The local adaptive Galerkin bases for large-dimensional dynamical systems, whose long-time behavior is confined to a finite-dimensional manifold, are optimal bases chosen by a local version of a singular decomposition analysis. These bases are picked out by choosing directions of maximum bending of the manifold restricted to a ball of radius epsilon. We show their geometrical meaning by analyzing the eigenvalues of a certain self-adjoint operator. The eigenvalues scale according to the information they carry, the ones that scale as epsilon(2) have a common factor that depends only on the dimension of the manifold, the ones that scale as epsilon(4) give the different curvatures of the manifold, the ones that scale as epsilon(6) give the third invariants, as the torsion for curves, and so on. In this way we obtain a decomposition of phase space into orthogonal spaces E-m, where E-m is spanned by the eigenvectors whose corresponding eigenvalues scale as epsilon(m). This decomposition is analogous to the Frenet frames for curves. We also discover a practical way to compute the dimension and focal structure of the invariant manifold.