FLRW embeddings in $\mathbb{R}^{n+2}$, differential geometry and conformal photon propagator

Abstract

This paper introduces differential-geometric methods to study $n$-dimensional locally conformally flat spaces as submanifolds in $\mathbb{R}^{n+2}$. We derive explicit formulas relating intrinsic and ambient differential-geometric objects, including curvature tensors, the codifferential and laplacian operators. We apply this approach to Friedmann-Lema\^itre-Robertson-Walker (FLRW) spaces using newfound embedding formulas, obtaining new and simplified expressions for the photon propagator in four dimensions.