\magnification=\magstep1

\def\eps{\varepsilon}
\def\frac#1#2{{#1\over#2}}
\def\mathcal#1{{\cal #1}}
\def\artanh{\mathop{\hbox{\rm artanh}}}

\def\CC{{\mathchoice
{\rm C\mkern-8mu\vrule height1.45ex depth-.05ex
width.05em\mkern9mu\kern-.05em}
{\rm C\mkern-8mu\vrule height1.45ex depth-.05ex
width.05em\mkern9mu\kern-.05em}
{\rm C\mkern-8mu\vrule height1.ex depth-.07ex
width.035em\mkern9mu\kern-.035em}
{\rm C\mkern-8mu\vrule height.65ex depth-.1ex
width.025em\mkern8mu\kern-.025em}}}

\def\RR{{\rm I\kern-1.6pt {\rm R}}}
\def\OO{{\rm \hbox{$\vert$}\kern-3.6pt {\rm O}}}
\def\MM{{\rm I\kern-1.6pt {\rm M}}}
\def\TT{{\rm T\kern-1.6pt {\rm T}}}

\def\Z{{\bf Z}} %JB*-triple
\def\E{{\bf E}} %Banach space
\def\D{{\bf D}} %unit ball
\def\U{{\bf U}} %domain for $c\mapsto T_c(0)$



%%%%%%%%%%%%%%%%%%%%%
%    REFERENCES     %
%%%%%%%%%%%%%%%%%%%%%

\def\Ist{1}         %Isidro-Stacho Oxford
\def\KaupM{2}       %Kaup 1977
\def\KaupR{3}       %Kaup Riemann mapping Thm
\def\KaupS{4}       %Kaup-Sauter
\def\Klingenberg{5} %book
\def\Sauter{6}      %Sauter's Dissertation
\def\St{7}          %Stacho Cluj 2007
\def\Upmeier{8}     %Upmeier book
\def\Werner{9}      %Werner Cluj 2007


%%%%%%%%%%%%%%%%%%%%%



\centerline{\bf JORDAN MANIFOLDS}

\bigskip
\centerline{\rm L.L. STACH\'O and W. WERNER}

\bigskip
\bigskip
{\narrower
\noindent
\bf Abstract. \rm
We introduce Jordan manifolds as
Banach manifolds whose tangent spaces are endowed with
Jordan triple products depending smoothly on the underlying points.
As chief examples we study in detail the natural complex geometry
bounded symmetric domains with their Harish-Chandra realizations as
unit balls of JB*-triples and we extend the results to
Jordan manifolds where the chart transition maps are locally generalized
M\"obius transformations.
\par}

\bigskip
\bf AMS Subject Classification: \rm
????????????????????????

\bigskip
\item{} \bf Key words: \rm
Jordan manifold,
Jordan triple, JB*-triple, Bergman operator,
\hfill\break
$\phantom{xxxxxxxxxxx}$M\"obius transformation.

\bigskip
\bigskip
\noindent
\bf 1. Introduction
\bigskip
\rm

Recently, in [\Werner] the second author investigated
the natural complex geometry of the unit ball of a
complex Hilbert C*-module.
Among others,
he proved the existence of a unique symmetry invariant (Levi Civit\'a) connection
in these structure along with its explicit form in terms of the
non-commutative scalar product.
Our first goal in Section 2 will be to show with
a different Jordan theoretical approach
that the results in [\Werner]
can be extended to the natural complex geometry of
any bounded symmetric domain
in a complex Banach space.
Actually, such domains are biholomorphically equivalent to
the unit balls of the so-called JB*-triples
that is Banach spaces equipped with a
three variable operation $(x,y,z) \mapsto\{ xyz\}$
satisfying the generalized C*-condition
$\Vert \{ xxx\} \Vert = \Vert x\Vert^3$
along with the Jordan identity
and the fact that
its elementary inner derivations are positive Hermitian operators.
Using the deep topological Jordan algebraic results in [\KaupR, Section 2]
we achieve simple explicit formulas for the Levi Civit\'a connection
in terms of the Bergman operator associated with the triple product $\{\ldots\}$.
Taking into account that the context in [\KaupR, Section 2]
extends to general real (not only complex) topological Jordan*-triples
(even Jordan pairs) our techniques can also be applied in a fairly more general setting.
Namely we establish analogous results in Section 3 for
real Banach manifolds
which admit an atlas consisting of maps into some Jordan*-triple such that
the transitions between charts around closely situated couple of points
are M\"obius transformations of the product triple product
(Jordan-M\"obius manifolds in out terminology).
These results apply automatically to all complex symmetric Hermitian Banach manifolds
treated in [\KaupM] but go far beyond the complex setting.
In Section 4 we start a more general approach.
We introduce the concept of Jordan manifold as a
real or complex Banach manifold equipped with
smoothly varying Jordan*-triple products on the tangent spaces.
This generality includes even classical Riemann spaces
(with triple product
$\{ uvw\}_p := {1\over 2}\langle u\vert v\rangle w +
{1\over 2}\langle w\vert v\rangle u$
in terms of the scalar product
on the tangent space at the point $p$)
in a natural manner.
To approach the real background of the previous results,
homogeneous and symmetric Jordan manifolds are more suited.
These concepts are at hand because
Jordan manifolds form a category with morphisms
being smooth mappings whose derivatives are homomorphisms for the
pointwise triple products.
Namely a Jordan manifold is homogeneous if its automorphism group is
transitive and symmetric if every point admits an automorphism
whose derivative there is minus-identity.
For the moment it seems to be an open question if
the automorphisms of a symmetric Jordan manifold form
a Banach-Lie group
with an analogous construction as Upmeier's topology
[\Upmeier]
for symmetric Hermitian Banach manifolds.
We close the paper with examples of
various Jordan manifolds.


\bigskip
\bigskip

\bf 3. The open unit ball of a complex JB*-triple \rm

\medskip
Throughout this work let $\Z$ denote an arbitrarily fixed
complex Banach space with norm $\Vert \cdot \Vert$.
We shall write ${\rm Ball}(\Z)$ for its open  unit ball
${\rm Ball}(\Z) := \{ z\in \Z:\ \Vert z\Vert <1\}$.
By a {\it Jordan triple product} on $\Z$ we mean a
continuous 3-variable operation
$(x,y,z)\mapsto \{ xyz\}$
being symmetric bilinear in the outer variables $x,z$
and conjugate-linear
in the inner variable $y$
satisfying the {\it Jordan identity}
$$\big\{ ab \{ xyz\} \big\} =
\big\{ \{ abx\} yz \big\} - \big\{ x \{ bay\} z \big\} + \big\{ xy \{ abz\}\big\}
\leqno({\rm J})$$
for all $a,b,x,y,z\in E$.
We say that the Jordan triple $(\Z,\{\ldots\})$ is a {\it JB*-triple}
if the {\it generalized C*-axiom}
$$\big\Vert \{ zzz \} \big\Vert = \Vert z \Vert^3 \ ,\qquad z\in \Z$$
holds and
all the operations
$$D(a) : z \mapsto \{ aaz \} \ , \qquad a\in \Z$$
are $\Z$-Hermitian with non-negative spectrum, that is
$$\big\Vert \exp \big( \zeta D(a) \big) \Vert \le 1 \quad
\hbox{for every \ \ $\zeta\in\CC$ \ with \ ${\rm Re}\, \zeta \le 0$ .}$$
\bf 2.1 Remark. \rm (1)
The Jordan identity is equivalent with the fact that
all the operators $i D(a)$ $(a\in E)$ are derivations of the triple product,
that is
we have the identities
$$iD(a)\{ xyz\} = \{ [iD(a)x]yz\} - \{ x[iD(a)y]z\} + \{ xy[iD(a)z]\} .$$
(2) C*-algebras are Jordan triples with the triple product
$$\{ xyz \} := {1\over 2} x y^* x + {1\over 2} z y^* x \ ,$$
moreover complex C*-algebras are JB*-triples with
their natural norm.

\medskip
\noindent (3)
Complex Hilbert C*-modules are also Jordan triples
with
%the product
$$\{ xyz\} := {1\over 2} \langle x \vert y \rangle z + {1\over 2} \langle z \vert y \rangle x .$$
(4) If the unit ball $\D:={\rm Ball}(\Z)$ is symmetric
(that is for each point $a\in\D$, there is a biholomorphic automorphism
$S_a : \D \leftrightarrow \D$ with Fr\'echet derivative
$S_a^\prime(a) = - {\rm Id}_{\Z}$ at $a$)
then $E$ can be equipped with a {\it unique} JB*-triple product.

\medskip
\noindent (5)
Any bounded symmetric domain in a complex Banach space
can be mapped biholomorphically onto
some bounded balanced convex symmetric domain
(a so-called {\it Harish-Chandra realization}), that is
onto the unit ball of some equivalent JB*-norm.

