A new construction of the Riemannian deformation sequence

Abstract

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 $\mathcal AM\to M$, 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 $(\mathcal OM,\omega)$, 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 $\omega$ (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.