# The Schwarzian derivative on symmetric spaces of Cayley type

Khalid Koufany

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Khalid Koufany. The Schwarzian derivative on symmetric spaces of Cayley type. RIMS Kôkyûroku Bessatsu, 2012, B36. hal-00582698

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
THE SCHWARZIAN DERIVATIVE ON SYMMETRIC SPACES OF CAYLEY TYPE

KHALID KOUFANY

ABSTRACT. Let $M$ be a symmetric space of Cayley type and $f$ a conformal diffeomorphism of $M$. We study a relationship between the conformal factor of $f$ and a generalized Shwarzian derivatine of $f$.

1. INTRODUCTION

Let $M = G/H$ be a Riemannian symmetric space and let $\underline{g}$ be a $G$-invariant metric on $M$. Let $f$ be a conformal transformation of $(M, \underline{g})$, i.e.

$$f^*\underline{g} = c_f\underline{g}$$

where $c_f$ is the conformal factor of $f$.

When $M = \mathcal{H}$ is the single sheeted hyperboloid (with the unique Lorentz metric), then it is well known that $\mathcal{H}$ is conformally equivalent to $S^1 \times S^1 \setminus \Delta_{S^1}$ where $S^1$ is the unit circle and $\Delta_{S^1}$ the null space. The group of (orientation preserving) conformal diffeomorphisms of $\mathcal{H}$ is $\text{Diff}(S^1)$, the group of diffeomorphisms of the circle $S^1$. In [KS] Kostant and Sternberg, pointed out an interesting relationship between the Shwarzian derivative of a transformation $f \in \text{Diff}(S^1)$ and the corresponding conformal factor $c_f$ (which is a singular function on the null space). More precisely, they prove that $c_f$ tends to 1 on $\mathcal{H}$ as we approach infinity, and that the Hessain of the (extended) $c_f$ is the Schwarzian derivative of $f$.

The single sheeted hyperboloid is the simplest example of a large class of para-hermitian symmetric spaces, the Cayley type spaces. Those spaces are characterized as an open orbit in $S \times S$ where $S$ is the Shilov boundary of a bounded symmetric domain of tube type $[K_1]$. In this paper, we will use this characterization to extend the results of Kostant and Sternberg to symmetric spaces of Cayley type.

<footnote>Key words and phrases. Symmetric space of Cayley type, Schwarzian derivative</footnote>
## 2. CAUSAL SYMMETRIC SPACES, SYMMETRIC SPACES OF CAYLEY TYPE

Let $M$ be a smooth $n$-manifold. A causal structure on $M$ is a cone field $\mathcal{C} = (C_p)_{p \in M}$ where

$$C_p \subset T_p M$$

is a causal cone $ie.$ non-zero, closed convex cone which is pointed $(C_p \cap -C_p = \{0\})$, generating $(C_p - C_p = T_p M)$ and such that $C_p$ depends smoothly on $p$.

If $M = G/H$ is a homogeneous space, where $G$ is a Lie group and $H \subset G$ a closed subgroup, then the causal structure is said to be $G$-invariant if for any $g \in G$

$$C_{g \cdot x} = Dg(x)(C_x), \text{ for } x \in M,$$

where $Dg(x)$ is the derivative of $g$ at $x$.

Let $M = G/H$ be a symmetric space, $ie.$ there exists an involution $\sigma$ of $G$ such that $(G^\sigma)^\circ \subset H \subset G^\sigma$ where $(G^\sigma)^\circ$ is the identity component of $G^\sigma$.

Let $\mathfrak{g} = \text{Lie}(G)$ be the Lie algebra of $G$ and put

$$\mathfrak{h} = \mathfrak{g}(+1, \sigma), \quad \mathfrak{q} = \mathfrak{g}(-1, \sigma)$$

the eigenspaces of $\sigma$. Then $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{q}$ and the tangent space $T_{x_0}M$ at $x_0 = 1H$ can be identified with $\mathfrak{q}$. In this identification, the derivative $Dg(x_0)$, $g \in H$ corresponds to $\text{Ad}(g)$. Therefore an invariant causal structure on $M$ is determined by a causal cone $C$ in $\mathfrak{q}$ which is $\text{Ad}(H)$-invariant.

