On the manifold of tripotents in JB∗-triples

The manifold of tripotents in an arbitrary JB*-triple Z is considered, a natural affine connection is defined on it in terms of the Peirce projections of Z, and a precise description of its geodesics is given. Regarding this manifold as a fiber space by Neher's equivalence, the base space is a symmetric Kähler manifold when Z is a classical Cartan factor, and necessary and sufficient conditions are established for connected components of the manifold to admit a Riemann structure.