A decomposition of a Banach space into a finite direct sum of subspaces can be represented by its projector frame, which is the ordered set of the corresponding linear projections. The authors consider the space \({\mathcal E}^ n({\mathcal X})\) of all n-frame projectors on a Banach space \({\mathcal X}\). This space can be given the structure of a differentiable manifold with an affine connection. It is proved that interpolation of the classical balanced transformation is close (to cubic order) to a geodesic path in the manifold structure. When a Hilbert space structure is available on the underlying space, the authors obtain a convenient differential equation for the Riemannian geodesics.