Suppose that $G$ is be semi-simple with a finite centre and that the pair $(\mathfrak{g}, \mathfrak{h})$ is irreducible ($ie.$ there is no non-trivial ideal in $\mathfrak{g}$ which is invariant under $\sigma$). Then, there exists a Cartan involution $\theta$ commuting with the given involution $\sigma$.

Let $K$ be the corresponding maximal compact subgroup of $G$. Let $\mathfrak{k} = \mathfrak{g}(+1, \theta)$, $\mathfrak{p} = \mathfrak{g}(-1, \theta)$. Then $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ is the corresponding Cartan decomposition of $\mathfrak{g}$.

Let $\text{Cone}_H(\mathfrak{q})$ be the set of $\text{Ad}(H)$-invariant causal cones in $\mathfrak{q}$. Then following Ólafsson (see [H-Ó]) the irreducible symmetric space $M$ is called
(CC): Compactly Causal space if there exists a $C \in \text{Cone}_H(\mathfrak{q})$ such that $C^\circ \cap \mathfrak{k} \neq \emptyset$.

(NCC): Non-Compactly Causal space if there exists a $C \in \text{Cone}_H(\mathfrak{q})$ such that $C^\circ \cap \mathfrak{p} \neq \emptyset$.

(CT): Cayley Type space if it is (CC) and (NCC).

The simplest example of Cayley type symmetric spaces is the space $M = SO_0(2, 1)/SO_0(1, 1)$ realized as the one-sheeted hyperboloid

$$ \{(x_1, x_2, x_3) \in \mathbb{R}^3 \mid x_1^2 + x_2^2 - x_3^2 = 1\}. $$

### 3. CAUSAL COMPACTIFICATION OF SYMMETRIC SPACES OF CAYLEY TYPE

Let $D = G/K \subset V_C = V + iV \simeq \mathbb{C}^n$ be a bounded symmetric domain in $\mathbb{C}$-vector space. The group $G$ is the identity component of the group of holomorphic automorphisms of $D$, and $K = \{g \in G \mid g \cdot o = o\}$ the stabilizer of the base point $o = 1K \in D$, which is a maximal compact subgroup of $G$.

Suppose that $D$ is of tube type,

$$ D \simeq T_{\Omega} = V + i\Omega $$

where $\Omega$ is the symmetric cone of the Euclidean Jordan algebra $V$.

Let $z \mapsto \bar{z}$ the complex conjugation in $V_C$ with respect to $V$ and $e$ the unit element of $V$. The set

$$ S = \{z \in V_C \mid \bar{z}z = e\} $$

is a connected submanifold of $V_C$ which is a Riemannian symmetric space of compact type

$$ S \simeq U/U_e, $$

where $U$ is the identity component of the group of linear transformations $g \in GL(V_C)$ such that $gS = S$, and $U_e$ is the stabilizer subgroup of $e \in S$.

There exists on $V_C$ a $U$-invariant spectral norm $z \mapsto |z|$, and one can prove (see [F-K]) that $D$ is the unit disc

$$ D = \{z \in V_C \mid |z| < 1\} $$

and $S$ is its Shilov boundary.

Let

$$ G(\Omega) = \{g \in GL(V) \mid g\Omega = \Omega\}. $$
It is a reductive Lie group which acts transitively on $\Omega$. Let $G_0 = G(\Omega)^\circ$ be the identity component of $G(\Omega)$.

Let $G^c$ be the identity component of the group of holomorphic automorphisms of $T_\Omega$. Then $G_0$ is a Lie subgroup of $G^c$. The subgroups $G_0$ and $N^+ = \{t_v : z \mapsto z + v, v \in V\}$, together with the inversion $j : z \mapsto -z^{-1}$, generate the group $G^c$.

For any $x \in V$ we define the quadratic representation $P(x)$ of Jordan algebra $V$ given by $P(x) = 2L(x)^2 - L(x^2)$, where $L(x)$ is the multiplication by $x$.

