Abstract. Given a complex Hilbert space $X$ and the von Nuemann algebra ${\cal L}(X)$, we study the Riemannian geometry of the manifold ${\cal P}(X)$ consisting of all minimal projections in ${\cal L}(X)$. To do it we take the Jordan--Banach triple approach (briefly, the JB$^*$-triple approach) because this setting provides a unifying framework for many other situations and simplifies the study previously made by other authors. We then apply this method to study the differential geometry of the manifold of minimal partial isometries in ${\cal L}(H, K)$, the space of bounded linear operators between the complex Hilbert spaces $H$ and $K$ with $\dim H \leq\dim K$.

AMS Subject Classification (1991): 17C36, 53C22

Keyword(s): Partial isometries, JB*-triples, Affine connections, Geodesics, Riemannian distance

Receved July 5, 1999, and in revised form November 12, 1999. (Registered under 2768/2009.)