Affine connection complexes (Q1973299)

The author abstracts the properties of exterior forms \(\{\bigwedge^\star {\mathcal E}, d\}\) where \({\mathcal E}^\star\) is the tensor algebra generated by the dual \(C^\infty (M)\)-module \({\mathcal E}\) of 1-forms on a differentiable manifold \(M\), to obtain geometric properties. In his words: ``An affine connection is traditionally regarded as a provider of covariant derivatives of smooth vector fields; it also happens to induce covariant derivatives'' on \({\mathcal E}\). ``By changing the emphasis one finds that \dots any classical torsion-free affine connection is equivalent to a connection complex \(\{\bigotimes^\star{\mathcal E},\bigtriangledown\}\) \dots with a natural projection to a de Rham complex. The compositions \(\bigtriangledown \circ \bigtriangledown\), slightly modified, induce \dots curvature homomorphisms, and the identities \(d \circ d = 0\) provide basic Bianchi identities.'' He then applies his results to Riemannian manifolds, obtaining Levi-Civita connection complexes. This is a carefully written and very readable paper. It is a build-up on the author's earlier work [mainly on Math. Ann. 170, 221-244 (1967; Zbl 0154.03704) and Math. Ann. 175, 146-158 (1968; Zbl 0156.04302)].