.

Proposition 3. Let

a) G be compact and simple,

b) Φ = −(Killing form). Assume that τ : G^{k+1} ⟶ I(G^{k}, Φ^{[k]}) has as image the full identity component of I(G^{k}, Φ^{[k]}). Then (G^{k}, Φ^{[k]}) is not l-symmetric for any l < k + 1.

Example. If we choose G = SO(3), Φ = −(Killing form), then the assumption of the Proposition 3 can be shown to be satisfied. Hence we get

Theorem 3. For each k ≥ 2, there is a generalized symmetric space (M, g) of order k, i.e., k-symmetric but not l-symmetric for ∀l < k. M = G/K with semisimple group G.

Classifications:

1) All generalized symmetric spaces of order 3 with semisimple (or reductive group) G by A. Gray [1].

2) All compact generalized symmetric spaces of order 4 by J.A. Jiménez [2].

An example of generalized symmetric Riemannian spaces of solvable type:

G = ∥ \begin{matrix}e^{u_{0}} & 0 & ⋯ & 0 & x_{0} \\ 0 & e^{u_{1}} & ⋯ & 0 & x_{1} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & e^{u_{n}} & x_{n} \\ 0 & 0 & ⋯ & 0 & 1 \end{matrix} ∥ ≅ ℝ^{2n+1}

u_{0}, u_{1}, ⋯, u_{n}, x_{1}, ⋯, x_{n} are variables, where

u_{0} + u_{1} + ⋯ u_{n} = 0.

We define a Riemannian metric

g = ∑_{i=0}^{n} e^{−2u_{i}}(dx_{i})^{2} + a ∑_{α,β=1}^{n} du_{α}du_{β}, a > 0.

Then

1) (G, g) is generalized symmetric of order 2n + 2 (even).

2) The group of all isometries preserving the origin is finite and isometric to

(ℤ_{2})^{n+1} × S_{n+1}

where S_{n+1} is the permutation group pf n + 1 elements. Thus, the group is of order 2^{n+1}(n + 1)!.

Remark. There are also examples of generalized symmetric spaces of solvable type and odd order. For any order k, there exists an example of solvable type. This is the main obstacle to the classification of all generalized symmetric spaces (in contrary to the ordinary symmetric spaces, where the classification is known).

3 Canonical connection

Let (M, g) be a Riemannian manifold and {s_{x} | x ∈ M} a regular s-structure on M. Then there is a unique affine conncetion ˜∇ on M such that

(i) ˜∇ is invariant under all s_{x}, (ii) ˜∇S = 0 (where S is the tangent symmetry field defined by S_{x} := (s_{x})_{∗x} x ∈ M).

In the explicit form, ˜∇ is given by the Ledger’s formula

˜∇_{X}Y = ∇_{X}Y − (∇_{(I−S)^{−1}X}S)(S^{−1}Y) for X, Y ∈ 𝔛(M)

where ∇ denotes the Riemannian connection of (M, g).

Note that since S has no engenvalue equal to 1 (s_{x} has only isolated fixed point), (I −S)^{−1} exists.

˜∇ is always complete.

Each tensor field P on M which is invariant by all s_{x} is parallel with respect to ˜∇, i.e., ˜∇P = 0. Therefore, the torsion tensor field T̃ and the curvature tensor field R̃ of ˜∇ are parallel:

˜∇T̃ = 0, ˜∇R̃ = 0.(∗)

Remark that if M is a symmetric space with standard {s_{x}}, ˜∇ = ∇.

Because of (∗), we have

˜∇R̃ = ˜∇T̃ = ˜∇S = ˜∇g = 0.

Therefore we can linearize Riemannian manifolds ((M, g), {s_{x} | x ∈ M}) with specific regular s-structure. We get algebraic objects (V, <, >, S, R̃, T̃), where S is a nonsingular linear isometry without fixed directions and R̃, T̃ are tensors of types (1, 3), (1, 2) respectively satisfying standard algebraic identities for the curvature and torsion. These objects are called infinitesimal models.

We have one-to-one correspondence between the set of simply connected Riemannian manifolds with regular s-structures and the set of infinitesimal models. Using the methods of linear algebra, we can classify regular s-structures and hence also generalized symmetric spaces (at least locally and for small dimensions).