The Lie algebra $\mathfrak{g}^c$ of $G^c$ is the set of vector fields on $V$ of the form

$$X(z) = u + Tz + P(z)v \approx (u, T, v),$$

where $T$ is linear and $u, v \in V$.

Consider on $G^c$ the involutions

$$\sigma^c(g) = \nu \circ g \circ \nu$$

$$\theta^c(g) = (-\nu) \circ g \circ (-\nu)$$

where $\nu : z \mapsto \bar{z}^{-1}$. We use the same letters for the corresponding involutions on the Lie algebra $\mathfrak{g}^c$.

If $X = (u, T, v) \in \mathfrak{g}^c$, then

$$\sigma^c(X) = (v, -T^*, u) \text{ and } \theta^c(X) = (-v, -T^*, -u),$$

Therefore,

$$\mathfrak{h}^c := \mathfrak{g}^c(\sigma^c, +1) = \{(u, T, u) \mid u \in V, T \in \mathfrak{k}_0\}$$

$$\mathfrak{q}^c := \mathfrak{g}^c(\sigma^c, -1) = \{(u, L(v), -u) \mid u, v \in V\}, (L(v)x = vx)$$

$$\mathfrak{k}^c := \mathfrak{g}^c(\theta^c, +1) = \{(u, T, -u) \mid u \in V, T \in \mathfrak{k}_0\}$$

$$\mathfrak{p}^c := \mathfrak{g}^c(\theta^c, -1) = \{(u, L(v), u) \mid u \in V\}.$$

Consider the convex cones in $\mathfrak{q}^c$

$$C_1 = \{(u, L(v), -u) \mid u + v \in -\bar{\Omega}, u - v \in \bar{\Omega}\},$$

$$C_2 = \{(u, L(v), -u) \mid u + v \in \bar{\Omega}, u - v \in \bar{\Omega}\}.$$

Then $C_1$ and $C_2$ are $\mathrm{Ad}(H^c)$-invariant causal cones and

$$C_1 \cap \mathfrak{p}^c \neq \emptyset, C_2 \cap \mathfrak{k}^c \neq \emptyset.$$

Let $c : z \mapsto i(e+z)(e-z)^{-1}$ be the Cayley transform corresponding to the bounded symmetric domain $D$. Then we have

**Theorem 3.1 ([K$_1$, K$_2$]).** (1) $H := c^{-1} \circ G_0 \circ c = H^c := G \cap G^c$.

(2) $M = G/H \simeq G^c/H^c$ is a symmetric space of Cayley type, and every Cayley type space is given in this way.
Let

$$ \Delta_S = \{(z, w) \in S \times S \mid \Delta(z - w) = 0\} $$

be the null space of $S \times S$, where $\Delta$ is the determinant function of $V$, extended to $V_C$.

The group $G$ acts diagonally on $S \times S$. Furthermore,

**Theorem 3.2** ([K₁, K₂]). $G$ acts transitively on $S \times S \setminus \Delta_S$ and the stabilizer of the base point $(e, -e) \in S \times S \setminus \Delta_S$ is the subgroup $H$. Therefore, $M = G/H \simeq S \times S \setminus \Delta_S$ and $S \times S$ is the (causal) compactification of $M$.

For example,

$$ \begin{aligned} D &= SU(n, n)/S(U(n) \times U(n)) \\ &= \{z \in \mathrm{Mat}(n, \mathbb{C}) \mid I_n - z^*z \gg 0\} \end{aligned} $$

$$ S = U(n) \qquad \text{and} $$

$$ \begin{aligned} M &= SU(n, n)/GL(n, \mathbb{C}) \times \mathbb{R}^+ \\ &\simeq \{(z, w) \in U(n) \times U(n) \mid \mathrm{Det}(z - w) \neq 0\}, \end{aligned} $$

$$ \begin{aligned} D &= Sp(n, \mathbb{R})/U(n) \\ &= \{z \in \mathrm{Sym}(n, \mathbb{C}) \mid I_n - z^*z \gg 0\} \end{aligned} $$

$$ \begin{aligned} S &= U(n)/O(n) \\ &= \{z \in U(n) \mid z^t = z\} \end{aligned} $$

$$ \begin{aligned} M &= Sp(n, \mathbb{R})/GL(n, \mathbb{R}) \times \mathbb{R}^+ \\ &\simeq \{(z, w) \in U(n) \times U(n) \mid z^t = z, w^t = w, \mathrm{Det}(z - w) \neq 0\}. \end{aligned} $$