\bigskip
\noindent
In the sequel let $(\Z, \{ \ldots\})$ denote any given JB*-triple and let
$$
\eqalign{
&D(a,b): z\mapsto \{ a,b,z\}, \qquad D(d):=D(a,a), \cr
&Q(a,b): z\mapsto \{ a,z,b\}, \qquad Q(a):=Q(a,a), \cr
&B(a,b):= 1 - 2D(a,b) + Q(a)Q(b), \quad B(a):=B(a,a) \cr} $$
be the usual
{\it skew-derivations},
{\it quadratic representations}
and
{\it Bergman operators}, respectively.
Recall that the transformation
$$T: c \mapsto
\Big[\exp\Big( \big[c - Q(z) c\big] {\partial\over\partial z} \Big)\Big] 0$$
that is \
$T(x) = \big[$the value $z_1$ for the initial value problem
${d\over dt} z_t = c - Q(z_t) c$, $z_0=0 \big]$ \
is a well-defined real bianalytic mapping
$$T: \Z \longleftrightarrow {\rm Ball}(\Z) \ .$$
Given any point $a\in {\rm Ball}(\Z)$,
it is well-known
[\Sauter, p. 27, \KaupS] that the mapping
$$g_a := \exp \Big( \big[ T^{-1}(a) - Q(z)T^{-1}(a) \big] \partial/\partial z \Big)$$
is a holomorphic automorphism of ${\rm Ball}(\Z)$
and we have
$$g_a (z) = a + B(a)^{1/2} [ 1 + D(z,a)]^{-1}  z, \qquad
\Vert z\Vert, \Vert z\Vert <1 .
\leqno(2.2)$$
\bf 2.3 Definition. \rm
In the sequel we shall call the mappings $g_a \circ L$ composed with
linear unitary operators of $\Z$ the
{\it M\"obius transformations}
associated with the triple product $\{\ldots\}$.
It is well-known [\KaupR]
the group
${\rm Aut}\ {\rm Ball}(\Z)$
of all holomorphic automorphisms of the unit ball of a
complex JB*-triple coincides with the set of all M\"obius transformations
of the underlying triple product.

\bigskip
\noindent
\bf 2.4 Remark. \rm
In [\KaupR] one can find
$g_a (z) = a + B(a)^{1/2} [ 1 - D(x,a)]^{-1}  z$
which is obviously incorrect in the sign of the term $D(z,a)$
as on can see on the 1-dimensional example of the classical M\"obius transformation
$g_a(z)=(z+a)/(1+\overline{a}z)$ \ (with $a,z\in{\rm Ball}(\CC))$.



\bigskip

Next we are going to consider ${\rm Ball}(\Z)$ as a complex manifold
equipped with the charts $\big\{ g_a^{-1}: \ a \in {\rm Ball}(\Z)\big\}$.
In this manner we get a natural generalization
of the complex Poincar\'e model on the unit disc ${\rm Ball}(\CC)$
for the real 2-dimensional hyperbolic geometry.

Due to the possible lack of non-trivial smooth functions vanishing outside a ball,
for a Banach manifold $M$,
it is no longer convenient to apply the usual definition of a connection as a mapping
$\nabla: TM\times TM\to TM$
as being a derivation for the first and linear in the second variable,
both with respect to multiplication with smooth functions.

\bigskip
\noindent
\bf 2.5 Definition \rm [\Klingenberg]. \
Let $M$ be a manifold, modeled over
the Banach space $E$, and denote the space of bounded bilinear
mappings $E\times E\to E$ by $L^2(E,E)$. Then $M$ is said to possess
a connection if there is an atlas $\mathcal{U}$ for $M$ so that for
each $(U,\Phi)\in\cal{U}$
(where $U$ is some open subset of $M$ and $\Phi$ is a homeomorphism of $U$  onto
some open subset of $E$)
there is a smooth mapping
$\Gamma_\Phi: \Phi(U)\to L^2(E,E)$,
called the
\it Christoffel symbol \rm
of the connection on $U$, which under a change of coordinates
$\psi: \Phi(U)\to E$
transforms according to
$$
\Gamma_{\psi\circ\Phi} (\psi'u,\psi'v)=\Psi''(u,v)+\psi'\Gamma_\Phi(u,v)
$$
for smooth vector fields \ $u,v$ \ on \ $\Phi(U) \subset E$.
The covariant derivative of a vector field $Y$
in the direction of the vector field $X$ is, locally,
defined to be the principal part of $$
\nabla_XY=dX(Y)-\Gamma(X,Y),
$$
%where, in a chart, the principal part of $(u,X)\in U\times T(U)$ is $X$.
that is, if $X,Y$ are smooth vector fields on $M$ with
$u:=\Phi^\# X \big(: \Phi(U)\ni \Phi(p) \mapsto \Phi^\prime(p) X(p)\big)$ and $v:=\Phi^\# X$
then
$$\Phi^\# \nabla_X Y = \nabla_{\! u} v + \Gamma_\Phi(u,v)  \ : \
\Phi(U) \ni q \mapsto
{d\over dt}\Big\vert_{t=0} v\big( q + t u(q)\big) + \Gamma_\Phi\big( u(q),v(q) \big) \ .$$

%\medskip
\rm
\noindent
If a Banach Lie group $G$ acts smoothly on $M$ then,
for each $g\in G$ a connection $g^*\nabla$ is defined by letting
$$
g^*\nabla_XY=\nabla_{g^*X}g^*Y, \qquad g^*X(gm)=d_mgX(m).
$$
The Christoffel symbols then transform as in the definition above,
$$\eqalign{
\Gamma_{\Phi\circ g} (g(m)) &\big( g'(m)X(m),g'(m)Y(m) \big) =\cr
&= g''(m)\big( X(m),Y(m) \big)+g'(m)\big[ \Gamma_\Phi (m) \big( X(m),Y(m)\big) \big],
\qquad m\in M \cr}$$
and we call
$\nabla$ invariant under the action of $G$
whenever $g^*\nabla=\nabla$ for all $g\in G$.

\bigskip
\noindent
\bf 2.6 Theorem. \it
On $U:= {\rm Ball}(\Z)$, there exists exactly one
M\"obius invariant connection
whose Christoffel symbol at $a$ is given by $$
\Gamma_{{\rm Id}} (a) (x,y)=2B(a)^{1/2}\{B(a)^{-1/2}x,a,B(a)^{-1/2}y\}=2\{x,g_a'(0)a,y\}_a, $$
\bf Proof.
\rm
Notice first the invariance of $\nabla$ with respect to the symmetry
$S_0(z):=-z$ $(z\in U)$ at the origin implies
$-\Gamma_{{\rm Id}} (0) (x,y)=S_0^\prime(0)\Gamma_{{\rm Id}}(0) (x,y) =
\Gamma_{{\rm Id}}(0) \big( S_0^\prime(0)x,S_0^\prime(0)y\big)=
\Gamma_{{\rm Id}}(0) (x,y)$
that is
$$
\Gamma_{{\rm Id}} (0) (x,y)=0.
$$
Then, for any M\"obius transformation $g$
leaving the origin fixed,
$g''=0$ and so $\Gamma_{g}(0)$ remains zero under the transformation $g$.
Recall [\Sauter,\KaupS] that the Fr\'echet derivative of $g_a$ can be expressed as
$$
g_a^\prime (z) = B(a)^{1/2} B(z,-a)^{-1} ,\qquad \Vert z\Vert < 1 .
$$
Using that $Q'(z)(h): x\longmapsto 2\{z,x,h\}$,
and writing the second derivative as a bilinear mapping, we similarly arrive at $$
\eqalign{
g_a''(z)(u,v)& = -B(a)^{1/2}B(z,-a)^{-1}\partial_1B(u,a)B(z,-a)^{-1}(v)\cr
             & =
2B(a)^{1/2}B(z,-a)^{-1}\big[D(u,a)-Q(u,z)Q(-a)\big]B(z,-a)^{-1}(v),\cr} $$
applying the chain rule. If we evaluate this expression at $z=0$, then $$
\eqalign{
\Gamma_{{\rm Id}}(a) (x,y)& = g_a''(0)(g_{-a}'(0)x,g_{-a}'(0)y)\cr
             & = 2B(a)^{1/2}D(g_{-a}'(0)x,a)(g_{-a}'(0)y),\cr
             & = 2B(a)^{1/2}\{B(a)^{-1/2}x,a,B(a)^{-1/2}y\} \ .\cr} $$
From the above calculations
it is immediate that a Christoffel symbol thus defined
gives rise to a connection being invariant under all M\"obius transformations.
Qu.e.d.