If (M, g) is a generalized symmetric space and simply connected, then we have the de Rham decomposition

M = M_{1} × ⋯ × M_{r},

here if {s_{x}} is an s-tructure of order k of M, each M_{i} is a generalized symmetric space of order k_{i} and k_{i} divides k (i = 1, ⋯, r).

4 Theory of eigenvalues for generalized symmetric spaces

This theory is useful for the classification procedure of generalized symmetric spaces.

Let {s_{x} | x ∈ M} be a regular s-structure. Then S_{x} : T_{x}M ⟶ T_{x}M has the same eigenvalues for ∀x ∈ M. S_{x} is real, orthogonal, without fixed vectors. Hence its system (θ_{1}, ⋯, θ_{n}) of eigenvalues must satisfy

|θ_{i}| = 1, θ_{i} ≠ 1 for i = 1, ⋯, n

and

(θ̄_{1}, ⋯, θ̄_{n}) = (θ_{1}, ⋯, θ_{n}) up to numeration.

Every n-tuple with these properties is called an admissible n-tuple. Denote by P_{n} the set of all admissible n-tuples.

We now make P_{n} a partially ordered set. We introduce characteristic equations (of two classes):

a) X_{i}X_{j} = 1, i, j = 1, ⋯, n,

b) X_{i}X_{j} = X_{k}, i ≠ j ≠ k ≠ i, i, j, k = 1, ⋯, n.

Each element (θ_{1}, ⋯, θ_{n}) ∈ P_{n} satisfies at least one equation of type a) (n ≥ 2). (If the eigenvalues of S_{x} on (M, g) do not satisfy any equation of type b), then (M, g) is locally symmetric.)

We introduce the partial ordering ”⪯” on P_{n} as follows: (θ_{i}) ⪯ (θ′_{i}) if all characteristic equations satisfied by (θ_{i}) are also satisfied by (θ′_{i}) after possible re-numeration.

(Here (θ_{i}) ⪯ (θ′_{i}) and (θ′_{i}) ⪯ (θ_{i}) is possible, e.g., if (θ′_{i}) is a renumeration of (θ_{i}).)

Proposition 4. Let {s_{x} | x ∈ M} be a regular s-structure on (M, g) with the corresponding eigenvalues (θ_{i}). Let (θ′_{i}) ∈ P_{n} and (θ′_{i}) ⪰ (θ_{i}). Then there is a regular s-structure {s′_{x} | x ∈ M} on (M, g) with the eigenvalues (θ′_{i}).

Theorem 4. In the partially ordered set (P_{n}, ⪯), there is only a finite set of maximal elements, which are all of finite order.

[figure: ⟵ maximal elements]

This set will be denoted by D_{n}.

The same procedure works without regularity condition.

We only have to add new type of characteristic equations

c) X_{i}X_{j}X_{k} = X_{l}. The corresponding set D′_{n} of maximal elements contains D_{n}, but it is bigger.

Now we go back to the regular s-structures. We have the following theorem.

Theorem 5. For every dimension n ≥ 2, there is a finite set D_{n} = {(θ_{1}^{α}, ⋯, θ_{n}^{α}) | α = 1, ⋯, r(n)} with the following properties:

a) all elements of D_{n} are of finite order,

b) if (M, g) admits a regular s-structure, then it also admits a regular s-structure with the system of eigenvalues contained in D_{n}.

(Warning: the same (M, g) can admit regular s-structures corresponding to more elements of D_{n}).

This theorem is important for the classification of generalized symmetric spaces in low dimensions.

4.1 Basic systems of eigenvalues in small dimensions

D_{2} : (−1, −1)  (symmetric space)
D_{3} : (−1, −1, −1)  (symmetric space)
(i, −i, −1)  (4-symmetric space of dimension 3)
D_{4} : (−1, −1, −1, −1)  (symmetric space)
(θ, θ, θ^{2}, θ^{2}), θ = e^{\frac{2πi}{3}}  (∃ new example of order 3)
(i, −i, −1, −1)  (direct product or space of order 3)
(θ, θ^{2}, θ^{3}, θ^{4}), θ = e^{\frac{2πi}{5}}  (only space of order 3)
D_{5} : (−1, −1, −1, −1, −1)  (symmetric space)
(i, −i, −1, −1, −1)  (direct product)
(i, −i, i, −i, −1)  (∃ new examples of order 4)
(θ, θ^{2}, θ^{3}, θ^{4}, θ^{5}), θ = e^{\frac{2πi}{6}}  (∃ new examples of order 6)
(e^{\frac{πi}{4}} , e^{− \frac{πi}{4}} , i, −i, −1)  (no additional examples)

An estimate: Let k(n) denote the maximum of all orders of the elements in D_{n}. Then

k(n) ≤ 5^{\frac{n}{4}}  n : even,
k(n) ≤ 2 · 5^{\frac{n−1}{4}}  n : odd.

