Warped Product Einstein Manifolds in Four Dimensions
Abstract
On four-dimensional (pseudo-)Riemannian manifolds M the curvature tensor (viewed as an endomorphism on 2-forms) admits a chiral 6 × 6 matrix representation which decomposes into four 3 × 3 blocks. M is Einstein if and only if the off-diagonal blocks vanish. If the manifold is a warped product M = F ×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-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.