Invariants of locally conformally flat manifolds

Abstract:

Let $M$ be a locally conformally flat manifold with metric $g$. Choose a local coordinate system on $M$ so $g = {e^{2h}}x dx \circ dx$ where $dx \circ dx$ is the Euclidean standard metric. A polynomial $P$ in the derivatives of $h$ with coefficients depending smoothly on $h$ is a local invariant for locally conformally flat structures if the expression $P({h_X})$ is independent of the choice of $X$. Form valued local invariants are defined similarly. In this paper, we study the properties of the associated de Rham complex. We show that any invariant form can be obtained from the previously studied local invariants of Riemannian structures by restriction. We show the cohomology of the de Rham complex of local invariants is trivial. We also obtain the following characterization of the Euler class. Suppose that for an invariant polynomial $P$, the integral $\int _{{T^m}} {P|d{v_g}|}$ vanishes for any locally conformally flat metric $g$ on the torus ${T^m}$. Then up to the divergence of an invariantly defined one form, the polynomial $P$ is a constant multiple of the Euler integrand.