We see that k(4) ≤ 5 (exact estimate) and k(5) ≤ 10 (not exact, in fact k(5) ≤ 8). It is a natural question whether there is an estimate to improve the above inequality or not.

4.2 Classification in dimension n = 4

All generalized symmetric spaces of dimension 4 are symmetric spaces or spaces of order 3 isometric to one of the following forms:

M = ∥ \begin{matrix}a & b & u \\ c & d & v \\ 0 & 0 & 1 \end{matrix} ∥ / ∥ \begin{matrix}cos t & − sin t & 0 \\ sin t & cos t & 0 \\ 0 & 0 & 1 \end{matrix} ∥

where det ∥ \begin{matrix}a & b \\ c & d \end{matrix} ∥ = 1

M = (group of all equiaffine trans- formations of the Euclidean plane) / \binom{subgroup of all rotations ar-}{round the origin}

M ≅ ℝ^{4}[x, y, u, v] and then

g = (−x + \sqrt{x^{2} + y^{2} + 1})du^{2} + (x + \sqrt{x^{2} + y^{2} + 1})dv^{2}

−2ydudv + λ^{2} [\frac{(1 + y^{2})dx^{2} + (1 + x^{2})dy^{2} − 2xydxdy}{1 + x^{2} + y^{2}} ]

where λ > 0 is a parameter.

A typical symmetry of order 3 at the origin is given by

u′ = cos \frac{2π}{3} · u − sin \frac{2π}{3} · v,
v′ = sin \frac{4π}{3} · u + cos \frac{4π}{3} · v,
x′ = cos \frac{2π}{3} · x − sin \frac{2π}{3} · y,
y′ = sin \frac{4π}{3} · x + cos \frac{4π}{3} · y.

4.3 Classification in dimension n = 5

A generalized symmetric space of dimension 5 is locally isometric to one of the followings:

• a symmetric space,

• 11 families of order 4,

• 1 family of order 6.

4.3.1 Examples of order 4 – selection:

Example 1. Nilpotent matrix group

∥ \begin{matrix}1 & 0 & 0 & x \\ 0 & 1 & 0 & y \\ u & v & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix} ∥ ≅ ℝ^{5}[x, y, z, u, v],

g = dx^{2} + dy^{2} + du^{2} + dv^{2} + λ^{2}(xdu − ydv + dz)^{2}, where λ > 0 arbitrary parameter.

Typical symmetry of order 4 at the origin is given by

x′ = −y, y′ = x, z′ = −z, u′ = −v, v′ = u.

Example 2. M = SO(3,)/SO(2) with the invariant Riemannian metrics depending on 3 real parameters. The symmetry at the origin of M is as follows

∥ \begin{matrix}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} ∥ ↦ ∥ \begin{matrix}b̄_{2} & −b̄_{1} & b̄_{3} \\ −ā_{2} & ā_{1} & −ā_{3} \\ c̄_{2} & −c̄_{1} & c̄_{3} \end{matrix} ∥ .

Example 3. Solvable complex matrix group

M = ∥ \begin{matrix}e^{λt} & 0 & z \\ 0 & e^{−λt} & w \\ 0 & 0 & 1 \end{matrix} ∥ ≅ ℝ^{2}[z, w] × [t]

where z, w: complex variables, t: real variable, λ: complex parameter, λ ≠ 0.

A family of invariant metrics g such that (M, g) is irreducible (i.e., not a product of Riemannian manifolds).

Symmetry at the origin is given by

z′ = iw, w′ = iz, t′ = −t.