\bigskip\rm

Next we proceed to the geodesic equation of the connection $\nabla$.
Recall that a smooth curve $\gamma: I\to {\rm Ball}(Z)$
on an open real interval $I$ around $0$ is a
$\nabla$-geodesic if its derivative satisfies the equation
$\nabla_{\dot\gamma(t)} \dot\gamma(t) = 0$.
According to Theorem 2.6, this means
$$\ddot\gamma (t) +
2 B\big( \gamma(t)\big)^{1/2}
D\Big( B\big( \gamma(t)\big)^{-1/2} \dot\gamma(t) , \gamma(t) \Big)
B\big( \gamma(t)\big)^{-1/2} \dot\gamma(t) = 0 .$$
First we look for particular solutions with the property $\gamma(0)=0$.
Let $0\ne v\in Z$ be any fixed vector and let
$F_v$ denote the real JB*-subtriple generated by $v$.
That is
$F_v = \overline{{\rm Span}_{\RR}} \big\{ D(v)^n v: \ n=0,1,\ldots\big\}$.
Since $(\Z,\{\ldots\})$ is JB*-triple, the spectrum of the operator
$D(v)\vert F_v$ is non-negative, and, by writing
$\Omega_v := \big[ {\rm Sp} D(v)^{1/2} \vert F_v\vert \setminus \{ 0\}$,
the commutative Gelfand-Naimark Theorem of JB*-triples asserts that
there is a real JB*-isomorphism
$H_v :{\rm Re}\, {\cal C}_0(\Omega_v) \to F_v$
such that
$H_v ({\rm Id}_{\Omega_v}) = v$
and
$H_v (\varphi\psi\chi) = \{ H_v(\varphi),H_v(\psi),H_v(\chi)\}$
for all functions
$\varphi,\psi,\chi\in{\rm Re}\, {\cal C}_0(\Omega_v)$.
In particular, given any solution
$\varphi: I\to {\rm Re}{\cal C}_0(\Omega_v)$,
of the equation
$${d^2 \over dt^2} \varphi(t,\omega) +
2 \big[ 1 - \varphi(t,\omega)^2 \big]^{-1} \varphi(t,\omega)
\Big[ {d\over dt} \varphi(t,\omega) \Big]^2 ,
\qquad \omega\in\Omega_v ,$$
the curve
$\gamma(t) := H_v \varphi(t)$, $t\in I$ is a $\nabla$-geodesic.
Notice that, for any real constant $\alpha$,
the function
$x(t) ={\rm artanh} (\alpha t) :=
(e^{\alpha t}-e^{-\alpha t})(e^{\alpha t}+e^{-\alpha t})^{-1}$
is the solution of the initial value problem
$\ddot x + (1-x^2)^{-1} x \dot x^2$, $x(0)=0$, $\dot x(0)=\alpha$.
Indeed, we have $\dot x = 1-x^2$
and hence
$\ddot x = - 2 x\dot x = -2 x (1-x^2)$
and
$\ddot x + (1-x^2)^{-1} x \dot x^2 =
- 2 x\dot x = -2 x (1-x^2) + 2 (1-x^2)^{-1} x (1-x^2)^2 =0$.
Therefore we get the following.

\bigskip
\noindent \bf 2.7 Theorem. \it
The $\nabla$-geodesic curves passing through the origin have the form
$$\gamma_{0,v}(t) = H_v \ {\rm artanh} \big( t\ {\rm Id}_{\Omega_v}\big) ,
\qquad t\in{\bf R}, \ v\in Z$$
in terms of the Gelfand-Naimark representations $H_v$.
\bigskip

\noindent
\bf 2.8 Corollary. \it
The $($unique$)$ maximal $\nabla$-geodesic $\gamma_{a,w}$ with the properties
$\gamma_{a,w}(0)=a$ and $\dot\gamma(0)=w$ has the form
$${\bf R} \ni t \mapsto g_a \Big( H_{g_a^\prime(0)^{-1} w} \
{\rm artanh}\big( t\ {\rm Id}_{\Omega_{g_a^\prime(0)^{-1} w}} \big) \Big) .$$
\bf Proof. \rm
It is a well-known consequence of
the M\"obius invariance of the connection $\nabla$
that $h\circ \gamma$ is $\nabla$-geodesic whenever $\gamma$ is a $\nabla$-geodesic and
$h\in{\rm Aut}\ {\rm Ball}(Z)$.
The curve in the statement of the Corollary is thus a $\nabla$-geodesic.
It is straightforward to see that
its starting point is $a$ and its starting speed vector is $w$.
The classical Piccard-Lindel\"of Theorem ensures
the uniqueness of the solution of the geodesic equation
with given starting point and starting speed vector. \ Qu.e.d.

\bigskip

The expression for the geodesic in the above can be conveniently rewritten in terms of the power series for $\artanh$. In fact, we have $$
\gamma_{0,v}(t) = \sum_{n=0}^\infty\frac{t^{2n+1}}{2n+1}D(v,v)^{2n}v =\artanh{tv}.
$$
In the same vein,
$$
\gamma_{a,w}(t)=a+
B(a)^{1/2}\big[1 + D(\artanh{tB(a)^{1/2}v},a)\big]^{-1}\artanh{tB(a)^{1/2}v}.
$$

\bigskip
\noindent
\bf 3. \vtop{
\hbox{Local M\"obius transformations in real Jordan-Banach triples}
\hbox{ and Jordan-M\"obius manifolds}} \rm
\bigskip

Throughout this section $\E$ denotes a real Jordan-Banach triple
with the norm $\Vert .\Vert$ and triple product $\{ \ldots\}$, respectively.
Thus we only assume that $\E$ is a real Banach space and $\{\ldots\}$
is a continuous real trilinear mapping $\E^3\to \E$ satisfying the Jordan identity
$({\rm J})$.
We can take over the notations $D(a,b),Q(a,b),B(a,b)$ in the real setting
without formal changes.
In particular,
all the operators $D(a,b)-D(b,a):z\mapsto \{ abz\}-\{ baz\}$
belong to
${\rm Der}(\E,\{\ldots\})$ the set of the
derivations of the triple product.
Though not explicitly stated,
a straightforward inspection
of [\KaupR, Corollary 2.20] establishes
the existence of a constant $\eps>0$
such that
the transformations
$$H_v := \exp V_v \quad \hbox{with the vector fields \ \
$V_v:= \big[ v-\{ zvz\} \big] \partial/\partial z$}$$
are well-defined on the ball
$\eps {\rm Ball}(\E)$ whenever $\Vert v\Vert <\eps$,
moreover they have the fractional linear M\"obius form
$$\eqalign{
&H_v (z) = g_{H_v(0)} (z) \quad\hbox{where}\cr
&g_a (z) := a + \lambda(a) [1+D(z,a)]^{-1} z \qquad
\hbox{where} \ \
\lambda(a) := H_v^\prime(0) \in{\cal L}(\E) \ .\cr}
\leqno(3.1)$$
Notice that in the case of JB*-triples we can write
$\lambda(a)= B(a)^{1/2}$ in termd of the Bergman operator.
Besides the vector fields $V_v$ of polynomial degree 2,
let us introduce also the linear vector fields
$$L_\ell :=
\big[ \ell z \big] \partial / \partial z,
\quad \ell\in {\rm Der}(\E,\{\ldots\}) .$$
For their Poisson commutators
$\big[ f(z)\partial/\partial z,g(z)\partial/\partial z \big] :=
\big( f^\prime(z) g(z) - g^\prime(z) f(z) \big) \partial/\partial z$,
it is straightforward to check that we have
$$[L_\ell , V_v] =V_{\ell v} ,\quad
[L_\ell , L_m] = L_{[\ell,m]} ,\quad
[V_u,V_v] = L_{D(v,u)-D(u,v)} .$$
Therefore the real linear space
$$\big\{ V_v + L_\ell :\ v\in Z,\ \ell\in {\rm Der}(\E,\{\ldots\}) \big\}$$
equipped with the norm
$\Vert V_v+L_\ell \Vert :=
\sup\{ \Vert v-\{ zvz\} + \ell z\Vert :\ \Vert z\Vert \le 1 \}$
is a Banach-Lie algebra with the Poisson commutator.
Thus, according to the Campbell-Hausdorff formula,
for some sufficiently small constant $\delta>0$ we have
$$\big[ \exp (V_u+L_\ell) \exp(V_v+L_m)\big] z =
\big[ \exp (V_{w(u,v,\ell,m)} + L_{\Delta(u,v,\ell,m)} \big] z$$
with two suitable real-analytic mappings
$$\eqalign{
&w: \big[\delta\, {\rm Ball}(\E) \big]^2 \times
\big[\delta\, {\rm Ball}\big({\rm Der}(\E,\{\ldots\})\big)\big]^2 \longrightarrow \E, \cr
&\Delta: \big[\delta\, {\rm Ball}(\E)\big]^2 \times
\big[\delta\, {\rm Ball}\big({\rm Der}(\E,\{\ldots\})\big)\big]^2 \longrightarrow
{\rm Der}(\E,\{\ldots\}) \cr}$$
whenever
$\Vert z\Vert, \Vert u\Vert, \Vert v\Vert, \Vert\ell\Vert, \Vert m\Vert <\delta$.

\bigskip
\noindent
\bf 3.2 Definition. \rm
By a
{\it M\"obius transformation}
of \ $(\E,\{\ldots\})$ \ we mean a bianalytic mapping \ $\Phi:U\to \E$ \
defined in a %connected
neighborhood \ $U$ \ of the origin in \ $\E$ \
such that for some \
$\ell\in{\rm Der}(\E,\{\ldots\})$ \ we have \
$\Phi(z)= g_a (\exp\ell z)$, $z\in U$ \
with the customary notation in \
$g_a(z) := a + \lambda(a) [1-D(z,a)]^{-1}$ \
established in (3.1).

\bigskip
The next two auxiliary results 3.3-4 establish in particular that
composition preserve M\"obius tranformations.


