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We describe the affine connections, geodesics and symmetries of various Banach manifolds of tripotents in JB*-triples which include the C*-algebras and Hilbert spaces where the nonzero tripotents are respectively the partial isometries and the extreme points of the closed unit ball. Classification (1991): 46G20, 46L70, 58B20, 17C65 Mathematics Subject

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Given a complex Hilbert space H and the von Neumann algebra L(H) of all bounded linear operators in H, we study the Grassmann manifold M of all projections in L(H) that have a fixed finite rank r. To do it we take the Jordan-Banach triple (or JB * -triple) approach which allows us to define a natural Levi-Civita connection on M by using algebraic tools. We identify the geodesics and the Riemann distance and establish some properties of M .

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