Example 4. M = SO(3) × SO(3)/SO(2)

SO(2) = \{(∥ \begin{matrix}cos t & − sin t & 0 \\ sin t & cos t & 0 \\ 0 & 0 & 1 \end{matrix} ∥ , ∥ \begin{matrix}cos t & sin t & 0 \\ − sin t & cos t & 0 \\ 0 & 0 & 1 \end{matrix} ∥) , t ∈\}

Symmetry of order 4 at the origin is induced by the following automorphisms of the group GL(3,) × GL(3,):

(∥ \begin{matrix}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} ∥ , ∥ \begin{matrix}ã_{1} & ã_{2} & ã_{3} \\ b̃_{1} & b̃_{2} & b̃_{3} \\ c̃_{1} & c̃_{2} & c̃_{3} \end{matrix} ∥) ↦ (∥ \begin{matrix}ã_{1} & −ã_{2} & −ã_{3} \\ −b̃_{1} & b̃_{2} & b̃_{3} \\ −c̃_{1} & c̃_{2} & c̃_{3} \end{matrix} ∥ , ∥ \begin{matrix}a_{1} & −a_{2} & a_{3} \\ −b_{1} & b_{2} & −b_{3} \\ c_{1} & −c_{2} & c_{3} \end{matrix} ∥)

Example 5. M = SU(3)/SU(2) ≅ S^{5} ((M, g) is not a standard sphere). For a special choice of the invariant metric, S^{5} has the metric induced from the Fubini–Study metric in P^{3} as a geodesic sphere in P^{3}.

4.3.2 A family of spaces of order 6

M = ∥ \begin{matrix}e^{−(u+v)} & 0 & 0 & x \\ 0 & e^{u} & 0 & y \\ 0 & 0 & e^{v} & z \\ 0 & 0 & 0 & 1 \end{matrix} ∥ ≅ ℝ^{5}[x, y, z, u, v] (A solvable Lie group)

We consider invarinat Riemannian metrics depending on two real parameters a > 0, b > 0:

g = a^{2}(du^{2} + dv^{2} + dudv) + (b^{2} + 1)(e^{2(u+v)}dx^{2} + e^{−2u}dy^{2} + e^{−2v}dz^{2})
+(b^{2} − 2)(e^{v}dxdy + e^{u}dxdz − e^{−(u+v)}dydz).

The typical symmetry at the origin is given by

x′ = y, y′ = −z, z′ = x, u′ = v, v′ = −(u + v).

This symmetry is of order 6!

The isotropy group of isometries at the origin consists of 8 elements.

Remark. Let (M, g) be not necessarily homogeneous in general. Let there exist a point p ∈ M such that p is an isolated fixed point for some isometry ∃s_{p} : M ⟶ M. Then there exists an isometry s′_{p} : M ⟶ M of finite order k:

k ≤ (4 + s(p))^{\frac{dimM+1}{2}}

where s(p) denotes the Singer number defined as follows: Put

G_{p}^{(l)} = {A ∈ Aut(T_{p}M) | A(g_{p}) = g_{p}, A(R_{p}) = R_{p}, ⋯, A((∇^{(l)}R)_{p}) = (∇^{(l)}R)_{p}},

l = 0, 1, 2, · · ·. Then there exists the minimal k ≥ 0 such that the sequence {G_{p}^{(l)}} stabilizes, i.e.,

G_{p}^{(0)} ⊃ G_{p}^{(1)} ⊃ ⋯ ⊃ G_{p}^{(k)} = G_{p}^{(k+1)} = ⋯

Such a number k = s(p) is called the Singer number.

References

[1] A. Gray, Riemannian manifolds with geodesic symmetries of order 3, J. Differential Geometry, 7(1972), 343–369.

[2] J.A. Jiménez, Riemannian 4-symmetric spaces, Trans. Amer. Math. Soc., 306(1988), 715–734.

[3] O. Kowalski, Generalized symmetric spaces, Lecture Notes in Mathematics, 805, Springer-Verlag, Berlin-New York, 1980.

[4] R. A. Marinosci, Classification of five-dimensional generalized pointwise symmetric Riemannian spaces, Geom. Dedicata, 57(1995), 11–53.

Oldřich Kowalski Faculty of Mathematics and Physics Charles University Sokolovská 83 186 00 Praha Czech Republic e-mail: kowalski@karlin.mff.cuni.cz

URL: http://www.karlin.mff.cuni.cz/~kowalski/