### 4. THE SCHWARZIAN DERIVATIVE ON THE ONE-SHEETED HYPERBOLOID

Recall the classical cross-ratio of four points in the complex plane

$$ [z_1, z_2, z_3, z_4] = \frac{z_1 - z_3}{z_2 - z_3} : \frac{z_1 - z_4}{z_2 - z_4} $$

Then the Élie Cartan formula for the cross-ratio is given by

**Theorem 4.1** (Élie Cartan). Consider $f: S^1 \to S^1$ and four points $x_1, x_2, x_3, x_4 \in S^1$ tending to $x \in S^1$. Then

$$ \frac{[f(x_1), f(x_2), f(x_3), f(x_4)]}{[x_1, x_2, x_3, x_4]} - 1 = \frac{1}{6} S(f)(x)(x_1 - x_2)(x_3 - x_4) + [\text{higher order terms}] $$

where $S(f)$ denote the Schwarzian derivative of $f$,

$$ S(f) = \frac{f''''}{f'''} - \frac{3}{2} \left( \frac{f''}{f'''} \right)^2. $$
Consider now the the single sheeted hyperboloid

$$
\begin{aligned}
\mathcal{H} &= SL(2, \mathbb{R})/\mathbb{R}_+^* \\
&\approx SU(1, 1)/SO(1, 1) \\
&\approx S^1 \times S^1 \setminus \Delta_{S^1} \\
&= \{(e^{i\theta_1}, e^{i\theta_2}) : \theta_1 \neq \theta_2\}.
\end{aligned}
$$

$\mathcal{H}$ carries a (unique up to multiplicative constant) Lorentz metric,
$\underline{\mathbf{g}}$, invariant under $SL(2, \mathbb{R})$,

$$
\underline{\mathbf{g}} = \frac{d\theta_1 d\theta_2}{|e^{i\theta_1} - e^{i\theta_2}|^2}.
$$

Let $f : S^1 \to S^1$ be a diffeomorphism viewed as a conformal trans-
formation of $(\mathcal{H}, \underline{\mathbf{g}})$, $f^*\underline{\mathbf{g}} = \mathbf{c}_f \underline{\mathbf{g}}$. Applying the Cartan formula, when
$\theta_1, \theta_2 \to \theta$, we get

$$
\begin{align*}
\mathbf{c}_f(\theta_1, \theta_2) - 1 &= \frac{f^*g(\theta_1, \theta_2)}{g(\theta_1, \theta_2)} - 1 \\
&= \frac{1}{6} S(f)(e^{i\theta})(e^{i\theta_1} - e^{i\theta_2})^2 + \dots
\end{align*}
$$

Then we have

**Theorem 4.2** (Kostant-Sternberg [KS]). *The conformal factor $c_f \to 1$ as $(\theta_1, \theta_2) \to \Delta_{S^1}$. In the other word $c_f$ tends to 1 on $\mathcal{H}$ as we approach the infinity.*

So let us extend $\mathbf{c}_f$ to be defined on $S^1 \times S^1$ by setting it equal to 1 on $\Delta_{S^1}$. Then $\mathbf{c}_f$ is twice differentiable on $S^1 \times S^1$, and has $\Delta_{S^1}$ as critical manifold and the Hessian $\mathrm{Hess}(\mathbf{c}_f)$ is equal to $S(f)$.

5. THE SCHWARZIAN DERIVATIVE ON SYMMETRIC SPACES OF
CAYLEY TYPE

The Kantor [Kan] cross-ratio for $z_1, z_2, z_3, z_4$ in $V_C$, is the rational function

$$
[z_1, z_2, z_3, z_4] = \frac{\Delta(z_1 - z_3)}{\Delta(z_2 - z_3)} : \frac{\Delta(z_1 - z_4)}{\Delta(z_2 - z_4)}
$$

where $\Delta$ is the determinant function of $V$ (extented to $V_C$).