\bigskip
\noindent
\bf 3.3 Proposition. \it
There exists $\delta^\prime >0$ such that
$$\Vert v\Vert,\Vert\ell\Vert < \delta^\prime \ {\rm and} \
[\exp (V_v+L_\ell)]0 =0 \quad \Longrightarrow \ v=0 .$$
\bf Proof. \rm
We can choose $\delta^\prime$ to be so small that
the mapping $u\mapsto [\exp(V_u)]0$ be injective on \
$\delta^\prime\, {\rm Ball}(\E)$ \ and
the terms
$$[\exp (t L_m)] [\exp(V_u+L_m)] [\exp(-tL_m)]z,
\quad \vert t\vert, \Vert z\Vert, \Vert u\Vert, \Vert m\Vert <\delta^\prime$$
be all well-defined.
Consider any couple $v,\ell$ with
$\Vert v\Vert,\Vert\ell\Vert < \delta^\prime$ and $[\exp (V_v+L_\ell)]0 =0$.
Then, for all $-\delta^\prime < t<\delta^\prime$,
$$\eqalign{
0 &= [\exp (t L_\ell)] [\exp(V_v+L_\ell)] [\exp(-tL_\ell)]0 =\cr
&= \Big[ \exp\Big( \big[ \exp (t L_\ell^\#) \big] (V_v+L_\ell) \Big)\Big] 0 =\cr
&= \Big[ \exp \big( V_{\exp t\ell)v} + L_\ell \big)\Big]0 .\cr}$$
%Here we have
%$V_{\exp t\ell)v} + L_\ell = (V_v + L_\ell) +t V_{\ell v} + o(t)$ and hence
%$$0 = \Big[ \exp \big( V_{\exp t\ell)v} + L_\ell \big)\Big]0 =
%\big[ \exp(V_v+L_\ell + t V_{\ell v})\big]0 + o(t) .$$
Notice that the points
$$x_{s,t} :=
\Big[ \exp \big( s [ V_{(\exp t\ell)v} + L_\ell] \big) \Big]0,
\quad \vert s\vert <1+\eps^\prime,\ \vert t\vert <\delta^\prime$$
are well-defined for some $\eps^\prime>0$
and they satisfy the differential equation
$${\partial\over \partial s} x_{s,t}  =
(\exp t\ell)v - \{ x_{s,t} [(\exp t\ell)v] x_{s,t} \} + \ell x_{s,t} .$$
Moreover, since
$\big[ \exp \big( V_{\exp t\ell)v} + L_\ell \big)\big]0 =0$
for $\vert t\vert< \delta^\prime$, it follows
$$\eqalign{
0 &= {\partial^2 \over \partial s \partial t}\Big\vert_{s=1,\ t=0} x_{s,t} =
{\partial^2 \over \partial t \partial s}\Big\vert_{s=1,\ t=0} x_{s,t} =\cr
&= \ell v - 2 \Big\{
\Big( {\partial\over\partial t}\Big\vert_{t=0,\ s=1} x_{s,t} \Big) v x_{s,t} \Big\} +
\ell {\partial\over\partial t}\Big\vert_{t=0,\ s=1} x_{s,t} .\cr}$$
However, here we have
${\partial\over\partial t}\Big\vert_{t=0,\ s=1} x_{s,t} =0$
which implies that necessarily $\ell v =0$.
It follows
$[ L_\ell, V_v] = V_{\ell v} =0$ and therefore
$$0 = [\exp (V_v+L_\ell)]0 = [\exp V_v][\exp L_\ell]0 = [\exp V_v]0 .$$
The choice of $\delta^\prime$ with $\Vert v\Vert< \delta^\prime$
ensures that $v=0$. Qu.e.d.

\bigskip
\noindent
\bf 3.4 Corollary. \it
There exists $0<\delta^{\prime\prime} (<\delta^\prime)$
such that given any $v\in Z$ and $\ell \in{\rm Der}(\E,\{\ldots\})$
with $\Vert v\Vert, \Vert \ell \Vert <\delta^{\prime\prime}$,
for some $m\in{\rm Der}(\E,\{\ldots\})$
we have
$$\big[ \exp (V_v + L_\ell)\big] z =
g_{a(v,\ell)}\big( [ \exp m ] z \big),
\quad \Vert z\Vert <\delta^{\prime\prime}$$
where $a(v,\ell) := \big[ \exp (V_v+L_\ell)\big]0$.
In particular, for sufficiently small vectors $p,q\in \E$,
$$g_p \circ g_q = g_{g_p(q)} \circ [\exp m_{p,q}]$$
with a real-analytic mapping
$(p,q)\mapsto m_{p,q} \in{\rm Der}Z,\{\ldots\})$.


\bigskip
\noindent
\bf 3.5 Definition. \rm
By a {\it Jordan-M\"obius manifold} modeled with \ $(\E,\{\ldots\})$ \
we mean a a real-analytic manifold \ $M$ \
with an {\it inverse atlas} \
${\cal X} =\{ X_p:\ p\in M\}$ \
such that for each point \ $p\in M$ \ we have
$$X_p:U_p\to M ,\quad X_p(0) =p ,\quad
X_p^{-1}\circ X_q\ \hbox{M\"obius transformation} \ \ {\rm for} \ \ q\in X_p(U_p)
\leqno(3.6)$$
where $U_p$ \ is an open connected
neighborhood of the origin in $\E$
the


A Jordan-M\"obius manifold \ $M$ \
with an inverse atlas \ ${\cal X} =\{ X_p:\ p\in M\}$ \
satisfying (3.6) is said to be a
{\it uniform Jordan-M\"obius manifold}
if
there exists a common constant $\eps>0$
such that
$${\rm dom}(X_p \circ g) \supset \eps\,{\rm Ball}(\E) \quad
\hbox{if} \ \ g\in\{\hbox{M\"obius transformations}\} \ \hbox{and} \ \Vert g(0)\Vert < \eps .$$
In particular the region $\bigcup_{L\in{\rm Aut}(\E,\{\ldots\})} \eps L {\rm Ball}(\E)$
is contained in ${\rm dom}(X_p)$ for all $p\in M$.

\bigskip
\noindent
\bf 3.7 Example. \rm
If $\E:=\Z$ is a JB*-triple then its unit ball
$M:={\rm Ball}(\Z)$ is a uniform Jordan manifold with the charts
$X_p := g_p\vert M$, $p\in M$.
In this case we may choose $\eps=1$.

\bigskip
\noindent
\bf 3.8 Example. \rm
Let $(\E,\{\ldots\})$ be any Jordan-Banach triple.
Then we can find a constant $\varrho>0$ such that
the sections of the mapping
$[\varrho {\rm Ball}(\E)]^2 \ni(p,z)\mapsto g_p(z)$
are real-bianalytic for any fixed $p$ and $z$, respectively.
Then the ball
$M:= \varrho {\rm Ball}(\E)$ with the topology from $\E$ and with the charts
$X_{p} (z) := g_p(z)$ defined for
$z\in U_p := \{ u\in \E:\ \Vert u\Vert, \Vert g_p(u) \Vert < \varrho\}$
is a Jordan manifold which is not uniform in general.


\bigskip
\noindent
\bf 3.8a Example. \rm (Special case of 3.8 with Lorenz space).
$E:={\rm Mat}(1,2,\RR)$, $S:=\big( {{1 \ \ \ 0}\atop{0\ -1}} \big)$;
indefinite scalar product $\langle x \vert y \rangle_S := x S y^* \in \RR$ $(x,y\in E)$,
tripe product
$\{ xyz\}_S :=
{1\over 2} \langle x \vert y\rangle_S z +
{1\over 2} \langle z \vert y\rangle_S x =
{1\over 2} xy^*Sz + {1\over 2} zy^*Sx$ on $E$.
Then the local M\"obius transformations have the form
$$M_a (x) \vtop{
\hbox{$=(1-aSa^*)^{-1/2} (x+a) (1+Sa^*x)^{-1} (1-Sa^*a)^{1/2} =$}
\hbox{$=(1-aSa^*)^{-1/2} (1+xSa^*)^{-1} (x+a) (1-Sa^*a)^{1/2} \ .$}}$$
Proof. By definition, $M_a(x)=a+B(a)^{1/2}[1+D(x,a)]^{-1}x$.
Here we have
$$\eqalign{
B(a)z &= z - 2D(a)z +Q(z)^2 = z - aSa^*z - zSa^*a + aSa^*zSa^*a =\cr
&= (1-aSa^*)z(1-Sa^*a) ,\cr
D(x,a)^n x &= (xSa^*)^n x = x (Sa^*x)^n \quad (n=0,1,\ldots) .\cr}$$
Thus $B(a)^{1/2}z = (1-aSa^*)^{1/2}z(1-Sa^*a)^{1/2}$ and
$[1+D(x,a)]^{-1}x =
\sum\limits_{n=0}^\infty (-1)^n D(x,a)^n x =
(1-xSa^*)^{-1} x = x (1-Sa^*x)^{-1}$.
Since also $(1 - aSa^*)^{1/2} a = a (1-Sa^*a)^{1/2}$ as one can see from the Newtonian expansion,
it follows
$$\eqalign{
M_a(x) &= a + (1 - aSa^*)^{1/2} x (1+Sa^*x)^{-1} (1-Sa^*a)^{1/2} =\cr
&= (1 - aSa^*)^{-1/2} a (1-Sa^*a)^{1/2}
 + (1 - aSa^*)^{1/2} x (1+Sa^*x)^{-1} (1-Sa^*a)^{1/2} =\cr
&= (1 - aSa^*)^{-1/2} \big[ a + (1 - aSa^*) x (1+Sa^*x)^{-1} \big] (1-Sa^*a)^{1/2} =\cr
&= (1 - aSa^*)^{-1/2} \big[ a (1+Sa^*x) + (1 - aSa^*) x \big]
(1+Sa^*x)^{-1} (1-Sa^*a)^{1/2} =\cr
&= (1-aSa^*)^{-1/2} (x+a) (1+Sa^*x)^{-1} (1-Sa^*a)^{1/2} .\cr}$$
The proof for the alternative expression of $M_a(x)$ is analogous.

