$L^p$ almost conformal isometries of Sub-Semi-Riemannian metrics and solvability of a Ricci equation

Abstract

Let M be a smooth compact manifold without boundary. We consider two smooth Sub-Semi-Riemannian metrics on M . Under suitable conditions, we show that they are almost conformally isometric in an L p sense. Assume also that M carries a Riemannian metric with parallel Ricci curvature. Then an equation of Ricci type is solvable in a specific sense, without assuming any proximity to a special metric.Lemma 1.1. Assume that G and G are two smooth SSR-metrics on M with equal signature. Let g be a smooth Riemannian metric on M , let p ∈ [1, ∞) and let ε > 0. Then there exist a smooth diffemorphism Φ and a smooth positive functionBefore moving on to an application to the Ricci equation, let us introduce some notation. Given a smooth Riemannian manifold (M, g), we let Ric(g) denote its Ricci curvature. Let Λ be a real constant. We consider the operator Ein(g) := Ric(g) + Λg.