Let A be a C * -algebra. In this paper the sets I of partial isometries and I ∆ ⊂ I of partial unitaries (i.e., partial isometries which commute with their adjoints) are studied from a differential geometric point of view. These sets are complemented submanifolds of A. Special attention is paid to geodesic curves. The space I is a homogeneous reductive space of the group U A × U A , where U A denotes the unitary group of A, and geodesics are computed in a standard fashion. Here we study the

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... lem of the existence and uniqueness of geodesics joining two given endpoints. The space I ∆ is not homogeneous, and therefore a completely different treatment is given. A principal bundle with base space I ∆ is introduced, and a natural connection in it defined. Additional data, namely certain translating maps, enable one to produce a linear connection in I ∆ , whose geodesics are characterized.