\bigskip
\noindent
\bf 3.9 Example \rm [\Ist,\St]. \
Let \ $(\Z,\{\ldots\})$ \ be any complex JB*-triple such that
the family \
${\rm Tri}(\Z,\{\ldots\}) :=\big\{ e\in \Z:\ \{ eee\} =e\ne 0 \big\}$ \
of its non-trivial tripotents is not void.
Two tripotents $e,f\in {\rm Tri}(\Z,\{\ldots\})$
are said to be {\it equivalent}
($e\sim f$ in notation) if $D(e,e)=D(f,f)$.
Actually $\sim$ is an equivalence relation on ${\rm Tri}(\Z,\{\ldots\})$.
Let $M$ be a connected component of ${\rm Tri}(\Z,\{\ldots\})$
(with respect to the topology inherited from $\Z$).
Then
the Peirce spaces
$Z_{1/2}(e) := \{ u\in Z:\ D(e,e)u=u/2\}$ $(e\in M)$
are all isomorphic and
the set $\MM := \big\{ {\bf e}: \ e\in M\big\}$ \
$({\bf e}:= \{ f\in M:\ f\sim e\})$
of its equivalence classes can be regarded as a
complex hermitian symmetric manifold modeled
on $Z_{1/2}(e_0)$ with any $e_0\in M$
and being such that the automorphisms
$\exp t[ D(e,u)-D(u,e)]$ $\big( t\in\RR, \ u\in Z_{1/2}(e) \big)$
of $\Z$ acting on $\MM$
form a continuous one-parameter subgroup
of the Banach-Lie group ${\rm Aut}(\MM)$ of all biholomorphic
automorphisms of $\MM$ (with Upmeier's topology [\Upmeier]).
Given any $e\in M$, there is a neighborhood ${\bf U}$
of its equivalence class ${\bf e}$ in $\MM$ along with a holomorphic chart map
$\Phi:{\bf U}\to Z_{1/2}(e)$ such that each map
$\Phi^\# \exp [ D(e,u)-D(u,e)]$ $\big( u\in Z_{1/2}(e) \big)$
is a M\"obius transformation with the triple product
$$
\{ u_1 u_2 u_3\} := \left. {\partial^3
\over
\partial \zeta_1 \partial \overline{\zeta_2} \partial \zeta_3}
\right\vert_{\zeta_1=\zeta_2=\zeta_3=0}
\Phi^\# \Big( \Big[  P_{\zeta_1 u_1} ,
\Big[ P_{\zeta_2 u_2} ,
 P_{\zeta_3 u_3} \Big] \Big]_e \Big)$$
in terms of the vector fields
$P_u ({\bf f}) := \big[ t\mapsto \exp t[ D(e,u)-D(u,e)] {\bf f} \big]
\in {\rm T}_{\bf e} \MM$.

\bigskip
\noindent
\bf 3.10 Lemma. \it
Assume $U,V$ are domains in a Banach space $W$ and let
$T:U\leftrightarrow V$ be a smooth diffeomorphism between them.
Then given any couple $X,Y:U\to W$ of smooth vector fields on $U$,
for their transforms
$\widetilde{X} := T^\# X$ respectively
$\widetilde{Y} := T^\# Y$ on $V$
we have
%\footnote{$^*$}{\rm For notation: $^\prime$ means Fr\'echet derivative.
%In particular,
%$T^\prime (u) w \!=\! {d\over dt}\big\vert_{t=0} T( u \!+\! t w)$,
%$T^{\prime\prime}(u) w_1 w_2 =
%{\partial^2 \over \partial t_1 \partial t_2} \big\vert_{t_1=t_2=0} T(u+t_1 w_1+t_2 w_2)$
% \ and \
%$T^\# X (v) = T^\prime \big( T^{-1}(v)\big) X\big( T^{-1}(v)\big)$.}
$$\widetilde{Y}^\prime \widetilde{X} (v) =
T^{\prime\prime} \big(T^{-1}(v)\big) X\big( T^{-1}(v) \big) Y\big( T^{-1}(v) \big) +
T^\prime \big(T^{-1}(v)\big) Y^\prime \big( T^{-1}(v) \big) X\big( T^{-1}(v) \big) ,
 \ \ v\!\in\! V .$$
\bf Proof. \rm
By definition,
$\widetilde{Y}^\prime \widetilde{X} (v) =
{d\over dt}\big\vert_{t=0} \widetilde{Y} \big( v + t \widetilde{X}(v)\big)$.
Therefore
$$\eqalign{
\widetilde{Y}^\prime \widetilde{X} (v) &= {d\over dt}\Big\vert_{t=0}
T^\prime \Big( T^{-1} \big( v + t\widetilde{X}(v) \big)\Big)
Y \Big( T^{-1} \big( v + \widetilde{X}(v) \big)\Big) = \cr
&= \Big[ {d\over dt}\Big\vert_{t=0}
T^\prime \Big( T^{-1} \big( v + t\widetilde{X}(v) \big)\Big) \Big]
Y \big( T^{-1} (v)\big) + \cr
& \ \ + T^\prime \big( T^{-1}(v)\big)
{d\over dt}\Big\vert_{t=0} Y \big( v + t \widetilde{X}(v)\big) =\cr
&= T^{\prime\prime} \Big( T^{-1} (v) \big) \Big[ {d\over dt}\Big\vert_{t=0}
T^{-1} \big( v + t\widetilde{X}(v) \big) \Big] Y\big( T^{-1}(v)\big) +\cr
& \ \ + T^\prime \Big( T^{-1} (v) \big) Y^\prime\big( T^{-1}(v)\big)
{d\over dt}\Big\vert_{t=0} T^{-1} \big( v + t\widetilde{X}(v) \big) =\cr
&= T^{\prime\prime} \Big( T^{-1} (v) \big)
\Big[ T^\prime \big( T^{-1} (v)\big)^{-1} \widetilde{X}(v) \Big]
Y\big( T^{-1}(v)\big) +\cr
& \ \ + T^\prime \Big( T^{-1} (v) \big) Y^\prime \big( T^{-1}(v)\big)
\big[ T^\prime \big( T^{-1}(v)\big)\big]^{-1} \widetilde{X}\big( T^{-1}(v) \big) =\cr
&= T^{\prime\prime} \Big( T^{-1} (v) \big) X\big( T^{-1}(v) \big) Y\big( T^{-1}(v) \big) + \cr
& \ \ + T^\prime \Big( T^{-1} (v) \big) Y^\prime \big( T^{-1}(v) \big) X\big( T^{-1}(v) \big) .
\ \ \hbox{Qu,e.d.}\cr}$$

\bigskip

In the sequel let \ $\big( M,\{ X_p:\ p\in M\}\big)$ \
be a Jordan-M\"obius manifold modeled on $(\E,\{\ldots\})$ \
with inverse charts \
$X_p: \E \supset U_p\to M$ \
satisfying \ $X_p(0)=p$, $p\in M$ \ and with M\"obius transition maps \
$X_p^{-1}\circ X_q$ \
for couples of point lying sufficiently close together.

