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ABSTRACT.
By a Jordan manifold modeled with a Jordan triple
$(Z,\{\ldots\})$
we mean a
real-analytic manifold $M$ with an atlas $
{\cal A} =\{ X_p:\ p\in M\}$
such that for each point $p\in M$
we have $X_p:U_p\to M$ with $X_p(0) =p$
where $U_p$ is an open neighborhood of the origin in $Z$
and the chart transition maps
$X_p^{-1}\circ X_q$
are M\"obius transformations in the sense of the
underlying triple product $\{\ldots\}$
whenever $q\in X_p(U_p)$.
The category of Jordan manifolds includes
all symmetric Hermitian Banach manifolds,
in particular the unit balls of JB*-triples.
We show some fundamental examples of Jordan manifolds and
describe in explicit algebraic terms a
natural Moebius invariant connection along with its geodesics.



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