2026/07/29 by Andreas Cap
Mathematics · #math.DG #msc:53B20 #msc:53C07 #msc:58H15 #msc:58J10 #msc:58J60
25 pages, comments are welcome
arxiv created 2026/07/29 · arxiv updated 2026/07/30
We obtain a new construction of a sequence of invariant differential operators on a Riemannian manifold (M,g) that governs the linearized deformation theory of g. Starting from an explicit linear connection on a natural bundle \mathcal AM→ M, we construct a twisted de Rham sequence and then apply an analog of the construction of BGG sequences. If g has constant sectional curvature, both sequences are complexes which compute the cohomology of the sheaf of local Killing fields, which are equivalent to parallel sections of \mathcal AM. In a second step, we relate the construction to the description of (M,g) as a (torsion-free) Cartan geometry (\mathcal OM,ω), where \mathcal OM is the orthonormal frame bundle of M. This provides a manifest relation of the twisted de Rham sequence to the deformation theory of the Cartan connection ω (which is easier do deal with than the deformation theory of g). The BGG-like construction can then be nicely viewed as interpreting the linearized deformation theory of torsion free Cartan geometries in terms of the underlying Riemannian metric.