\def\X{{\bf X}}
\def\Y{{\bf Y}}
\def\R{{\bf R}}

\bigskip
\noindent
\bf 3.11 Proposition,
\it
Let $p\in M$ be an arbitrarily given point, let $\X,\Y$
be two smooth vector fields on $M$ and define $\R := \nabla_{\X} \Y$.
Then, for any vector $w\in U_p$,
the image $R:=[X_p^{-1}]^\# \R$ of the vector field $\R$
by means of the local coordinate $X_p$
can be expressed in terms of the image vector fields
$X:=[X_p^{-1}]^\# \X$ and $Y:=[X_p^{-1}]^\# \Y$ as
$$R(w) = H_w^\prime(w)^{-1} H_w^{\prime\prime}(w) X(w) Y(w) + Y^\prime (w) X(w)
\quad \hbox{\rm where} \ \ H_w := X_{X_p(w)}^{-1} \circ X_p .$$
\bf Proof. \rm
Let $w\in U_p$ be arbitrarily fixed and write $q:=X_p(w)$.
Then, with the mapping $T:= H_w$
we have
$$T(w) = X_q^{-1}\circ X_p(w) = X_q^{-1} (q) =0 .$$
Also in terms of $T$ we can write
$$\eqalign{
[X_q^{-1}]^\# \X &= [X_q^{-1}]^\# X_p^\# [X_p^{-1}]^\# \X =\cr
&= [X_q^{-1}\circ X_p]^\# [X_p^{-1}]^\# \X =\cr &= T^\# X \cr}$$
and similarly
$[X_q^{-1}]^\# \Y = T^\# Y$ and
$[X_q^{-1}]^\# \R = T^\# R$.
Hence
$$\eqalign{
[X_q^{-1}]^\# \R (0) &= \big[ T^\# \big( [X_p^{-1}]^\# \R \big) \big] (0) =\cr
&= T^\prime \big( T^{-1}(0)\big) \big[ [ X_p^{-1}]^\# \R \big] \big( T^{-1}(0)\big) =\cr
&= T^\prime(w) \Big[ \big[ [X_p^{-1}]^\# \R \big] (w) \Big] =\cr &= T^\prime(w) R(w) .\cr}$$
By axiomatic assumption we have \quad
$[X_q^{-1}]^\# \R (0) =
\Big[\big[ [X_q^{-1}]^\# \Y\big]^\prime (0)\Big] \Big[\big[ [X_q^{-1}]^\# \X\big] (0)\big]$.
That is
$$\eqalign{
R(w) &= [X_p^{-1}]^\# \R (w) =\cr
&= T^\prime(w)^{-1}
\Big[ \big[ [X_q^{-1}]^\# \Y\big]^\prime (0) \Big] \Big[ \big[ [X_q^{-1}]^\# \X\big] (0) \Big] =\cr
&= T^\prime(w)^{-1}
\big[ (T^\# Y)^\prime (0)\big]
\big[ T^\# X (0) \big] =\cr
&= T^\prime(w)^{-1} T^{\prime\prime}(w) X(w) Y(w) + Y^\prime(w) X(w) \cr}$$
in view of Lemma 3.7 and the fact that $w=T^{-1}(0)$. \ \ Qu.e.d.

\bigskip
Our technical results 3.3-4, 3.10-11,
establish immediately that the calculations
for the proof of Theorem 2.6 can be carried out locally
even in the setting of general real Jordan-M\"obius manifolds.
As a first straightforward consequence we get the following.

\bigskip
\noindent
\bf 3.10 Theorem. \it
Let $M$ be a Jordan-M\"obius manifold modeled on $(\E,\{\ldots\})$
with a system of inverse charts
$\{ X_p:\ p\in M\}$ having the properties $(3.6)$.
Then there exists a $($necessarily unique$)$ connection $\nabla$
on $M$ such that its Christoffel symbol $\Gamma$ with the charts
$\Phi_p := X_p^{-1}$ satisfies
$$\Gamma_{\Phi_p} (0) =0 \ ,\qquad p\in M .$$
Namely, if $p\in M$ is any point and
the constant $\delta>0$ is so chosen that
$\delta\,{\rm Ball}(\E) \subset {\rm dom}(X_p)$ then
for any couple of vectors $x,y\in\E$ and any point $a\in \delta\,{\rm Ball}(\E)$ we have
$$\eqalign{
\Gamma_{\Phi_p} (a) (x,y) &= g_a''(0)(g_{-a}'(0)x,g_{-a}'(0)y) =\cr
&= 2 \lambda(a) \big\{ [\lambda(a)^{-1}x]a[\lambda(a)^{-1}y] \big\} \ .\cr}$$
In particular \ $\Gamma_{\Phi_p} (a) (x,y) =
2B(a)^{1/2}\{[B(a)^{-1/2}x]a[B(a)^{-1/2}y]\}$ \
in the case of \ $(\E,\{\ldots\})$ \ being a complex JB*-triple.
\rm

\bigskip
In the above theorem we could not include a statement about symmetry invariant
connections because, unlike the unit ball of a complex JB*-triple,
Jordan-M\"obius manifolds need not be necessarily symmetric.
We close this section by showing that the assumption of uniformness implies this property.

\bigskip
\noindent
\bf 3.11 Theorem. \it Connected uniform Jordan-M\"obius manifolds are symmetric
and they admit a unique symmetry invariant connection.

\bigskip
\noindent
\bf Proof. \rm
SKETCHED

Let $M$ be a connected uniform Jordan-M\"obius manifold modeled with $(\E,\{\ldots\})$
along with a constant $\eps>0$ and a system
${\cal X} =\{ X_p:\ p\in M\}$
as in Definition 3.5.
By Corollary 3.3,
there exists $\delta\in(0,\eps)$
such that
$g_a\big(\delta\,{\rm Ball}(\E)\big) \subset \eps{\rm Ball}(\E)$
for any vector $a\in \delta\,{\rm Ball}(\E)$.
Observe that, given any couple of points $p,q\in M$,
we can find a finite sequence $v_1,\ldots,v_N \in\delta\,{\rm Ball}(\E)$
such that the recursively defined sequence
$$p_0:=p, \quad p_{n+1} := X_{p_n} (v_n)$$
ends in $q=p_N$.
Let us fix any point $p\in M$.
We can see by induction that
there exists a sequence $q_0=p,q_1,\ldots,q_N\in M$ of points
along with (linear) automorphisms
$L_0={\rm Id}_\E , L_1,\ldots,L_N \in{\rm Aut}(\E,\{\ldots\})$
such that for the modified charts
$$Y_n := X_{q_n} \circ L_n ,\qquad n=1,\ldots n=0,\ldots,N$$
we have
$$q_{n+1} = Y_{n} (-v_n), \quad Y_{n+1}^{-1} \circ Y_{n} =
X_{q_{n}}^{-1} \circ X_{q_{n+1}} .$$
In view of 3,3-4, if we have another sequence
$\widetilde{v}_1,\ldots,\widetilde{v}_{\widetilde{N}}$
and consider the corresponding points
$\widetilde{p}_1,\ldots,\widetilde{p}_{\widetilde{N}}$
respectively
$\widetilde{q}_1,\ldots,\widetilde{q}_{\widetilde{N}}$
with the above construction
then the coincidence $p_N = \widetilde{p}_{\widetilde{N}}$
of the endpoints
implies the coincidence $q_N = \widetilde{q}_{\widetilde{N}}$
as well.
It is not hard to check that
the thus well-defined transformation
$S_p: p_N \mapsto q_N$ is a symmetry through the point $p$
such that the maps
$X_r^{-1} \circ S \circ X_p$ $(r\in M)$ are M\"obius transformations.
Finally we notice that $S_p\circ S_q$ is a M\"obius transformation if
its domain contains the origin of $\E$.
Hence the statement of the theorem is immediate.
Qu.e.d.

\bigskip
\bigskip
\noindent
\bf 4. Jordan manifolds
\rm
\bigskip
\noindent
4.1 Definition. \rm
A manifold $M$ modeled on a (real) Jordan-Banach triple $(\E, \{\ldots\})$
is a {\it Jordan manifold} if its tangent spaces $T_p M$ $(p\in M)$
are endowed with a
triple product $\{ \ldots \}$ being isomorphic to $\{\ldots\}$ on $\E$
and
the mapping $(p,u,v,w)\mapsto \{ uvw\}_p$ is continuously differentiable.
In the sequel we shall write
$(M,\overline{{\cal A}},{\cal P})$ for the triple of the carrier space, atlas and system
of triple products.
A morphism $F:M\to \widetilde{M}$ between two Jordan manifolds is a smooth mapping
such that
its derivatives give rise to triple product homomorphisms on the tangent spaces.

Given two Jordan manifolds
$(M,\overline{{\cal A}},{\cal P})$ and $(N,\overline{{\cal B}},{\cal Q})$,
\hfill\break
a continuously differentiable map $S:M\to N$ is a {\it Jordan morphism} if
\hfill\break
$\forall\ p\in M \quad \forall\ u,v,w\in T_p(M) \qquad
S^\prime(p) \{ uvw \}_p = \{ [S^\prime(p)u] [S^\prime(p)v] [S^\prime(p)w] \}_{S(p)}$.
\hfill\break
Isomorphism and automorphisms can be defined as usually.
\hfill\break
${\rm Aut}(M) := \{ S : M\leftrightarrow M$ with $S, S^{-1}$ Jordan morphisms$\}$.

\medskip
\item{(D3)} Given
$(M,\overline{{\cal A}},{\cal P})$ and $(N,\overline{{\cal B}},{\cal Q})$
with $U\ \hbox{open}\subset M$ and $V\ \hbox{open}\subset N$,
\hfill\break
a continuously differentiable map $S:U\leftrightarrow V$ is a {\it local Jordan automorphism} if
\hfill\break
$S$ is a Jordan isomorphism between $U$ and $V$ as Jordan submanifolds.
\hfill\break
${\rm Aut}_{{\rm loc}} (M) := \{$local Jordan automorphism in $M\}$.
\hfill\break
Notice that $S_1\circ S_2 \in {\rm Aut}_{{\rm loc}} (M)$ whenever
$S_1,S_2 \in {\rm Aut}_{{\rm loc}} (M)$ with ${\rm ran}(S_1) \cap {\rm dom}(S_2) \ne\emptyset$.

\bigskip
\bigskip
\noindent
\bf Convention. \rm
Henceforth $(M,\overline{{\cal A}},{\cal P})$ stands for a Jordan manifold.