The cross-ratio is invariant under the group $G^c$ (when it is well de-
fined) : The invariance under translations is clear. The invariance
under the group $G_0$ follows from the relation $\Delta(gz) = \chi(g)\Delta(z)$ where
$\chi$ is a character of $G_0$. The invariance under the inversion follows from
the Hua identity $\Delta(w^{-1} - z^{-1}) = \Delta(z)^{-1}\Delta(z - w)\Delta(w)^{-1}$.
On the Cayley type symmetric space $M \simeq S \times S \setminus \Delta_S$ there exists a $G$-invariant measure

$$g = |\Delta(z - w)|^{-2\frac{n}{r}} d\sigma(z)d\sigma(w),$$

where $n$ is the dimension of $V$ and $r$ its rank.

Let $\langle \cdot, \cdot \rangle$ be the inner product of the Euclidean Jordan algebra $V$ extented to a Hermitian inner product of $V_C$.

Let $f: V_C \to V_C$ be a map of class $C^3$. Let $z_j = z + taju$ be four points tending to $z \in \bar{D}$, where $t \in \mathbb{R}$ and $a_i \in \mathbb{R}$ for $j = 1, 2, 3, 4$.

**Theorem 5.1.** For any $\alpha \in \mathbb{R}$ we have

$$\frac{[f(z_1), f(z_2), f(z_3), f(z_4)]^\alpha}{[z_1, z_2, z_3, z_4]^\alpha} - 1 = \alpha t^2 (a_1 - a_2)(a_3 - a_4)S(f)(z) + o(t^3)$$

where

$$S(f) = \frac{1}{6} \langle f'''', f'^{-1} \rangle - \frac{1}{4} \langle P(f'')f'^{-1}, f'^{-1} \rangle$$

with $f' = Df(z)u, f'' = D^2f(z)(u, u)$ and $f''' = D^3f(z)(u, u, u)$

One can also prove

**Theorem 5.2.** Let $f$ be an orientation-preserving diffeomorphism of $(M, g)$. Then

(1) $c_f(z, w) \to 1$ as $z \to w$ and $c_f$ extends smoothly to $S \times S$ and has, moreover, $\Delta_S$ as its critical set.

(2) The Schwarzian $S(f)$ completely determines $c_f$.

The complete proofs will appear in a forthcoming paper.

## REFERENCES

C. Cartan, É., *Leçons sur la théorie des espaces à connexion projective*. Gauthiers-Villars, Paris 1937.

D-G. Duval, C.; Guieu, L., The Virasoro group and Lorentzian surfaces: the hyperboloid of one sheet. *J. Geom. Phys.* **33** (2000), 103–127.

F-K. Faraut, J.; Korányi, A., *Analysis on symmetric cones*. Oxford Math. Monog. The Clarendon Press, Oxford University Press, New York, 1994.

Kan. Kantor, I. L., Non-linear groups of transformations defined by general norms of Jordan algebras. *Dokl. Akad. Nauk SSSR*, **172**, (1967) 779–782.

KS. Kostant, B.; Sternberg, S., The Schwartzian derivative and the conformal geometry of the Lorentz hyperboloid. M. Cahen and M. Flato (eds.), *Quantum Theories and Geometry*, 113–125, Kluwer Academic Publishers, 1988.

K₁. Koufany, K., *Réalisation des espaces symétriques de type Cayley*. *C. R. Acad. Sci. Paris*, **318** (1994), 425–428.
K₂. —, Semi-groupe de Lie associé à un cône symétrique. *Ann. Inst. Fourier*, **45** (1995), 1–29.

H-Ó. Hilgert, J.; Ólafsson G., *Causal symmetric spaces. Geometry and harmonic analysis*. Perspect. Math., **18**. Academic Press, Inc., San Diego, CA, 1997.

KHALID KOUFANY – INSTITUT ÉLIE CARTAN, UMR 7502, UNIVERSITÉ HENRI POINCARÉ (NANCY 1) B.P. 239, F-54506 VANDŒUVRE-LÈS-NANCY, FRANCE
*E-mail address: khalid.koufany@iecn.u-nancy.fr*