Warped Product Einstein Manifolds in Four Dimensions

Abstract

On four-dimensional (pseudo-)Riemannian manifolds $\mathcal{M}$ the curvature tensor (viewed as an endomorphism on 2-forms) admits a chiral $6 \times 6$ matrix representation which decomposes into four $3 \times 3$ blocks. $\mathcal{M}$ is Einstein if and only if the off-diagonal blocks vanish. If the manifold is a warped product $\mathcal{M} = F \times_f B$, then there exists an alternative matrix representation relative to the decomposition of the 2-forms into spaces induced by the exterior algebra on both the base and the fiber. These two representations are not independent and a similarity transformation can be found between them. We construct these matrices and associated transformations for $1+3$, $2+2$ and $3+1$ warped products, giving classifications for the Einstein limits from this algebraic perspective. Using this, one can easily Petrov classify the Einstein warped products for each case considered: $3+1$ are generically type-\RNum{1}, $2+2$ are type-D while $1+3$ are constrained to be type-O. In the closed Riemannian case, there are a number of topological restrictions on these manifolds that we discuss: in the half-conformally flat limit, each of these Einstein warped products must be flat.