\bigskip
\bigskip
\noindent
\bf Example. \rm (E0) Riemann spaces can be regarded as Jordan manifolds:
\hfill\break
If $\langle .\vert .\rangle_p$ is the inner product on $T_p M$ then
we take
\hfill\break
$\{ uvw\}_p :=
{1\over 2} \langle u\vert v\rangle_p w + {1\over 2} \langle w\vert v\rangle_p u$.


\medskip
\noindent
\bf Example. \rm
The triple products in ${\cal P}$ need not be isomorphic to each other:
$$\hbox{$M:=\CC$ \ \ with \ \ $\{ uvw\}_p := {\rm Re}(p) u\overline{v} w$.}
\leqno{\rm (E1)}$$

\noindent
\bf Conjecture. \rm (HARD) \it

If we assume all the $\{\ldots\}_p$ to be
$JB^*$-triple products then they are isomorphic.


\bigskip
\bigskip
\noindent
\bf Lemma. \it If for every couple of points $p,q\in M$ there are neighborhoods
$U,V\subset M$
along with a diffeomorphism
$S:U \leftrightarrow V$
such that
$q=S(p)$ and
$S^\prime(p)$ is an isomorphism between $\{\ldots\}_p$ and $\{\ldots\}_q$
then there is an atlas ${\cal A} \subset \overline{{\cal A}}$ with the properties {\rm (N4)}.

\bigskip
\noindent
\bf Proof. \rm
Fix any $o\in M$ and a local coordinate $X\in \overline{\cal A}$
with $X(o)=0$.
For any $p\in M$ choose a local diffeomorhism
$S_p$ with $S_p(o)=p$ and the properties in (D2).
We redefine the triple product $\{\ldots\}$ on $Z$
as the image of $\{\ldots\}_o$ by $X^\prime(0)$ and we set
$X_p := X\circ S_p^{-1}$ $(p\in M)$.


\bigskip
\noindent
\bf Definition. \rm

\item{(D4)} A Jordan manifold $M$ is {\it homogeneous} if
\hfill\break
${\rm Aut}(M)$ is transitive on it:
$\forall\ p,q\in M \quad \exists\ S\in{\rm Aut}(M) \qquad S(p)=q$.

\medskip
\item{(D5)} A Jordan manifold is {\it locally homogeneous} if
\hfill\break
${\rm Aut}_{{\rm loc}}(M)$ is transitive on it.

\bigskip
\bigskip
\noindent
\bf Examples. \rm

\item{(E2)} [Corach-Porta-Recht].
\hfill\break
$M:={\bf A}^+$ the positive invertible elements
of a unital $C^*$-algebra ${\bf A}$.
\hfill\break
$E:= {\bf A}$ with $\{ xyz\} := {1\over 2} xy^*z + {1\over 2} zy^*y$ as usually.
\hfill\break
$U_a := M$ and $X_a(b) := b-a$  for all $a,b\in M$.
\hfill\break
$\{ uvw\}_a = \{ uvw\}$ \quad $(a\in M; \ u,v,w \in T_a M \equiv E)$.
\hfill\break
\it Then, for any invertible $g\in{\bf A}$ we have
$L_g := [a\mapsto (g^*)^{-1} a g^{-1}] \in {\rm Aut}(M)$.
\hfill\break
In particular $M$ is homogeneous because $\{ L_g(1):\ e\in M\} = \{ g^{-2}: e\in M\} =M$.
\rm
\hfill\break
\item{(E2$^\prime$)} [Corach-Porta-Recht$_2$].
\hfill\break
${\bf A}$ $C^*$-algebra with unit $1$,
\hfill\break
$E:=\{ a\in A:\ a=a^*\}$,
\quad $\{ xyz \} := {1\over 2} xyz + {1\over 2} zyx$
\hfill\break
${\cal M} := \{ a\in A :\ a > 0\} = \{ g^*g : \ g$ invertible$\}$
\hfill\break
$L_g: {\cal M} \leftrightarrow {\cal M}, \ x\mapsto g^* x g$ \ linear
\hfill\break
$S: p\mapsto p^{-1}$ \ \ symmetry on ${\cal M}$
\hfill\break
$S_{g^*g} := L_g \circ S \circ L_{g^{-1}} : p \mapsto g^* g p^{-1} g^*g$
\hfill\break
We can define
$$\{ uvu\}_{g^*g} : \vtop{
\hbox{$= L_g \{ (L^{-1}_g u) (L^{-1}_g u) (L^{-1}_g u) \} =$}
\hbox{$= u (g^* g)^{-1} v (g^*g)^{-1} u$.}}$$
{\bf Lemma.} {\it The transformations $L_g,S_p$ are ${\cal P}$-automorphisms.}
\hfill\break
$\ $
\hfill\break
Define
$X_v:= {d\over d\tau} \big\vert_{\tau=0+} S_{\exp(\tau v/2)} \circ S_1$.
\hfill\break
Notice that \ $\exp(X_v) p = \exp(v/2)p\exp(v/2)$.
\hfill\break
$\ $
\hfill\break
{\bf Proposition.} {\it In the chart \ \
$Y: \big[\exp (v-\{ xvx\}) {\partial\over\partial x}\big] 0
\mapsto
[\exp X_v] 1$
\hfill\break
the transformations $\exp(X_w)$ $(w\in E)$
are not in general $\{ \ldots \}$-M\"obius type.}
\hfill\break
\bf Proof. \rm
We would have $Y: {\rm tanh} \,v \mapsto \exp v$
that is
$Y(x) = \exp\big( {\rm areath}\, x \big)$
and
$Y^{-1}(y) = {\rm tanh} \, \log y$.
This implies
$$\eqalign{
Y^{-1} \circ [\exp X_w] \circ Y :
x &\buildrel{Y}\over\mapsto\cr
&\exp({\rm areath}\, x)
\buildrel{\exp X_w}\over\mapsto\cr
&\exp(w/2) \big[ \exp({\rm areath}\, x) \big] \exp(w/2)
\buildrel{Y^{-1}}\over\mapsto\cr
&{\rm tanh}\, \Big[ \log \Big(\exp(w/2)
\big[ \exp({\rm areath}\, x) \big] \exp(w/2)\Big)\Big] .\cr}$$
Such mappings are of M\"obius type only in the commutative case.


\medskip
\item{(E3)} [Chu-Isidro-Kaup-Stach\'o]. A case with equivalence classes of tripotents.
\hfill\break
${\bf H}$ complex Hilbert space with inner product $\langle .\vert .\rangle$. \
${\bf S}:=$[unit sphere].
\hfill\break
Triple product
$\{ xyz\} := {1\over 2} \langle x\vert y\rangle z + {1\over 2} \langle z\vert y\rangle x$
\ $(x,y,z\in {\bf H})$.
\hfill\break
$E := e_0^\perp= \{ z\in{\bf H}: \ \langle z\vert e_0\rangle=0$ with a fixed $e_0\in{\bf S}$.
\hfill\break
$M:= \{ \TT e: \ e\in{\bf H}, \ \langle e\vert e\rangle=1\}$ with
$\TT := \{ \zeta\in\CC:\ \vert\zeta\vert=1\}$.
\hfill\break
For $e\in {\bf S}$, let us write $e^\sim := \TT e$.
\hfill\break
$T_{e^\sim} M \equiv e^\perp = \{ z\in {\bf H}: \ \langle z\vert e\rangle=0\}$.
\hfill\break
$\{ uvw\}_{e^\sim} := - \{ uvw\} \in T_{e^\sim} M$ for $u,v,w\in T_{e^\sim} M$.
\hfill\break
Given $e\in{\bf S}$, let ${\bf R}_e$ be a rotation of ${\bf H}$ with ${\bf R}_e(e)=e_0$.
\hfill\break
$\Phi_{e^\sim}^{-1}: e\perp z\mapsto [\CC {\bf R}_e(e+z)]\cap {\bf S}$ \ \ $(e\in {\bf S})$.
\hfill\break
\it Then ${\cal X} := \{ \Phi_{e^\sim} : \ e^\sim \in M\} = \{ \Phi_{e^\sim} : \ e \in {\bf S}\}$
is an atlas on $M$
\hfill\break
with holomorphic coordinate transitions.
\hfill\break
${\cal G} ^* \subset {\rm Aut}_{{\rm loc}}(M) \cap \{ {\cal X}$-holomorphic maps$\}$
is transitive on $M$.
\hfill\break
${\rm Aut}(M) \cap \{ {\cal X}$-holomorphic maps$\} =\{ {\rm Identity}\}$.


\bigskip
\noindent
\bf Case (E2) \rm

\noindent
$Z := \{ z\in{\bf A} : \ z^* =z\}$ self-adjoint elements.

\noindent
$\{ xyz \} := {1\over 2} xyz + {1\over 2} zyx$ \quad
$(x,y,z\in Z)$.

\noindent
$G := \{ g\in Z:\ g\ \hbox{invertible}\}$, \quad
$M = {\bf A}^+ = \{ g^*g:\ g\in G\}$.

\noindent
$\sigma: a\mapsto a^{-1}$ symmetry on $M$ with fixed point $1$.

\noindent
$L_g(a) := g^* a g$ \quad $(a\in M,\ g\in G)$; \quad
$L_g^{-1}=L_{g^{-1}}$, \ $L_g:M\leftrightarrow M, \ 1\mapsto g^* g$.

\noindent
\bf Definition. \rm
$\sigma_a := L_{a^{1/2}} \circ \sigma \circ L_{a^{-1/2}}$ \quad symmetry at $e\in M$.

\noindent
\bf Remark. \rm
$a = g^*g \Rightarrow L_g \circ \sigma \circ L_{a^{-1}} = \sigma_a$ \qquad $(g\in G)$.

\noindent
\bf Definition. \rm
Infinitesimal translation on $M$ with vector $h\in Z \equiv T_1 M$ at $1$:
$$X_h (a) := {d\over d\tau}\Big\vert_{\tau=0} \sigma_{1+(\tau/2) h} \circ \sigma_1 (a)
\qquad (a\in M).$$
\bf Lemma. \it We have
$${\rm 1)}\ X_h(a) = {1\over 2} (ha+ah) = \{ h1a\},
\qquad
{\rm 2)}\ X_h (a) = {d\over d\tau}\Big\vert_{\tau=0} \sigma_{\exp(\tau h/2)} \circ \sigma (a) .$$
\bf Proof. \rm
1) $\sigma_{1+(\tau h/2} \circ \sigma_1 (a) =
\hfill\break
= \sigma_{1+\tau h/2} (a^{-1}) =
(1+\tau h/2)^{1/2} \big[ (1+\tau h/2)^{-1/2} a^{-1} (1+\tau h/2)^{-1/2} \big]^{-1}
(1+\tau h/2)^{1/2} =
\hfill\break
= (1+\tau h/2) a (1+\tau h/2)$.

\noindent
2) $\exp(\tau h/2) = \sum\limits_{n=0}^\infty {1\over 2^n n!} \tau^n h^n =
1 + \tau h/2 + o(\tau)$.

\bigskip
\noindent
\bf Lemma. \it
$\exp X_h (a) = [\exp (h/2)] a [\exp  (h/2)]$.

\bigskip
\noindent
\bf Proof. \rm
The left and right multiplications
${\rm Mult}_\ell^h: {\bf A}\ni x\mapsto hx$ and
${\rm Mult}_r^h: {\bf A}\ni x\mapsto hx$ commute.
Observe that
$H_h = {1\over 2} [{\rm Mult}_\ell^h + {\rm Mult}_r^h]$.
Therefore
$\exp X_{2h} (a) =
\hfill\break
= \exp [{\rm Mult}_\ell^h + {\rm Mult}_r^h] a =
\exp [{\rm Mult}_\ell^h] \exp [{\rm Mult}_r^h] a =
[{\rm Mult}_\ell^{\exp h}] [{\rm Mult}_\ell^{\exp h}] a$.
% = [\exp h] a [\exp h]$.

\bigskip
\noindent
\bf Remark. \rm Since all elements from $M={\bf A}^+$ admit a spectral resolution
over some compact interval $0<[\alpha,\beta]<\infty$,
each element $a\in M={\bf A}^+$ can be written as $a=\exp h$ with a {\it unique}
self-adjoint $h\in Z$ such that $(\log\alpha) \cdot 1 \le h \le (\log \beta)\cdot 1$.

\bigskip
\noindent
\bf Definition. \rm
We write
$${\rm Exp}_Z (h) := \exp(h) \quad (h\in Z), \qquad
{\rm Log}_M a := {\rm Exp}_Z^{-1} (a) \quad (a\in M).$$
Let $h\in Z$ and set $a:=\exp(h) = {\rm Exp}_Z(h) = \exp(X_h)1$.
We call the mapping
$$M_a := \exp(X_h) = \exp X_{{\rm Log}_M a} : M \ni x \mapsto a x a$$
the
{\it pseudo-M\"obius transformation}
of the point $a$.
Natural atlas on $M$:
$$\Phi_a := M_a \circ {\rm Log}_M \circ M_a^{-1} \qquad (a\in M).$$
\bf Remark. \rm $M_a^{-1} = M_{a^{-1}}$.

\medskip
\noindent
Our next aim is to calculate the image
$\Phi_1^\# X_h : Z\to Z$
of the vector field in the atlas page of $\Phi_1$.
By definition,
$$\eqalign{
\big[ \Phi_1^\# X_h \big] (z) &:= \big[ \Phi_1^\prime \big( \Phi_1^{-1} (z)\big) \big]
X_h\big( \Phi_1^{-1}(z)\big) =\cr
&= {d\over d\tau}\Big\vert_{\tau=0}
\Big[ \Phi_1^\# \big[ \sigma_{\exp(\tau h/2)} \circ \sigma \big] \Big] (z) = \cr
&= {d\over d\tau}\Big\vert_{\tau=0}
\Big[ \Phi_1^\# \sigma_{\exp(\tau h/2)} \Big] \circ \Big[\Phi_1^\# \sigma \Big] (z) .\cr}$$
Notice that $\Phi_1^\# \sigma (z) =-z$. Therefore
$$\eqalign{
\Big[ \Phi_1^\# X_h \Big](z) &=
{d\over d\tau}\Big\vert_{\tau=0}
\Big[ \Phi_1^\# \sigma_{\exp(\tau h/2)} \Big] (-z) =\cr
&= {d\over d\tau}\Big\vert_{\tau=0}
{\rm Log}_M
\Big( \exp\big( {1\over 2} \tau h\big)
\exp\big(z\big)
\exp\big( {1\over 2} \tau h\big) \Big) .\cr}$$
Let us fix $z,h\in Z$ and define
$$w(\tau) :=
{\rm Log}_M
\Big( \exp\big( {1\over 2} \tau h\big)
\exp\big(z\big)
\exp\big( {1\over 2} \tau h\big) \Big) .$$
Since $w(0)=z$ trivially, we can write
$w(\tau) = z  + \tau b + o(\tau)$ with a unique vector $b\in Z$.
In terms of $b$, we have
$${d\over d\tau}\Big\vert_{\tau=0} \exp\big( w(\tau)\big) =
{d\over d\tau}\Big\vert_{\tau=0}
\exp\big( {1\over 2} \tau h\big)
\exp\big(z\big)
\exp\big( {1\over 2} \tau h\big) .$$
On the right hand side we simply have
$${d\over d\tau}\Big\vert_{\tau=0}
\exp\big( {1\over 2} \tau h\big)
\exp\big(z\big)
\exp\big( {1\over 2} \tau h\big) =
{1\over 2} h \exp(z) + {1\over 2} \exp(z) h \ .$$
On the left hand side the derivation can be carried out
with the power series of the exponential map termwise.
That is
$$\eqalign{
{d\over d\tau}\Big\vert_{\tau=0} \exp\big( w(\tau)\big) &=
\sum_{n=1}^\infty {1\over n!}
{d\over d\tau}\Big\vert_{\tau=0} w(\tau)^n =
\sum_{n=1}^\infty {1\over n!}
\sum_{k=1}^n z^{k-1} b z^{n-k} =\cr
&= \sum_{k,\ell=0}^\infty {1\over (k+\ell+1)!}
\big[ {\rm Mult}_\ell^{z} \big]^k \big[ {\rm Mult}_r^{z} \big]^\ell b \ .\cr}$$






\bigskip
\bigskip
\noindent
\bf References
\bigskip
\rm

\item{[\Ist]} J.-M. Isidro - L.L. Stach\'o,
\it On the manifold of complemented
principal inner ideals in JB$^*$-triples,
\rm Quart. J. Math. (Oxford) \bf 57, \rm 505-525 {2006}.


\smallskip
\item{[\KaupM]} W. Kaup,
\it Algebraic characterization of symmetric complex Banach manifolds,
\rm Math. Ann. \bf 228, \rm 39-64 (1977).

\smallskip
\item{[\KaupR]} W. Kaup,
\it A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces,
\rm Math Z. \bf 183, \rm 503-529 (1983).

\smallskip
\item{[\KaupS]} W. Kaup -- J. Sauter,
\it Boundary structure of bounded symmetric domains,
\rm Manuscripta Math. \bf 101, \rm 351-360 (2000).

\smallskip
\item{[\KaupS]} W. Klingenberg,
A course in differential geometry;
Graduate texts in mathematics Vol. 51,
Springer, Berlin-Heidelberg-New York, 1978.

\smallskip
\item{[\Sauter]} J. Sauter,
Randstrukturen beschr\"ankter symmetrischer Gebiete, Dissertation,
T\"ub\-ingen, 1995.

\smallskip
\item{[\St]} L.L. Stach\'o,
\it On the manifold of tripotents in JB*-triples,
\rm in: D. Andrica - S. Morianu (Editors), Contemporary Geometry and
Topology and Related Topics,
The 8th Worksop on Differential Geometry and its Applications,
Cluj-Napoca, Aug. 19-25, 2007, Cluj University Press, pp. 351-264
(2008).

\smallskip
\item{[\Upmeier]} H. Upmeier,
Symmetric Banach manifolds and Jordan $C^*$-algebras.
North-Holland Mathematics Studies, 104. Notas de Matem�tica 96.
North-Holland Publishing Co., Amsterdam, 1985.


\smallskip
\item{[\Werner]} W. Werner, \it ??????
\rm Cluj-Napoca (2007).








\bye
